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‚ƒ€e}‚‚€„‚u ∈ R n‹ u > 0 ‹ €F Ì?> Ì/@ÔJ¥ÒaÐ¢bHBAK¡ŒºzŸZ¥z¤ Õ Ï^Î5Ð£Ž§¥z¤C ¢ED!H 8»?‡ô'@ò@ý#ó–ÿòüó–ñ‹÷@ñùñ@ò@ÿ#ÿNõ#ý@þaöù÷@ñò¥ó/ ý¥÷@þaÿò¥ú ô:õ#ö÷@ñ#¢ óôaþò@ý¡¡A =( a ij ) ∈ KR m×n ¤žô¤( m)∑‖A‖ := max | a ij | ,1≤j≤ni=1ýbõ¥ý√ óô¤ý¥óô:õ#ö÷@ñ ÿ]ñ#þaúÿ5÷@÷@ñ ÿI5÷¥ô@ÿ÷¥ý@ÿ‡óôaþaò@ý¡ (‖A‖ := |A| |A|⊤ ) ,ýbõ¥ýÅŽÿ^¤bô'¤‖A‖ := max1≤i≤m⎛⎝⎞n∑| a ij | ⎠ .j=1ò@ñõ¥ö#ÿ5þùý¥øò¥ôþaö ÷@ÿ¥¤žñþñò¥ôaû@ÿ@ý¥ôaõ¥ö÷¥ôaû!óôaþò@ý¡@÷¥ôaû%÷¥ñò¥óô$ÈŽô:õ¥ÿ5ÿ%ú ÷¥ôN8ý.§@ñ#þaò@ñÿ5÷@ýbû.ñ8Nñ#C—#„6…dc¡ggB±npo&q&rstr#u/r8v&w&r3xzyéèËyY{AÑ }“…Z}K…(‰i!„”#€…B†Œ…OŠA = ( a ij ) ∈ KR n×n Š‖A‖ u := max1≤i≤n(1u in ∑i=1| a ik | u k).…–Ôh9g'„°c iÒ’‘B…BŠ¡huTÒt¤HBU§Ÿ¥¤#B¢#¥ËJB¹8¤Î€ÎKJŽ³Òt¤'º!HZ¢€ÒD€ÒTÎ J¥Ò JBŽ3ic Š'g¥}”#€}Ki!îÿ\#ý@þö#uûü8@þañÔ¥õ@û!ú ÿ¥RŽÿNþúÿ÷¥÷¡ ).Hóôaþaò@ý4¨5þaý¥óñ#@ò@ÿ#ÿNõ¥ÿ¥?¾ÿþaò¡#÷¥ñ5ú ô÷@÷@ñ¢Hóô¤Ný¥ó/bó&V÷@ñò¥ó–ÿVóô#N8þ:ôý@ò@ñu¸¾ò@ñ ó–ÿ8þañø5ñ¥üu‖x‖ u := max1≤i≤n| x i |u i.÷@ý@ÿ=ó#ÿ¥¢þaúý¥þaÿNõ¥ö ÷@ñ3=ôôÉþuû óôaþaò¥ý¡@÷¥ôaû÷@ñò¥óô¥üË@ñ(K@ý@÷¡É5÷@÷¥ô:û úÿ¤@þañ ò÷@ñ¢üT÷¥ô' ú ôÿþuûù÷¥ôaý¡ ñaõ¥ö#N8ÿ5ÿŽý¡!ô'#ñaõCÆþa÷¡ ).Ì5÷¥ô@ÿ5÷@ý¡¢óôNŽþ:ô'ý¥ò@ñúôa÷@÷¥ôaûŒóô'¤Ný#ó/žó&V÷@ñ ò¥óô‹ý@÷@þÿò@úô:õ¥ö ÷@ñ¢óôþBò@ý¡¡û@úaõ¥û@ÿþuûô'#ñaõCÆþa÷@ñ#¢!÷@ñ ò¥ó–ñ¢Ë$?‡ô'@ñó–÷@ý¥ó

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Õ Ï^ÎKJBŽ¡ÇTÒt¤ŸZ¥ ¢¡¦K¡ŒŽ ‹ ¹8¤‚‚–‚ia€„‚”‚i „€ƒ‚}‚„”‚œ‚A ∈ IR n×n €‚A ∈ IR n×n „5g'—…BŠ'g ‹ ˆ„6…„pfd“ ¢€Î Ÿ¡~¥T ‚A ∈ A ŠB± “A i A €!‚‚ÊÕ ¤'¹8¤ F ÌžÍÎ Ÿ¡Ö¥ ¹8¤ Õ Ï^Î5ÐI¡°¹¡\º ŸJ ¥+Ð£¢FÒt¤ŸŒ¥ ¢¡¦5Ð??RÈ(I − α −1 A) v ≥ v,Av ≤ 0 ü@þñQ@ò@ñþý@úñ ò@ÿ¥@ý¥þe÷#ô'N8ÿ=ó/HýZ.ñ(#÷@ñ ó/5ñ¡R8ÿ5÷@ý¡C6$ %ñ øZô6Žÿý@ñþañó/g(‘B’tŠ'gnpo&q&rstr#u/r8v&w&r3xzy©œËyéèå˜^ÉæÓÈ gB„ScKiÒKgf¥„5Šg¥“=hÔgB„°c iÒKgf~}Ki—#g‰›šg¥“AŠ„pfd“¡Ç_±Yh9gB„ScKi!Ò‚¥€€}“A ∈ IR n×n “ Š¥}“ó

F Ì¸ËšÖÌ‹ÍÎ Ÿ¡Ö¥ ¹8¤ Õ Ï^Î€ÐI¡Ô×TÒt¤ŸZ¥ ¢¡¦5Ð ? ?Õ÷@ÿ¥¢N8ý¥óŒññ#RŽÿ÷@ý@ÿ=ó¡Çóôaþaò¥ý¡¾û@úaõ@û¡CÆþuû¾þ¡ô¤¾÷¥ô' ú ôaÿ=ó/ ÿ¤×9±Yh9g'„°c iÒ’üò¥ôó–ñ þaò@ÿ÷¥ý¡C_¤žñþañ ò¡ ).²ó/ @ñ8Núaû@þý¥óŒñ8Nþ:ôaú#NÔCIuû]¥ôþaö£¥ôò¥ôaø5ò¥ôP‡ô$ ¼ ÷¥ýÈŽô:õ¥öý¡PÇóôaþaò¥ý¡¹ñ#Nõ#ô'õ¥ÿ÷@ý¥ÿ5óñ#ò@ÿ=ó–ÿ÷@ý¥þaÿNõ¥ö ÷@ñø5ñb8Lõ¥ñ úýbû ÷¥ô@ñaõ!8#ô'C‰þ#uû‚€ij ‚h ±–ÅíŒ}‚„‚ €‚〈A〉 ‹ „pgB—#g¥“bˆ„5…„ˆ€ ‚−| a ij |, ‚˜ $»@? ˜ $» ˜ ˜ $»»‚à â#ØÀ2æ%ïÆê0"è)í*)¢1‡æéåênÙ5ÂJEí‡ï9"/$%o%éúeóôþaò@ý¡¡ ‡ü¤bô¤Q#ý¥ôøñ ÷¥ô:õ¥ö ÷¡ ).Füþ:ô'¤!ý!ú ÷@ÿ\ý¥ôaø5ñ÷¥ôaõ¥ö÷¡ ).Fü¤žñþañ ò¡ ÿ)÷@ÿû@ú÷@ñH÷#ô'¤bõ#ôB! ú ôC‰þuû ¼ @ò¥ÿ\#ÿõ¥ÿ÷@ý¥ÿ5ó ˜ $ $˜ $!7¾ò@ý3¨þñ ó ñ¤bô¥ )÷¥ô¤@ý=¨õ¥ÿ5ó–ÿ5÷@þañôÿþuûüB@þñ)÷@ÿ¥¤žñ þañò¡ ÿ

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”A ∈ IR n×n “ Š¥}“£¢4¥ò@ý¥ó–ÿò%ý@÷@þÿò@úô:õ¥ö ÷¡ ).¡×póôaþaò@ý4¦*¨5þañ6¥ò¥ôaú ýbõ¥ö÷ ÿÆ÷@ÿ5ñ#ñ'¥›ô'‹÷¡ÿþaò¥ÿ¥žøñõ¥ö÷¡ ÿóôþaò@ý¡ ‡üú ÿò8.b÷@ý@ÿ ýbõ#ýH÷@ý‹÷@ý@ÿ6å»¥?\æ $8?¾ÿ5þaò¡#÷@ñ)þ:ô¤Žÿ6@ñ÷bû@þö¥üÿ5÷@÷¡ªO#ý¥ôaø5ñ÷¥ôaõ¥ö÷ óâ@ò@ÿñ#aõ#ôBôa÷¥ý@ÿ5ó þ:ô'¤eÿ)û@úõ@û@þañ%óôaþaò@ý¡‚ˆ€ˆ‚”…i€„‚A ý Λ $g‚š‚‚A ∈ IR n×n i„5…(±iF Ì¸ËšÖÌ‹ÍÎ Ÿ¡Ö¥ ¹8¤ Õ Ï^Î€ÐI¡Ô×TÒt¤ŸZ¥ ¢¡¦5Ð?¡˜b?øZô6Žÿ‡úQNýbõ!=Nú ñ¢þaú ôÅ ˜ $»#ž%ñ〈B〉 u ≥ 〈A〉 u,ý@ñ¨5þañ ó/B þ:ô'¤eÿ¾û@úaõ@û@ÿ5þuû%×QVóôþaò@ý¡@ÿ¢z$mid A ×9±YhÔgB„ScKiÒKgpiρ ( 〈mid A〉 −1 rad A ) < 1.–;„6…B†\šgpiV r8Ï!q/rW^ÙbxzyY{ ËyY{È g'„°c iÒ!g„pfd“]×9±Yh9g'„°c iÒ—#…Q„6…B†\šg ‹ —…B†\šg„6…Œ}C‰þuû¢×póôaþaò¥ý¡¥ôó–ýË$Λ ∈ R n×n „pgB—…BŠB’ ‹ ˆ„5…]iA fi} ”€V r8Ï!q/rW^ÙbxzyY{ Ëy®xå»¥?\æ ¯ e#f„c Š'g¥}g¥“phÔgB„°c iÒKg”#€g¥“9h9g'„°c iÒ!gcK…i‘'Š„5fZ“´§ƒ‚€‚ ΛA “ Š¥}“ ‚€K‚‚€!€×9±YhÔgB„°c iÒ–0[…B†\š(g…Bf…aÏØzÙ¡Ú#ÙÛ¡r#u/ÜKÝÛÞKÏ`y ¼ ñ#÷¥ô@ý#óA¥õ@û#ñ8Nþaúô×B = ΛA.¼ @ÿú ý#÷@ñ#ümid B = Λ mid Arad B = |Λ| rad A.ó›¤bô'¤Ê5ôóôA mid B #!@ýÛ×póôaþaò@ý¥ôó–ýü;û@úõ@û¡C‰þuû÷@ÿñ8Nñ#ÿ÷¥÷¡ ó–ýz$%xô¤Žÿ‡÷@ÿ5ñ#ñÿ5÷@÷¡7¾ò@ý_¨þñÈŽô:õ¥ÿ5ÿü∣ ∣ üTþ:ô¤ý ∣ ǜ ((mid A)−1 = Λ −1 (Λ mid A) ) −1 ∣ ∣ = (mid B) −1 Λ ∣ ≤ ∣ ∣ (mid B) −1∣ ∣ |Λ| ≤ 〈B〉 −1 |Λ|.ôþaÿNõ#ö÷@ñ#ü á1õ¥ÿ\ñúρ ( ∣ ∣ (mid A)−1 ) ≤ ρ ( 〈mid B〉 −1 |Λ| rad A )= ρ ( 〈mid B〉 rad B ) < 1.

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Ü´õ!ö ÞG÷ø ñ2ð ß˜Þ`ù5Þ Ü Ý¡ú_Ý Ü øQÞ9ß ñ2ðï¡û Ü¡ü|ý ÷£Ý]÷øQÞ`þî6ï9ð3ñ6ðò6óÌôM¦!¼¡®½¡®¬µ¯!µ§¡·¯w°¡±¡¯¨Ä±9¸!¨®¡½Ë¡¨¯®»¨#Ç²9¯¡½¡¯;°µ¯!±¡ḠÀK¯!¡¡·|»]Ê´³z¨¨§®¯!§³!§®¹6ØDÅo½»V¡¯!¡¹G¬µ¯®·¥À«¦!¬µ¯ÀK¯¡§3À«®a ∃ F ¡¯,°¡¯!±µ¯³z¯¼ØD§³ÏËA²9±¡¹6ª³½Q¦!§½%®»2¨#Ç²9¯!¡½¡ËA®8±9¸!µ·|»A¼2¨2ÀK°¡¨µ¯!¡§33»¨§²9¯!¬µº¡·¥ÀÑÂ¬µ¯µÀ¿¼6¨¡çgØq¡¼¡¾¡½µ©©¨ ªb«NÍÜß@§Ξ αβ (F, a, b)= { x ∈ R n | (∀â ∈ a ∀ )(∀ˆb ∈ b ∀ )(∃ǎ ∈ a ∃ )(∃ˇb ∈ b ∃ )( F (a, x) = b ) }= { x ∈ R n | (∀â ∈ a ∀ )(∀ˆb ∈ b ∀ )(∃ǎ ∈ a ∃ )( F 1 (a, x) = b 1 & · · · & F s (a, x) = b s &F s+1 (a, x) ∈ b s+1 & · · · & F m (a, x) ∈ b m ) }{x ∈ R n | (∀â ∈ a ∀ )(∀ˆb ∈ b ∀ )(∃ǎ ∈ a ∃ )( F 1 (a, x) ≥ b 1 & F 1 (a, x) ≤ b 1 & · · · &F s (a, x) ≥ b s & F s (a, x) ≤ b s &F s+1 (a, x) ≥ b s+1 & F s+1 (a, x) ≤ b s+1 & · · · &F m (a, x) ≥ b m & F m (a, x) ≤ b m ) }⊆ { x ∈ R n | (∀â ∈ a ∀ )(∀ˆb ∈ b ∀ )((∃ǎ ∈ a ∃ )(F 1 (a, x) ≥ b 1 ) & (∃ǎ ∈ a ∃ )(F 1 (a, x) ≤ b 1 ) &· · · &(∃ǎ ∈ a ∃ )(F s (a, x) ≥ b s ) & (∃ǎ ∈ a ∃ )(F s (a, x) ≤ b s ) &(∃ǎ ∈ a ∃ )(F s+1 (a, x) ≥ b s+1 ) & (∃ǎ ∈ a ∃ )(F s+1 (a, x) ≤ b s+1 ) &· · · &(∃ǎ ∈ a ∃ )(F m (a, x) ≥ b m ) & (∃ǎ ∈ a ∃ )(F m (a, x) ≤ b m ) ) } .Î §ÀK¯!§½µÀ|ÊÂµ§¨l°¡¨³z¬µ¯`²:¡¯!¯®¼¬µØDÂµ¯!¡½¡¯D¨Ä±9Åo¯!§³ÏËo®l§¨Â¡µ¨¯±µ®¯!9³z§®¨µÊ3¯³zÁ¬µ½V®¯±¡¡·â¹³`¬9¨®½¡ËgÊ29¼¬93²9·®¯ÀK·¯wµF (a, x) ®¨,®§¨±¡¨©ÌÂ9³z§½8¹2³`¬µ¨®½Ë§¯!¨±¡¯ÀK·lÍgèlÀK¯¡¡¨9Ê]¯³`¬9½Q¼Ço²´3Ëk¼6¨ÀK°µ¨¡¯¡§3F ½µÀK¯!¯§8¬µ½¡ÈºV¯z²9½¡µ³!§®¯Á²9±¡¹6ª!¨2À|Ê¡§¨TÀK·â½µÀK¯!ḠÀb°¡±µ®Ï3°¡±¡¨µ¨³z½µ§ºJ¼¡®2¡§¨±¡·ä³!¹6Å¯³!§®¨®¡½Ë∃ ¼Ho3¬µ¯¯Ê*²µ¬µËl¬µØDÄ¨©lÐw¹6¡¼¡¾¡½¡½f Ê¨°¡±µ¯`²9¯!¬µÆ!¡µ¨©wµ¡¯¼6¨§¨±¡¨ÀÌ½¡¡§¯±¡®¬µ¯

a Ê¡½9ÀK¯!ØD§TÀKḠ³z§¨T³z¬µ¯`²:¹2ØqÅ½¡¯w¦¼¡®½¡®23¬µ¯¡§¡¨2³z§½ M²9½¡¼§¨ÀÑ³z¬µ¯`²:¹2ØqÅ½µÀ¿¨Ä±µ¸¨Àåw¨¦§¨ÀK¹kÀK·ŸÀK¨#ÇoḠÀÕ°¡±µ¨G²9¨¬¡Ç½¡§º89È½Q®·¼¡¬9#²9¼µ½%³®·²9¯z¬¡Ë¡ØqÅ½µÀä°¡±¡¯!ÁMò2ó}ó €Ýð ß ðl‚ øQÞ9ßDÝ=ƒ ðl„ Ý

Ξ αβ (F, a, b){⊆ x ∈ R ∣ n (∀â ∈ a ∀ )Üß2Üî6ï9ð3ñ6ðò6óÌô Ü´õ!ö ÞG÷ø ñ2ð ß˜Þ`ù5Þ Ü Ý¡ú_Ý Ü øQÞ9ß ñ2ðï¡û Ü¡ü|ý ÷£Ý]÷øQÞ`þ§3¼8Â¡§¨½µÀK¯¯À( (maxǎ∈a ∃ F 1 (a, x) ≥ b 1)&(minǎ∈a ∃ F 1 (a, x) ≤ b 1)&· · · &( ) ( )max F s (a, x) ≥ b s & min F s (a, x) ≤ b s &ǎ∈a ∃ ǎ∈a ∃( ) ( )max F s+1 (a, x) ≥ b s+1 & min F s+1 (a, x) ≤ b s+1 &ǎ∈a ∃ ǎ∈a ∃· · · &(maxǎ∈a ∃ F m (a, x) ≥ b m)&(minǎ∈a ∃ F m (a, x) ≤ b m) ) } .¬l¼6¨µ¯!¾gÊ½ÞÀK·ä°¡¨¬µ¹6ÂµḠÀ{Ξ αβ (F, a, b) ⊆ x ∈ R ∣ n (minâ∈a ∀(minâ∈a ∀(minâ∈a ∀(minâ∈a ∀(∀ a ∈ a)( f(a) ≥ b ) ⇐⇒ min f(a) ≥ b,a∈a(∀ a ∈ a)( f(a) ≤ b ) ⇐⇒ max f(a) ≤ b,maxǎ∈a ∃ F 1 (a, x) ≥ b 1)&(maxâ∈a ∀· · · &maxǎ∈a ∃ F s (a, x) ≥ b s)&(maxâ∈a ∀maxǎ∈a ∃ F s+1 (a, x) ≥ b s+1)&(maxâ∈a ∀· · · &maxǎ∈a ∃ F m (a, x) ≥ b m)&(maxâ∈a ∀a∈a)min F 1 (a, x) ≤ b 1 &ǎ∈a ∃)min F s (a, x) ≤ b s &ǎ∈a ∃)min F s+1 (a, x) ≤ b s+1 &ǎ∈a ∃minǎ∈a ∃ F m (a, x) ≤ b m) } ,

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“U“( Ax = b ) } “YnU¥¢/ḧ §©$U Xu‹ m Y?]^[l`badt?]EceWsU:˜Y%]u¡lžbU/X£V=U”T(f£]lŴÙl¼T½Ì±µ¡¯¯Ê6³`¬µ¯z²9¹6ØDÅo½¡¯D»2¨±¡¨È¨l½¡¸®Ḡ³z§¡·¯wÀK¡¨#Ç¯³z§®l±¡¯È¯¡½¡©F½µÁ¡ó(…ˆ‡‰ ß˜Þ*ùBÞ Ü Ý?® Ý Ü øQÞß ñ6ð3ï¡û Ü¡ü|ý ï Ý Ü Þ*ú Üµü|ýS¯ ß ð3ñ Ü Þ Ü Ý¡ú ÜßEŠò2óm × nAx = b m × n |v@[^]lV/Wr`ŸYnV=hÈX ¡E]ôb]lV/h|gceU:ceW¡XgfhdR:U=V:Xea/|α U m |g[^X£vEWs`ŸY β US]^c£c£`¤¢=U=U^Y:`l[^]lV/V=hÈXcV/U-fÜ{Y?]w¡^žbU/XeV/U-TSU=V/obX£v@ce|î¡ï .Ì+EðÈ+@B0+-ñÌòÌ+d476ON76ONþ£i-ceWs_†opR%TºU=V=W¡XZY\[^]£R:_@V=`baV=hG•ÿ]£R¡ kX£ž÷Y?]lU/t?X£cev@U”•úikY?]^[lV/X£V=U/af¨]lW{Y:U£¢/]XwceW¡[f£]lW{Y:U¥¢/h [^X£vEWs`÷Y?]W¡X’• qxX0Y?]w¡£f£XZYn`b[3V:]V/X m X—YnX£cwXev-]l|V=hG•¡f¨V=ùq U`lopf¨V=ùq XwceW¡[^]ˆΓŽrU:X£c’T m = { ˆγ1 , . . . , ˆγ p } ˇΓ = { ˇγ1 , . . . , ˇγ q p + q = mn“}“ˆ∆ = { ˆδ 1 , . . . , ˆδ r } ˇ∆ = { ˇδ1 , . . . , ˇδ s U r + s = m˜U }“Ax = b¦r]u¡l`l[E‹—f þ Ü´õ!ö ÞG÷ø ñ õ þ …{‡‰ ß¥Þ*ù5Þ Ü Ý¡ú8øÝ æ:ðαβ U/V=W¡XZYn[E]£R:_EV/ùasR:U=V/X£a=V/ùa{x ∈ R n |X£c£Ws[^` fV/ùqΞ αβ (A, b) :=ceU:ceW¡Xgfh(∀aˆγ1 ∈ aˆγ1 ) · · · (∀aˆγp ∈ aˆγp ) (∀bˆδ1 ∈ bˆδ1) · · · (∀bˆδr ∈ bˆδr)(∃aˇγ1 ∈ aˇγ1 ) · · · (∃aˇγq ∈ aˇγq ) (∃bˇδ1 ∈ bˇδ1 ) · · · (∃bˇδs ∈ bˇδs )Û¡Í¤š@šu›U-R:Ut%Ws` õ v@[Û/[^]£RnX£V=WsV=`fV/ẁqxX£c£Ws[l`Ξ αβ (A, b) := { x ∈ R n | (∀Â ∈ A∀ )(∀ˆb ∈ b ∀ )A = A ∀ + A ∃(∃Ǎ ∈ A∃ )(∃ˇb ∈ b ∃ )( ( Â + Ǎ) x = ˆb + ˇb ) } ,b = b ∀ + b ∃ ‘ c£`bùW¡[^X£Wsc£Ws[li@ŽrU:X(ouU%¡lülV/vEWsV=hÈXkobX¦§¯!±µ®3¬9º¡·|»¬µ½µ¡¯!©¡µ·|»8³z½9³z§ḠÀ•`bžbü^X£ouU=Vn‹kV/V=`lXSfV/ẁqxX£c£Ws[l`sY\Xe~¡XeV/U=a ¡¯±¡¯`²9¼6¨Qµ¸·®¯ÀK¨¯F°¡±¡¨³!§¨›Ξ uni (A, b) = { x ∈ R n | (∃A ∈ A)(∃ b ∈ b)(Ax = b) },I3ÀKµ¨#Ço¯³!§®¨2ÀÉ±¡¯È¯¡½¡©J• ou` m i-ceWsU-f`lXfV=ùqxX£ceW¡[l`{YnX£~sX£V=U=a ³!À|Í¥2¬9®¹©ª@›Ξ tol (A, b) = { x ∈ R n | (∀A ∈ A)(∃ b ∈ b)(Ax = b) },• i m Y?]^[£R%TnXgf`lXˆf¨V=ùq XwceW¡[l`ˆY\Xe~¡XeV/U=a ³À|ÍgÚ*Û¡ÍÛ@›Ξ ctrl (A, b) = { x ∈ R n | (∀ b ∈ b)(∃ A ∈ A)(Ax = b) }.

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ceWsU=t?XwcevEU”f$f¨]lW{Y:U£¢/XU([EXevEWs`ŸY:i m Y?]^[l`baSt?]^c£WrU9ÊG½°¡½µ³§ºΞ (F, A c , b c ) Ê¡¯¬w½Ço¯lÀK·â°¡±¡½¡®¨G²9½µÀÑ±µ¸®Æ!±µ¡¹6§¨¯q²9¨¼6¸§¯z¬µº2³z§®¨¦§¨ª¨±µ¯!¸!¹G¬µº3§3§#µÍ»6±µ¼¡§¯±¡½¡¸¾¡½Ë©±€¨µ,ÀK¡¨#ÇoḠ³z§®ÓDÔEÁ1±µ¯!È¯¡½¡©/›*+-,/.0+-1325476ON76}‡“¥¼¬µØDÂµ¯!¡½¡¯p ⊆ q ²µ¬¡Ë¿°µ±µ®½¬µºµ·|»É½¡¡§¯±¡®¬µº¡·|»®¯¼¡§¨±µ¨®;5,=@2%A?+@B0C:DEA%F:,G6p q ±µ®¡¨2³z½¬µº¡¨µÊ6¼¼8½¡¸®¯³!§¡¨9Ê¡¯!±µ®¯µ³z§®¹½ò2ó ¡ó(…ˆ‡‰ ß˜Þ*ùBÞ Ü Ý?® Ý Ü øQÞß ñ6ð3ï¡û Ü¡ü|ý ï Ý Ü Þ*ú Üµü|ýS¯ ß ð3ñ Ü Þ Ü Ý¡úÜá6æ¼6¨§¨±¡¨¯w³z¨®°µ3²:¯§o³ Û¡Í¤šuª@›*Í¥©®¯`²9Æ¡¡¨¯ Î °¡±¡¯z²9¯z¬µ¯¡½¡ḠÀÉÛÍ4ÜÍßl°¡¨¡Ë¡§½¡¯D9³z§¨¬µº¼2¨®3Ç¡¨l®o±µ¸®½¡®23Á¡¯!¯Íåw¨G²9Â¡¯!±µ¼¡¡ÆÀ|ÊÂ¡§¨b¹6¼¸¡½¡¯Þ»6±µ¼¡§¯±¡½µ³z§½¡Â¡¯³!¼¡½»m®¯¼¡§¨±¡¨®¿°¡¨¬µµ¨Á¯ÀK¨©ÞµÀK½Þ§¯¨±¡½µ½gÊ9Âµ§¨Tµ¯ª!¨T¨Ä³!¹Ç²9¯¡½¡½Þ³z§¨½¡§T¨2³z§3µ¨®½µ§º³ÏË8°¡¨G²9±¡¨ÄÁ¡¯!µ½¡©gÊ]µ±Ë6²:¹A³§±µ½µ#²9¨©gÊ]¨°µ½µ³!¡¡¨©k®jÚÏáµÍ4Ü3®9Ê9§#Í¯Í]¼¡®2¡§¨±¡¡·¥ÀK½QÀE§±¡½¡Á³z§ºØ×¨°¡±¡¯`²:¯z¬¡Ë¡Øq§Ó¥ÔEÁÀK¡¨#Ç¯³z§®¨;±¡¯È¯¡½¡©F½µ¡§¯!±µ®3¬9º¡¨©T³z½µ³!§¯ÀK·m¹6±µ®Áα β ÊY±µ¸Ä½¡¯¡½¡ḠÀs½¡6²:¯!¼³z¡·|»bÀK¡¨#ÇoḠ³z§®Q½¡¡§¯±¡®¬µºµ·|»°µ±9ÀK¯!§±¡¨®l³z½9³z§ḠÀK·lÊw§3¼Ç¯|½»;²9½µ¸!çgØq¡¼¡§¡·¥ÀK½F±µ¸!¬µ¨#Ço¯¡½ËµÀK½gÍ^¬w¨lĢ3Á½b®¯¼¡§¨±µ¨À ¾¡¯!©½¡µÐw¨±µÀE¾¡½µ½gÊ¨G²9µ¨®±µ¯ÀK¯!µ¡¨;¹6¼¸!·®23Ë¼6¼T§½¡°F¡¯!¨°¡±¡¯`²:¯z¬µÆ¡¡¨³!§½gÊ2§3¼T½²:µ½¡¯Ý»2±9¼¡§¯±¡½µ³!§½¡Â¡Ḡ³z¼¡½»8ÀE§±¡½µ¾¡·ä½V°¡±9®¨©FÂµ³z§½F²:Æ!§o²:3Ç¯qÄ¨¬µºÈ¯±¡¯!¼µ§¡·¥ÀQª!¨®¨±µ½¡§ºw¨;ÀK¡¨#ÇoḠ³z§®¯Ó¥ÔEÁN±¡¯!Èo¯!¡½µ© ¡¯¼2¨§¨±µ¨©/›€½¡µ§¯!±¡®23¬µº¡¨©³!ÀK½½µ¡§¯!±µ®3¬9·¿°µ±µÀK¯§±¡¨®Ý®l³!½µ³z§ḠÀK¯Í2åw¨¦!§¨ÀK¹;Ä¹!²:¯!§l³z¨®¯!±µÈ¯!µ¡ẅ¼2¨±Á•:]pY%]lv@WsX—Y:U/|³z½µ³!§¯ÀK·Õ¬µ½¡¡¯©¡¡·|»T¹6±µ®¡¯!µ½¡©gÊ\ce`b`bWs[EXeW¡ceW¡[liEŽsU”•dou]^V=V=hLf¡¯!©[Í¡åw±¡½Ì±9³!³ÀK¨§±µ¯!¡½¡½V½¡µ§¯!±¡®23¬µº¡·|»¬µ½¡µ¯!©¡¡·|»V³z½µ³!§¯À¿¹6±µ®¡¯!µ½¡©ÌÀK·¹6¼6¸!·®23ËlË¡®µ¨q¦§¹o³!½µ³z§ḠÀK¹;½±µ³!°¡±¡¯`²:¯z¬µ¯¡½¡¯|§½¡°¡¨®w¡¯!¨°¡±¡¯`²:¯z¬µÆ¡¡¨³!§¯!©®½V§¯!±µÀK½µ¡¨¬µ¨ª!½¡½[Í°¡¨¬9¹2Âµ½µÀQ²9¯©µ³z§®½¡§¯!¬µº¡¨¨Åo¹6§½µÀK¹6Øä®·ª!¨G²9¹¨§o®®¯`²9¯¡½ËT¡¨®·|»°¡¨¡Ë¡§½¡©é¬µ¯`²9¹6ØqÅo3ËV¸ÀK¯!Âµ§¯z¬9ºµËF»2±µ¼µ§¯!±¡½µ¸¾µ½Ë8Ä·|¬9°¡±¡¯z²µ¬µ¨#Ço¯µÂ¡¯!ÈÁ¼6¯!¯!®·¥ÀÉ®,±µÄ¨§¯s¨ßÜb«1Í Î 9Ë¡®¬¡Ë¡¯!§³ÏËÌ°¡¯±¡¯Ðw¨±µÀK¹G¬µ½¡±¡¨®¼6¨©©°g¯!¨±¡¯ÀK·sÛ¡ÍÜ¡ÍÜ³z¼¡½9Àä½9³!³`¬9¯`²9¨®§¯z¬µḠÀ×è,Í7±K¨µ¨Àÿ¨á/šk«½Ã®°µ¯!±¡®·¯½9³z°¡¨¬µº¸¨®µ8ë;Í¥qÍ[ê;3Á®Ḡ³z§¡·¯±µ¯!¸!¹G¬µº3§3§· Î ¯§§G¬µ½Áåw±µª¯!±µ¥²µ¬¡Ë¨Äçg¯z²9½¡¡Æ¡¡¨ª¨ÝÀKµ¨#Ço¯³!§®Ý±¡¯!Á®,®½6²9¯w¬9½¡¡¯!©µ¡·|»8µ¯!±µ®¯!9³z§®³lÀK¨G²9¹G¬¡ËµÀK½gÊµ¨Ä¨ÄÅ3ËgÊ6§3¼¡½µÀ¿¨Ä±µ¸¨À|Ê½¡¸!ÁÈ¯¡½¡©x¨ß6æw«D½Õ³ÀK¨ª¨5±K¨µj²µ¬¡Ë¿²9¨°µ¹³z§½µÀK¨ª¨%ÀKµ¨#Ço¯³!§®Q±¡¯È¯¡½¡©ú¨ßEŠb«NÍx m Y:U/V/]ôkRnXeqU=WΞ αβ (A, b)n=ùt%v-]fV/ẁqxX£c£Ws[li y{z|}Y\Xe~¡XeV/U=aWs` kou]ÇU∣ (mid A)·x − mid b∣ ∣ ≤ ( rad A ∃ − rad A ∀ )·|x| + ( rad b ∃ − rad b ∀ ).Wr`pR:_Ev@`†Wr` kob]vE` kob]Û¡Í¤šGí@›| mid q − mid p | ≤ rad q − rad p

½¦ ¡½¡© ¹6§®¯±Ç²9¯¡½¡¯ ÛÍ4Ü^§-›x ∈ Ξ uni (A, b) ⇐⇒ ∣ ∣ (mid A) x − mid b ≤ rad A · |x| + rad b¦ Âµ³z§¨Qµ¸!·®ØD§©•:]pY%]lv@WsX—Y:U%¡^]¢/U/X£aEˆXeWsW3R:U/|9þˆY%]R kX—Y%] ³!À|Í¨ßÛÊKß2æu«1ÊEh (i) ∈ R p , ξ i ∈ R U M ‘“R p “î6ï9ð3ñ6ðò6óÌô Ü´õ!ö ÞG÷ø ñ2ð ß˜Þ`ù5Þ Ü Ý¡ú_Ý Ü øQÞ9ß ñ2ðï¡û Ü¡ü|ý ÷£Ý]÷øQÞ`þÜ3á2íÛ¡Í¤š#ÛE›YÀK¨#Ç¯!§oÄ·§ºo°¡¯!±¡¯!Á°¡½µ³µ®³`¬µ¯z²9¹6ØDÅo¯À¿®½6²9¯³À|Í2¬:®¹šb›`Íµé¬µ¯`²:¨®§¯z¬9º¡¨9Ê»2±µ¼µ§¯!±¡½µ¸¾µ½ËM∣ mid( b ∃ − A ∃·x) − mid(A ∀·x − b ∀ ) ∣ ≤ rad( b ∃ −A ∃·x) − rad(A ∀·x − b ∀ ).Û¡Í¤šuŠ@›Ho3¬µ¯¯Êrad (p ± q) = rad p + rad q,mid (p ± q) = mid p ± mid q.ÛÍ}šwŠ-›Y®·°¡¨¬9Ë¡¯!§³ÏË8®,§¨À¿½8¬µ½¡Èº®,§¨2À¿³z¬µ¹6Âµ¯Ê¯³z¬µ½åw¨¦§¨ÀK¹ ∣ mid b ∃ − mid(A ∃·x) − mid(A ∀·x) + mid b ∣ ∀ ≤ rad b ∃ + rad(A ∃·x) − rad(A ∀·x) − rad b ∀ , ÛÍ}ší-›*Ê¡§3¼8¼¼Â¡§¨¦¼¡®½¡®23¬µ¯¡§¡ö»2±µ¼µ§¯!±¡½µ¸¾µ½¡½mid(A ∃·x) = ( mid A ∃ )·x, mid(A ∀·x) = ( mid A ∀ )·xrad(A ∃·x) = ( rad A ∃ )·|x|,rad(A ∀·x) = ( rad A ∀ )·|x|.Jl³z§¡·©³z¬µ¹6Âµ©¡°g¯¨±¡ḠÀK·«Û¡ÍÜ¡ÍáD²µ¬¡Ë¨Äçg¯`²9½µ¡Æ!¡µ¨ª!¨ÝÀK¡¨#ÇoḠ³z§®2D±µ¯!È¯!Á§3¼Ç¯s¨ßl«NÊ´Ú*Ü¡Í4æÍÛ@›`Í.Ì+EðÈ+@B0+-ñÌòÌ+d476ON76}‡ÿ¨¤šu§b«Qô Ü´õbë#õlë ß ð Ü

ùx ∈ KR n V/]l|α U β m XZY\X£c£X£t?XeV/U/XfV/ẁqxX£c£Ws[^]“Ax = b c A ∈ vert A Ux 2x 1ò2ó ¡ó(…ˆ‡‰ ß˜Þ*ùBÞ Ü Ý?® Ý Ü øQÞß ñ6ð3ï¡û Ü¡ü|ý ï Ý Ü Þ*ú Üµü|ýS¯ ß ð3ñ Ü Þ Ü Ý¡úx 2Üá@Šx 1+R ‰= ~ 4=&,e#)4=CŠ,'*9TH*TH4&?9'='&CogN,;'&*/.6#)12+3&î¡ï .Ì+EðÈ+@B0+-ñÌòÌ+d476ON76O4E‹ Þ9ß´ù¤Ý Ü ð þ=Ý U/V=W¡XZYn[E]£R:_EV/`¤ p`[^Xev@Wr`ŸY?]¡lhÈ[E]lWsc’T5Wr`bt%vEU©fV=ùqxX£ceW¡[^]vert x := { x ∈ R n | x i ∈ { x i , x i }, i = 1, 2, . . . , n } .pU/t%V=hLf `bž÷Y?]w¡^`’f ` m YnXwobXgR%T:Wsc’TÐ[^X—Y:~sU=V/h U/V=W¡XZY\[^]£R:_EV/`ba f¨]lW{Y:U/|V/]£R:`¢=hµ˜Ξ αβ (A, b) cv@]lq ouhLfÿU”¡r`ŸY:W¡]lV=Ws`l[ m Y:`lc£Ŵ Y?]lV/c£Ws[E]R n Tn[£R%TnXeWs|*+-,/.0+-1325476ON76O4QPSR%TR:žuhG•Kv@[^]lV/Wr`ŸYn`b[YnX£~sX£V=U=ab ∈ vert b˜cŸT©[lh m iEvuR:hLfÿfV/`¤ p` ÃY?]lV=V/hLfúfV/ẁqxX£c£Ws[l`pft%_EU©[^XZYn~rU/V=h ‘½õ Ws`YnX£~sX£|V=U”T×øli& gR:`l[lhG•Wr`bt?X£t%V=hG•QR:U/V/X£a=V=hG•–ceU:ceW¡Xgf;5,=@2%A?+@B0C:DEA%F:,G6[åw±¡½¡9#²µ¬µ¯!Ç¡¨³!§º®¯Å¯³!§®¯¡¡¨ª!¨,®¯¼¡§¨±9x ¼¼6¨ÀK¹2Á¬µ½¡Ä¨F¨±µ§#¡§¹V¨°µ±¡¯`²9¯!¬¡Ë¡¯!§³ÏËA¹2¼¸µ½¡¯À«¸µ¼6¨®F¯!ª!¨Ì¼6¨ÀK°¡¨¡¯!¡§#Í\Þ]ÀK¯§½µÀ(C · x) i =°¡±¡¯z²:³z§3®¬µ¯!µ·×®³`¬µ¯z²9¹6ØDÅo¯À¿®½6²9¯M=[n∑ ∑ nc ij x j = c ij x j ,j=1[∑ nc ′ ij x j ,j=1j=1n∑j=1c ′′ij x j],]n∑c ij x jj=1Û¡ÍÜ?šu›m × n C ¼6¨2ÀK°¡¨¡¯¡§·°¡±¡¨½¡¸!®¯`²9¯¡½Ë§3¼Ç¯ÊÂµ§¨Ã²µ¬µËm¬µØDÄ¨©m½¡¡§¯!±¡®23¬µº¡¨©C · x = ( (C · x) 1 , (C · x) 2 , . . . , (C · x) m ) ÁÀE§±¡½µ¾¡· ⊤ÀK¨ª!¹6§ÑÄ·§º

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ò2óFÌGóÍ ß˜Þâñ6ð ßDÝgøQÞ ï¡û Ü:õ Þ õ „ Þ Ü Ý ñ6ð Ü ÝDÞDþ Ü´õ!ö ÞG÷ø ñ ß˜Þ*ùBÞ Ü ÝµúÝ §¨§Ñ°µ±µª!±µÐ Ä¹²9¯!§É°µ¨³z®Ë¡Å¯m¨Ä¸¨±¡¹bÀK¯!§¨G²:½¡¼gÊ¼2¨§¨±µ·Ā°¡¨¸®¨¬¡Ë¡ØD§Ü2æ”š(√ )1 2 − 1√2 − 1 −1°¡±¡½¡ÂµÆÀ¿¦!§¨§,ÀE¼³z½µÀK¹ÀÉ²9¨³!§½¡ªz¯§³ÏËÌ9o®¯!Åo¯³z§®¯!µ¡¨©8³z½µ³!§¯ÀK¯x =(22),°µ±¡½V¦!§¨2ÀxÀE§±µ½¡¾µ,¼6¨§¨±¡¨©Ì¡¯wË¡®¬µË¡¯!§³ÏË8¼¡±µ©µ¡¯!©2 = −1ÿ¡ 1Î Ï §UÐd£¥§(©¸º¹¨ ©©¨¥©©›zÍÇ§&'(©)¼¡½TÀK¡¨#Ç¯³z§®l±¡¯!Èo¯!¡½¡©½µ¡§¯!±µ®3¬9º¡·|»o¬µ½¡¡¯©¡¡·|»³z½µ³!§¯À|Í6¥¨ÁN°¡¯!±¡®·|»]Ê3¨¡½Ä·¥³!§±¡¨«µ3»2¨G²9½¡§º«¼¼µ½¡¯zÁN§¨µÊ®¨ĢÀK¨#Ç,µ¨b®¯³!ºÀE¿ª!±¡¹6Ä·¯Ê¥®µ¯!Èµ½¡¯%¨¾¡¯Á½µÀK¯!Øq§k³!ÀK¨³!§¨#Ë¡§¯z¬9º¡¹6Øu¾¡¯¡¡¨2³z§º9Ê]²:®3Ë%½¡¡¨ªÏ²:Þ°¡¨¬µº¸¨®2§¯!¬µØ£²9¨³!§33Á§¨Âµ¡¹6Ø¤³z¨G²:¯!±2Ç§¯z¬µº¡¹6Øu½¡µÐw¨±µÀE¾µ½¡Ø ¨j±¡¯!Èo¯!¡½¡½b³z§¨#Ë¡Åo¯!©É°µ¯!±¡¯z²¿¡½µÀ¸3²:#Â¡½gÍ:è,Êµ®¨Á1®§¨±µ·|»´Êµ¡¯!¨Ä#»¨G²9½µÀK¨2³z§º®F°¡¨G²9¨Ä¡·|»8¬µª!¨±¡½¡§ÀE»Þ®·¸!®2µ§¯ÀÕ¨Ä³!§¨#Ë¡§¯!¬µº³!§®¨2À|Ê9Â¡§¨8ÀK¡¨ª!½¡¯½¡¡§¯!±¡®23¬µº¡·ōÀK¯§¨G²9·ã²µ¬µËk®¡¯È¡¯ª!¨¨¾¡¯¡½¡®2¡½ËTÀK¡¨#Ç¯³!§®±¡¯È¯¡½¡©TË¡®¬¡Ë¡ØD§³ÏËgÊ®;²9¯!©µ³!§®½µ§¯z¬µº¡¨³!§½gÊ6¹6§¨ÂµËÁØDÅo½µÀK½m°¡±¡¨¾¡¯`²9¹6±µÀK½gÊK²µ¬µË«¹2³z°¡¯È¡¨ª!¨Qµ#Âµ¬9Q±µÄ¨§·¤¼2¨§¨±µ·|»¿¡¹2Ç¡¨¸!µ§º»¨§º,¼6¼6¨¯zÁN¡½¡Ä¹²9ºµ#Â93¬µº¡¨¯Ý®¡¯È¡¯¯w°¡±¡½¡Ä¬µ½Ço¯¡½¡¯ÍýÞ·Ñ±µ¸!Ä¯!±¡ḠÀV§±¡½,ÀK¯!§¨G²9½µ¼¡½o°µ±¡¯`²9®2±¡½µ§¯z¬µº¡¨ª¨D®¡¯!Èo¡¯!ª¨D¨¾µ¯!¡½¡®2¡½¡Ë¨Äçg¯`²9½µ¡Æ!¡µ·|»ÀK¡¨#ÇoḠ³z§®w±¡¯È¯!µ½¡©èé©ê;ëKìÍåw¯!±¡®23Ë½,°¡±µ¨³z§¯!©¡È3Ë ¦ ¦§¨ÀK¨G²9½µÐw½¡¼¾¡½Ë¿9Ã½¡¡§¯±¡®¬µºµ·©«³z¬µ¹6Âµ©b»2¨±¡¨È¨Q½¡¸!®¯³z§¡·|»b±9³!³!¹2Ço²:¯zÁ¡½¡©gÊ°¡¨¼¸!·®ØDÅo½»±¡¨¬µºlÂ¡½µ³z¬9l¨Ä¹³z¬µ¨®¬µ¯!¡µ¨³z§½gÍ Ý §3;ÀK¯!§¨G²:½¡¼w®¯³!ºÀEª!±¡¹6Ä2¡Ê#¡¨Ý°¡±¡½µÀK¯¡½µÀEw¼½¡¡§¯±¡®¬µº¡·¥ÀA¬µ½¡¡¯©¡¡·¥ÀQ³z½µ³!§¯ÀEÀ|Ê3¨§µ¨³z½µ§¯z¬µºÁ¡¨,¼6¨§¨±¡·|»T§±¡¯Ä¹¯§³ÏËF¬9½¡Èº,¡¯¨³z¨Ä¯¡¡¨³!§ºoÀE§±¡½¡¾µ·lÍ”H®¯D²9±¡¹6ª!½¡»ÌÀK¯!§¨Á²9½¡¼¡½[Ê°¡±¡½µµ#²µ¬9¯zÇÅ½¡¯q³z¨¨§®¯§³z§®¯!µ¡¨¨¥¯!¼¡¹½S¬w¨©µÀE©¡¯!±µ¹ÊÄ¨¬µ¯¯§¨Â¡µ·lÊ¡¨l¸§¨l§±¡¯Ä¹6ØD§#ÊÂ¡§¨Ä·¿½µ¡§¯!±µ®3¬9ºµËÀE§±¡½¡¾µ;³!½µ³z§¯ÀK·«Ä·|¬9wµ¯!¨³!¨Ä¯!Á¥³!°¡¨ÀKµ½µÀã»2±µ¼µ§¯!±¡½µ¸¾µ½¡Ø Î ¯§§G¬µ½Áåw±µª¯!±µ ÛÍ4Ü^§-›q²µ¬¡ËÑ¨Äçg¯z²9½¡¡ÆÁ¡¨©[Í¡¨ª¨ÀKµ¨#Ço¯³!§®±¡¯È¯¡½¡©V½¡¡§¯±¡®¬µºµ·|»T¬µ½¡¡¯©¡¡·|»Ì³!½µ³z§¯ÀMx ∈ Ξ uni (A, b) ⇐⇒ |(mid A) x − mid b| ≤ rad A · |x| + rad bx = ˆx + δxìqÀK¡¨#Ç,3Ë¨Ä¯Âµ³!§½o¦§¨ª¨qµ¯!±µ®¯!9³z§®2Dµq¡¯¨§±¡½µ¾µ§¯!¬µºµ¹2ØmÀE§±¡½¡¾¡¹åw¹³z§º|(mid A) −1 | Êµ°¡¨¬µ¹6Â¡½9À|(mid A) −1 | |(mid A) δx| ≤ |(mid A) −1 |(rad A|ˆx + δx| + rad b),

UAx = b A ceU”R:_EV=òV:Xe`lce`bžlXeV/V/]£T “ˆx f¨]lW{Y:U£¢/] Y\Xe~¡XeV=U:XSø^c—YnX£ouV:XeadceU:ceW¡Xgfh‘[ ˆx − ∆, ˆx + ∆ ]“Ó £ (¥§©() ÕÔ£¸¥¨£ ÿJÍùî6ï9ð3ñ6ðò6óÌô Ü´õ!ö ÞG÷ø ñ2ð ß˜Þ`ù5Þ Ü Ý¡ú_Ý Ü øQÞ9ß ñ2ðï¡û Ü¡ü|ý ÷£Ý]÷øQÞ`þÜæÜ¨§¼µ¹!²´|δx| ≤ |(mid A) −1 | rad A |δx||(mid A) −1 |(rad A ˆx + rad b),§¨,¯³!§º(I − |(mid A) −1 |rad A)|δx| ≤ |(mid A) −1 |(rad A + rad b).A §±¡¯Ä¨®µ½¡½8³!½¬µº¡¨©8¡¯!¨2³z¨Ä¯!µ¡¨Á³z§½ÀE§±¡½¡¾µåw±¡½Vµ3¬9¨#Ço¯!µ¡¨ÀÑµÀK½VµÀE§±¡½¡¾¡¹(I −|(mid A) −1 |rad A)ÀE§±µ½¡¾µ,¡¯¨§±¡½µ¾µ§¯!¬µº9¡Í?°[¼¡½µÀb¨Ä±µ¸¨À|ÊÀK·×²9¨¼Ģ3¬µ½8³`¬µ¯z²9¹6ØDÅo¯!¯§3¼Ço¯|¡¯¨³z¨Ä¯¡µ¡Ê3Ý¨Ä±µ§µ3Ë¼µ¯!©¿.Ì+Eð0B0,À5+@ñÌò0+54\6ÒÑ\6Ã8Ç¨4Ü3áE«œcgR:U [jU=V=W¡XZY\[^]£R:_@V=`ba–R:U=V/X£a=V/ùaúc£U/ceW¡Xgf¨XikY?]^[lV:XeV=U/aWs`d`bžbü^X£ouU=Vn‹kV/V=`lXfV/ẁqxX£c£Ws[l`£YnX£~sX£V=U/a õ WsùajceU:ceW¡Xgfh–¡^]lvbR:t?XeV/`[dU=V=W¡XZY\[^]£R:_@V=`’f¦[EXevE|Wr`ŸY\XkobX(mid A) x = mid b∆ = ( I − |mid A) −1 |rad A ) −1|(mid A) −1 |((rad Aˆx + rad b)¿.Ì+Eð0B0,À5+@ñÌò0+54\6ÒÑ\6 Nx¨ßÛl«ÖØ×@476ON¥ÙÎ °¡±¡¯`²:¯z¬µ¯¡½¡¯w»2±µ¼µ§¯!±¡½9³z§½¡Âµ¯³z¼6¨©VÀE§±¡½¡¾µ·â½V»6±µ¼¡§¯±¡½µ³z§½ÁÂ¡¯³!¼6¨ª!¨®¯¼¡§¨±µ°¡±µ®¨©ÌÂµ³!§½Þèé©ê;ëKìâ®°¡¯!±µ®·¯w°¡±¡¯z²µ¬µ¨#Ço¯¡¨®§¨±¡¨2ÀÃ®¨áæ2Êµá2íÊµá-Šb«NÍ:¥¯!±2»¡½¡©Þ½µ6²9¯!¼³ I¹VÀE§±¡½µ¾¡·A b ¦ ¦!§¨Tª¨§½¡ÁIG»6±µ¼¡§¯±¡½µ³z§½¡Â¡¯³!¼¡½¡© cJ Í½Þ®¯!¼¡§¨±µ JÂ¡¯³!¼3ËÌ°µ¯!±¡®23ËFÄ¹2¼µ®³`¬µ¨®ºIÚ1Û¥Ü>Ý)Ü=ÚH%ÞÝ ßàß ß ÚJ ¦¥©¨ÕÀKµ¨ª!½¡»âÀE§¯ÀE§½¡Â¡¯³!¼¡½»â§¯!¼³z§3» 9°¡±¡½9ÀK¯!±gÊq®b¼µ¡½¡ª!¯ ±KÍ±€¨¼3ÁÐw¯z¬¡¬:±µÐ¨¤š@šk«½É²9±µ¹2ª½»=›ÀKµ¨ª!¨ª!±µ¡¡·¯VÀK¡¨#ÇoḠ³z§®2Aµ¸!·®2ØD§³ÏËI#°¡¨¬9½ÁI¤ámâ>ã ä&Û¥ÞåÝ1ÜRã¦Ï²9±µ¬µº¡·¥ÀK½JÊ§#Í¯Í¼3¬µº¼2¨©Õ³k³!¨¨§®¯!§³z§®¹2ØqÅ¯ª!¨Ñ¡ª*¬µ½¡©µ³!¼6¨ª!¨Ñ§¯!±µÀK½µµýÞ¡¨#ÇoḠ³z§®¨%±¡¯!Èo¯!¡½¡©m½µ¡§¯!±µ®3¬9º¡¨©b³!½µ³z§ḠÀK· šEšÍßá-›o®°µ¯!±¡®·¯ÞÄ·|¬µ¨¨4Ü3ál« ½V²:Ço¯l®¨³z°µ±¡¨½¡¸®¯z²9¯!¡¨µ¯!Æ;³!¹2°µ¯!±¡¨Ä¬µ¨#Ç¼6ū›zÊ¡¨®o²9±µ¹2ª¨©8°¡±¡¨¯!¼¡¾¡½µ½gÊ¡¡¯!Ço¯z¬9½©±½µ³Í9Û¡ÍßµÍµ±µ½µ³z¨®µ¨,ëÍ/¬w¨©µÀE©¡¯±¡¨À¿®F³!®¨¯©V¼¡¡½¡ª¯

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î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ý· J ¦ ½¡¡§¯±¡®¬µºµ¨ōÀE§±¡½¡Âµ¡¨ō°¡±¡¨½µ¸!®¯z²9¯!µ½¡¯ÍnýÞ¨#Ço¯§²:3Ç¯°¡¨¼3Á([1, 2])[−1, 1]( )[1, 3][−1, 1] [1, 2]x =[1, 3] ª¡Í ª@›Üí2Ü¡¨ÀK¯±¡¡¨ª!¨¹2±9®µ¯!¡½Ë ªµÍ¤šb›|³A = [1, 2] Ê b = [2, 3] Í@H®¹ÀK¯!±¡93Ë8³!½µ³z§¯ÀE²:Æ§oÄ¨¬µ¯¯Ý³z¬µ¨#Ç¡·©F°µ±¡½µÀK¯!±V°¡¹2³z§¨ª!¨²9¨°µ¹³z§½µÀK¨ª¨ÀK¡¨#Ç¯³!§®o±¡¯!Èo¯!¡½µ©gÍ¥â°¡¨G²9¨Ä¡·|»8³z¬µ¹6Âµ3Ë»VÄ¹²9ḠÀ«ª!¨®¨±µ½¡§ºµÊ9Â¡§¨F¬µ½¡¡¯©¡µ3ËAĢ#²:#Â9¨,²9¨°¡¹³!¼6»V/X—Y%]w¡pYnX£~rU”f¨]l½¡¬µ½5V:X£c£`b[wf¨X£c£WrV:]3Ê§3¼F¼¼T§¨ªÏ²:o½µ³Ï»¨G²:µ3Ë°¡¨2³z§3¡¨®¼lĢ3Á²:#Â¡½ ªµÍÛ@›|§¯±Ë¡¯§T³!ÀK·¥³z¬[Í]é©¨¨§®¯!§³!§®¯¡¡¨9Ê½¡¸;¨ÄÅ¯©8°µ¨³z§3¡¨®¼¡½8¬µ½¡¡¯©Á¡¨©¸3²:#Â¡½o¨D²:¨°¡¹2³z¼3»;®·²9¯z¬¡Ë¡¯§³ÏËoÄ¨¬µ¯¯©¹6¸¼6Ëo°¡¨G²:¸#²´#Âµ¥½µ³³`¬µ¯z²9¨®2¡½¡Ë¯!Æq±µ¸±¡¯!Èo½µÀK¨³!§½gÍ6¥¿¦§¨©Fª*¬9®¯DÀK·m¹²9¯z¬µ½9ÀÃ¸!µ#Âµ½¡§¯z¬9º¡¨¯D®µ½µÀE¡½µ¯D±µ¸zÁ®½¡§½¡Ø£ÀK¯§¨G²9½¡¼Þ²µ¬¡ËÞ²9¯§33¬µº¡¨ª¨½µ³!³z¬µ¯`²9¨®¡½ËÞ±µ¸±¡¯!Èo½µÀK¨³!§½A¬9½¡¡¯!©µ¡¨©¸3²:#Â¡½F¨²:¨°¡¹2³z¼3»½V®¨°¡±¡¨³ÀÃ¯Æl¼6¨±¡±¡¯¼¡¾¡½¡½[Í°¡½¡±µËµ³zºD9D³z®¨©µ³!§®¥½¡¡§¯±¡®¬µº¡·|»lÀE§±¡½¡Â¡¡¨Á1®¯!¼¡§¨±¡¡·|»l¨°¡¯!±µ¾¡½¡©³À|Í„2¬9®¹jÜ@›`Ê[ÀK¨#Ç¡¨Þ¨°¡±¡¯`²:¯z¬µ½¡§º8²9¨°¡¹2³z§½9ÀK¨¯ÀK¡¨#ÇoḠ³z§®¨V±¡¯!Èo¯!¡½¡©Ã½µÁÎ ªµÍ¤šb›|³`¬µ¯z²9¹6ØDÅ½9ÀÑ¨Ä±µ¸!¨ÀM§¯!±µ®3¬9º¡¨©T¬µ½¡µ¯!©¡¡¨©Þ³z½µ³!§¯ÀK·Ξ tol (A, b) = { x ∈ R n | A · x ⊆ b },ªÏ²9¯©I¸§º³ÏËgÊ6Â¡§¨½9ÀK¯!¯!§TÀKḠ³z§¨,§¨Â¡¡¨¯l±µ®¯!µ³!§®¨{ Ax | A ∈ A, x ∈ Ξ tol (A, b) } = b.ÀK½¬w¨k¦§¨%¡¯!®¯!±¡¡¨9ÊK¼6¼Ñ°¡¨¼¸!·®2¯§A°¡±¡½µÀK¯±¿Ģ#²:#Âµ½Ñ¨k²9¨°¡¹2³z¼3»É³T²:µ¡·|ÁA = [−1, 1] b = [0, 1]Ξ tol (A, b) y ≠ 0 y ∈ Ξ tol (A, b) (−sgn y)Ê3µ ½¡¸½¡¡§¯!±¡®23¬9[−1, 1] = A −|y| /∈°µ±¡½¡®¯!¬µ¨VÄ·£¼%¨§±¡½µ¾µ§¯!¬µºµ¨ÀK¹AÂ¡½9³`¬µ¹ Ê Ê3°¡¨³!¼2¨¬µº¼µ¹q½¡9#Â¡¯©¹ÀK¡¨#Ç¯!¡½µ¯ ÍÞ9²9Ḡ³zºÑ¬µ½¡ÈºÑ¡¹G¬µº¿ÀK¨#Ç¯!§ÉÄ·§ºÑ¦!¬µ¯ÀK¯¡§¨ÀÊ[0, 1]{ Ax | A ∈ A, x ∈ Ξ tol (A, b) } = 0 ≠ [0, 1] = b.Í9é¬µ¯z²9¨®2§¯!¬µº¡¨9Ê{ Ax | A ∈ A, x ∈ C } = b,C ⊆ R n ¹²9¨®¬µ¯!§®¨±2Ë¡¯!§³z®¨©µ³!§®¹¥

âõ æ ¯ ÷ø ë þ õlë#õ þ è õ!öOé ÷!ø ñ6ð.ê é ù é è¤ë úÜíßÁ2ó ¡óZ¦ ø ê õé è¤ë é{ Ax | A ∈ A, x ∈ C } ⊆ b.¨Ä±µ¸½¡§ºµÊÂ¡§¨±¡¯!Èo¯!¡½µ©A½¡¡§¯±¡®¬µºµ·|»Ì¬µ½µ¡¯!©¡µ·|»Þ³!½µ³z§¯À|Í\¬w¯!§±µ¹!²:¡¨µÊ:§¯ÀÕ¡¯ÀK¯!µ¯!¯Ê]³!¨ÁΞ tol (A, C¹²9¨®¬µ¯!§®¨±2Ë¡ØDÅo½» b)¯³!§ºµ½¡Ä¨¬µºÈo¯!¯¥°µ¨®¼¡¬µØDÂ¡¯¡½¡Øâ½¡¸qÀKµ¨#Ço¯³!§®¥ÒÚ*Û¡ÍÜoÀK·ãÀK¨ª*¬9½Ì®½6²9¯!§ºµÊÂ¡§¨Ξ tol (A, b) = { x ∈ R n | (∀ A ∈ A)(∃ b ∈ b)(Ax = b)}⊆ { x ∈ R n | (∃ A ∈ A)(∃ b ∈ b)(Ax = b)} = Ξ uni (A, b),¡¨s±Y½µ³Í¡Û¡Í¤šq½©ªµÍÜw²9¯ÀK¨µ³z§±µ½¡±¡¹6ØD§#ÊÂ¡§¨ÀK¡¨#Ç¯³!§®Ξ tol (A, b) ½ Ξ uni (A, b)ÖD¨§GËÞ®µ³z§¨#Ë¡Åo¯!©8¼¡¡½¡ª¯ÀK·ã®³zØ²:¹F±µ³!³!ÀE§±¡½¡®2ḠÀÉ®µ¹2§±¡¯!¡µ¯!¯;°¡±µ½ÁÀK¨ª¹2§¸!9#Â¡½¡§¯!¬µº¡¨,¨§¬9½¡Âµ§º³ÏËT²:±¡¹6ªl¨§²9±¡¹6ªz±µĢÀK¯!±µÀK½gÍ§#Í¯Í2Ä±¡¹³ÀK½T³!ö³z§¨±¡¨µÀK½gÊ2°µ±µ3¬¡¬9¯z¬µº¡·¥ÀK½T¼6¨¨±²:½¡µ§¡·¥À%¨³ÏËµÀ|Ê6³`¬µ¯z²9¹2ÁÄ¬µ½¡Ço¯!µ½¡Ȳ²:¨°¡¹2³z§½µÀK¨ª!¨ÝÀK¡¨#ÇoḠ³z§®2¥±¡¯È¯!µ½¡©½¡µ§¯!±¡®23¬µº¡·¥ÀK½®¯!¼¡§¨±µÀK½gÊ¦!§¨ª!¨É³!°¡¨³!¨ÄQ¨¾¡¯!¡½µ®µ½ËgÍ Î gÊY®ÑÂµ³!§¡¨³!§½gÊÀK¨#Ço¯!§%¨¼6¸§º³ÏËb®¯³!ºÀE¯!§»¨±µ¨Èö¨³!¨¸!9®§ºl¼¼Ì°¡±¡¯!½9ÀK¹2Åo¯³!§®µÊ2§3¼F½F®¨ĢÀK¨#Ç,µ·¯D¡¯z²9¨³!§#§¼¡½¼k¨ÄçgÆGÀK¹VÀKµ¨#Ço¯³!§®T±¡¯!Èo¯!¡½¡©¡¯!®·ª!¨G²9µ·¥Àj®§¨ÀÃ³ÀK·¥³`¬µ¯ÊÂ¡§¨¨§¡¨Èo¯!¡½¡¯D¨Äç[ÆÀEw¨¾µ¯!¡½¡®2ØqÅ¯!ª¨lÄ±¡¹2³!Ξ tol (A, b) ÀK¨#Ço¯§FÄ·§ºÌ³!¼2¨¬µºF¹6ª¨G²9¡¨FÀEÁ³!À|Í?±½µ³Í¡á9Í4æE›`ÊÂ¡§¨°µ±¡¨½µ³Ï»2¨G²9½¡§;²µ¬¡ËÌ²µ¬µ½¡¡µ·|»F®·§GË¡¡¹6§·|»ÌÀKµ¨#Ço¯³!§®±¡¯!Èo¯!¡½µ©8èé©ê;ëKìÍ¥«¦!§¨ÀÑ³z¬µ¹6Âµ¯Ý½µÀK¯¯!§³!ÀK·¥³z¬Ì±µ³!³!ÀK¨§±¡¯§º,¨¾¡¯¡½¡®¡½¡¯¬µ·¥À²9¨°µ¹³z§½µÀK¨ª¨8ÀK¡¨#Ç¯³!§®8±¡¯È¯!µ½¡©jµ¯Ä±¡¹2³!ÀK½[ÊgÌ¼¼¡½µÀK½Á1¬µ½¡Ä¨T²9±µ¹2ª½µÀK½ÀK¡¨#ÇoḠ³z§®2ÀK½gÊ9°¡±¡½9ÀK¯!±gÊ¡¼6¨2³z·¥ÀK½8°µ±µ¬¡¬µ¯!¬µ¯!°¡½µ°¡¯`²:ÀK½5¨4Ü%še«1Í’² § {§¨( ÐdŽ (¿¨{ ¡µÃSÇ£¥§©&'(©)Ξ tol (A, b) = ⋂A∈AΞ tol (A, b), ªµÍí-›Ùl¼8Âµ³!§¡·©V³`¬µ¹6Âµ©º°g¯!¨±¡¯ÀK·ÒÛÍ4ÜÍ}šÝ½µÀK¯!¯§TÀK¯³z§¨°¡±¡¯`²´³z§3®¬µ¯!¡½µ¯Ξ tol (A, b) = ⋂{ A −1 b | b ∈ b },A∈A ªµÍ Š-›½gÊ¡Ä¨¬µ¯¯q§¨ª!¨9ÊAx b²µ¬¡Ë5v-]bq ou`b|Â¡§¨Ä·Õ°¡±µ¨½¡¸®¯`²:¯!¡½¡¯A ∈ A Ax ∈ b°¡¨°µ#²´3¬µö®o®¯!¼¡§¨±F°¡±9®¨©FÂµ³z§½ ÊÀK·Õ°¡¨¼Ço¯À|Ê2Â¡§¨²9¨2³z§3§¨Â¡¡ö®·°¡¨¬µ¡¯!µ½ËT®¼¬9ØDÂ¡¯¡½Ëp`¡¯¨³!¨Ä¯¡µË8¼¡®2#²9±9§µËkÀE§±µ½¡¾µ¡Ê´§#Í¯Í´¼6¨ªÏ²:µ¯!¨³!¨Ä¯!¡¡·ã®2³z¯A ¦ ¯³z¬µ½A ∈ A Í©ÖD¨§Ëm¨°¡±¡¯`²:¯z¬µ¯¡½¡¯V²9¨°¡¹³!§½µÀK¨ª!¨ÉÀK¡¨#Ç¯³z§®%±¡¯È¯!µ½¡©m§±¡¯Ä¹¯§#Ê

î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýÜíá¬µ½¡ÈoºÝ²µ¬¡ËTÀE§±µ½¡¾A ½¡¸¡¯¼6¨§¨±¡¨ª¨sv@`bV/Xet%V/` p`q°¡¨G²´ÀK¡¨#ÇoḠ³z§®2q® A ÍèlÀK¯!¡Á¡¨µÊ½9ÀK¯!¯!§TÀKḠ³z§¨N\6Ã8¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76Ξ tol (A, b) = { x ∈ R n | (∀ V ∈ vert A)(V x ∈ b) }˜;5,=@2%A?+@B0C:DEA%F:,A³!®¨G²9½µ§³ÏËÌ¼V°¡±¡¨®¯±¡¼6¯q§¨ª!¨9ÊÂ¡§¨Ξ tol (A, b) ⊇ { x ∈ R n | (∀ V ∈ vert A)(V x ∈ b) },åw±¡¯`²9°µ¨¬µ¨#Ç½µÀ|ÊÂ¡§¨¡¯!¼6¨§¨±¡·©8®¯¼¡§¨±§3¼8¼6¼Ì¨Ä±µ§¡¨¯Ý®¼¬µØqÂ¡¯!¡½µ¯Ý¨Â¡¯!®½6²9¡¨9Íx ∈ R n ¹²9¨®¬µ¯§®¨±2Ë¡¯!§V x ∈ bV ∈ vert A A ¦ ¼¼6ËÁ1§¨qÀE§±¡½¡¾µÝ½¡¸A Í¥Ã³!¨¨§®¯!§³z§®½¡½³w¨°¡±µ¯`²9¯!¬µ¯!¡½µ¯À²µ¬¡Ë®2³z¯!»vert A λ V ≥ 0 ¨ÄÅ½µÀ¿Â¡½µ³zÁÍåw¹³z§º ³!¹2Åo¯³!§®¹6ØD§¼6¨¦ÐlÐw½¡¾¡½µ¯!¡§·2 mn Ê¡§3¼µ½¡¯wÂ¡§¨ ¬µ¨À∑V ∈vert Aλ V = 1½A = ∑V ∈vert Aλ V V,A A Í°g¨ªÏ²:§#Í¯Í( )°¡±¡¯z²:³z§3®½µÀEÝ®w®½6²9¯®·°¡¹6¼¬µ¨©o¼6¨ÀKÄ½¡µ¾¡½¡½¼¡±µ©µ¡½»oÀE§±¡½¡¾½¡¸ ∑Ax =V ∈vert Aλ V V· x = ∑V ∈vert Aλ V V x,Ax V x V ∈ vert A V x ∈ b®;³z½¬9¹³ ²9¯!¬9µ¡¨ª¨Ý°µ±¡¯`²9°µ¨¬µ¨#Ç¯!¡½ËgÊ³z¬µ¯`²:¨®§¯z¬9º¡¨9Ê3½»®·°¡¹6¼¬9Ë¼6¨ÀKÄ½Á§#Í¯Íb ÍÍ@¬w¨;®2³z¯ Ê Ë¡®¬µË¡¯!§³ÏË®·°µ¹2¼¡¬µ¨©¼6¨ÀKÄ½¡µ¾¡½¡¯!© µ¾µ½ËÌ§3¼¡Ço¯w°¡±¡½¡9#²µ¬µ¯!Ç½¡§®·°¡¹6¼¬9¨ÀK¹FÀK¡¨#Ç¯³!§®¹Ξ tol (A, b) =⋂V ∈vert AΞ tol (V, b), ªµÍ¤šu§@›±€¯¸!¹G¬µº3§3§åw±¡¯z²µ¬µ¨#Ç¯!¡½Ë©ª¡Í4ÜÍ}šÝ®¬µ¯!ÂµÆ!§#Ê®Âµ³!§¡¨2³z§½Ξ tol (A, b) =⋂A ¦ ¼¡®3²9±µ§µ3Ëo½o®2³z¯§¨Â¡¯Â¡¡·¯ÀE§±¡½¡¾¡·A ∈ vert A ¡¯¨³!¨Ä¯¡¡·²µ¬µË8®2³z¯z»DḠ³`¬µ½A ∈ A ¡¯¨³z¨Ä¯¡¡¨³!§º¡¯w§±¡¯!Ä¹¯§³ÏË/›`Ê¡§¨V ∈vert A{ V −1 b | b ∈ b },

x = x ′ − x ′′ “âõ æ ¯ ÷ø ë þ õlë#õ þ è õ!öOé ÷!ø ñ6ð.ê é ù é è¤ë ún |g[^X£vEWs`ŸY:h x ′ UMx ′′ i@ou`l[£RnXeW¡[l`ŸY%T:W"ceU:ceW¡Xgf¨XR:U=V:Xea=V/hG•Á2ó ¡óZ¦ ø ê õé è¤ë éÜíÛÊ¥¬µ¯!ª¼2¨mÀK¨ª!¹6§«Ä·§º«®·°µ½µ³!¡· ®bË¡®µ¨Àu®½6²9¯Ê¥§¨mÀK¨#Ç,Á ª¡Íí@›©½ ªµÍ Š-›*ÍÙw¨¬µºÉ³!¼2¨±¡¨Ã¹6±µ®¡¯!µ½Ëbª½¡°¡¯±¡°¬µ¨2³z¼6¨³!§¯!©gÊ¨ª!±9¡½¡Âµ½¡®ØDÅ½¡»bÀK¡¨#Ç¯zÁ§#Í¯ÍÊµÀK·ã³z¹6Åo¯³z§®¯!µ¡¨,¹³z½¡¬µ½¬µ½8°¡±¡¯z²:³z§3®¬µ¯!µ½ËΞ³z§®¨tol (A,¡Ä°¡¨°¡·§3§º2³ÏËÃ°¡±¡¯`²´³z§3®½¡§ºb)Ξ tol (A, b) ¼¼ÉÀK¡¨#Ç¯³!§®¨V²:¨°¡¹2³z§½µÀK·|»%±¡¯!ÁÈ¯¡½¡©kµ¯!¼6¨§¨±¡¨©j¸3²:#Â¡½Þ¬µ½µ¡¯!©¡µ¨ª!¨8°¡±¡¨ª!±µÀEÀK½¡±¡¨®µ½ËgÍ7ûÝ®µËkÐw¨±µÀE°¡±¡¯z²:³z§3®¬µ¯!9oè,Í?±K¨¡¨ÀÑ®ĢÀK¯§¼6¯†¨ßßb«³z¨¨§®¯!§³!§®¹6ØDÅo¯!©%¸3²:#Â¡½Q¬µ½¡µ¯!©¡¡¨ª!Ä°µ±¡¨ª±µÀEÀK½¡±µ¨®¡½Ë%®°¡¯±¡®·¯FÄ·|¬9§¯!¨±µ¯ÀE¡±K¨µö²9¨°¡¹³!§½µÀK¨À¿ÀK¡¨#ÇoḠ³z§®¯w±¡¯!Èo¯!¡½¡©:›*+-,/.0+-132ËÇL6ON7698x ∈ R n m Y:U/V/]ôkRnXkq U=W ou` m i@c£WrU”f`’fïdfV=ùqxX£ceW¡[li YnX£~sX£V=U=a¢U=V/|n=ùt%v-]Ax = b Ws`¤ kob]©U Ws`’R:_Ev@`5Ws` kob] “V/X—Y%]^[EXeV/c£Ws[⎧⎪⎨⎪⎩Ax ′ − Ax ′′ ≤ b,−Ax ′ + Ax ′′ ≤ −b,x ′ , x ′′ ≥ 0. ªµÍ¤š@šu›WsX—Y%]wR:_EV=`baQR:U=V:Xea/V=`bajc£U/c£WsXgf¨hvE` kob]kobX;5,=@2%A?+@B0C:DEA%F:,G6LÞ]ÀK¯§½µÀ|Ê¡Âµ§Äx = [ (mid A) x − (rad A) |x|, (mid A) x + (rad A) |x| ] ,x ∈ Ξ½«°¡¨¦§¨ÀK¹¿°¡±¡½¡µ3²µ¬µ¯!Ç,µ¨³z§ºtol (A, b)Ax ⊆ b Ê¡®¬µ¯!ÂµÆ!§b ≤ (mid A) x − (rad A) |x| Ê©±µ®¡¨³!½¬µºµ3ËÑ®¼¬µØqÂ¡¯!¡½µØ(mid A) x + (rad A) |x| ≤ b .½x = x + − x − ½|x| = x + + x − ½®·°¡¨¬µËËQ¡¯³z¬µ¨#Ç¡·¯°¡±¡¯¨Ä±9¸!¨®¡½ËgÊ:¹2Ä¯zÇ²:¯ÀE³ÏËgÊ]Â¡§¨Þ®¯!¼¡§¨±¡·H¯z¬9Ë¿®Ã¦§½»¿µ¯!±µ®¯!9³z§®23»É¸ÀK¯!¡·x ′ =x + x ′′ = x − ¹²9¨®¬µ¯!§®¨±2Ë¡Øq§ ª¡Í}šEšu›`Í¬l¨Ä¨±¡¨§#Ê°¡¹³!§ºm¡¯!¼6¨§¨±¡·¯½x ′ x ′′ Ë¡®¬¡Ë¡Øq§³ÏËÒ±¡¯!Èo¯!¡½¡ËµÀK½s³z½µ³!§¯ÀK·Êd i = min{ x ′ i , x′′ }, i = 1, 2, . . . , n.i ªµÍ¤š@šu›*Í Î °¡±¡¯z²9¯z¬µ½9ÀV®¯!¼µ§¨±d ∈ R n §3¼gÊÂµ§¨Ä·Ã¯!ª!¨D¼2¨2ÀK°¡¨µ¯!¡§·Ä·|¬µ½VÀK½¡¡½9ÀK¹6ÀEÀK½Þ½¡¸;³z¨¨§®¯!§³!§®¹6ØqÅ½»T¼6¨ÀK°µ¨¡¯¡§¡¯!±9®¯µ³z§®x ′ x ′′ Ê¡§#Í¯Í ½d ≥ 0 x ∈ R n Ê§3¼2¨ª!¨,Â¡§¨°g¨ªÏ²:x = x ′ − x ′′½Ì²µ¬¡ËV®¯¼¡§¨±9 ,ÀK·ä½9ÀK¯!¯Àx + i= max{ 0, x ′ i − x′′ i } = x′ i − min{ x′ i , x′′ i } = x′ i − d i,

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é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýî6ï9ðñ2ðÆÁ2ó¨í³`¬µ½%ÀK·sĢ¼¬µØqÂ¡½¬µ½jµÌ¨³zµ¨®¡½¡½d°[¯!¨±µ¯ÀK·ÿªµÍßµÍ¤šo¨F¡Ḡ³z¨®ÀK¯³!§¡¨2³z§½³z·k ∈ {1, 2, . . . , m}®É¦§½»Õ³z§±µ¨¼3»mÀE§±µ½¡¾¡·A Ê§#Í¯Í©¹6®¯z¬µ½¡Âµ½¡§ºÄ¨¬µ¯!¯ÞÈo½¡±¡¨¼µ½¡¯k¦!¬µ¯ÀK¯¡§·max 1≤j≤n χ(a kj ) χ(b k ) ÍÊ¬µ½µÄ¨±µ³!È½¡±¡½µ§ºo°µ±µ®¹2ØâÂµ³z§º9Ê2§#Í¯Í¡¹ÀK¯¡ºÈo½¡§ºI#®®¯`²9¯¡½Ë8®Ä2¸½µ³Jm×n |9f£]lW{Y:U¥¢:] A UÜ^ŠáA = ( a¬µ½¡¡¯©¡¡¨©bĢ#²:#Âµ½Ñ¨j²9¨°µ¹³z¼»Ñ³ÞÀE§±¡½µ¾¡¯!©ij )®¨©ÑÂµ³z§½b = (b i ) ÊE§¨j²µ¬¡ËÑ²:3¬µº¡¯!©µÈ¯!ª¨k½µ³³`¬µ¯z²9¨®2¡½¡ËÑ¸3²:#Â¡½¿ÀK¨#Ç¯!§¨¼Ģ§º2³ÏËT°¡¨¬9¯!¸!µ¨©8®¯z¬9½¡Â¡½¡µM Ê¡®®¯`²9Æ¡µ3ËÌ®ÌÚ`ÛÍ4ÛÍ¡åw±¡½V¦§¨À½b®¯!¼¡§¨±¡¨2Às°¡±µÁM =min1≤i≤m⎧⎨⎩ χ(b i) − max1≤j≤na ij≠0⎫⎬χ(a ij )⎭ ≥ 0,M ²9¨q¡¯!¼6¨§¨±¡¨©³z§¯°¡¯!µ½oÀK¨#Ço¯§»2±µ¼µ§¯!±¡½µ¸!¨®2§ºI3ÀK¯!±µ¹¡¯±µ¸±¡¯!Èo½µÀK¨³!§½½;¼6¨¡¼µ±¡¯!§µ¨¯¸µ#Â¡¯¡½¡¯±µ³³!ÀE§±¡½¡®2¯ÀK¨©oĢ#²:#Âµ½MÂ¡¯Àj¨9qÄ¨¬µºÈ¯Ê3§¯ÀJÊG²µ¬¡Ë,¼2¨§¨±µ·|»;®·°¡¨¬µ¡¯!µ¨Ý¹2³`¬µ¨®½¡¯ àß}ß ›0°g¯!¨±¡¯ÀK·$ª¡Íß¡Í}šÊÄ¨¬µ¯!¯q²:¬µ¯!¼¸3²:#Âµo¨§±µ¸!±¡¯È½µÀK¨2³z§½gÊµ½89¨Ä¨±¡¨§#Í6Ùw±µ¨ÀK¯l§¨ª!¨µÊµ½¡6²9¯¼ÁA ½¿³z¨¨§®¯!§³!§®¹6ØqÅ½¡¯Ì¼2¨2ÀK°¡¨µ¯!¡§·®¯¼¡§¨±9¹6¼6¸!·®2ØD§Þ9A§¯V³z§±µ¨¼¡½bÀE§±¡½µ¾¡·b Ê¼6¨§¨±¡·¯%®¡¨2³ÏË¡§Ã²9¨2ÀK½¡¡½¡±¡¹6ØqÅ½¡©ã®¼¡¬9#²ã®¿¡¯±µ¸±¡¯!Èo½µÀK¨³!§º±µ³³!ÀE§±¡½¡®2¯ÀK¨©%¸#²´#Â¡½%¨V²9¨°µ¹³z¼»´Í\H;¬¡ËÃ§¨ª¨µÊ[Âµ§¨Ä·£¹ÀK¯¡ºÈo½¡§ºA¨§3Á¼¬µ¨¡¯!¡½µ¯¸3²:#Â¡½Þ¨§T±µ¸±¡¯!Èo½µÀK¨ª¨F³z¨2³z§¨#Ë¡µ½ËgÊ:³z¬µ¯`²:¹¯!§T¬µ½¡Ä¨Ì³z¹6¸!½¡§ºFµ½ÁÃS ¬« Ï 3¸¨ È¸ºUÐd£¨¥©© §¥§&© ¨{ÀK¨#Ço¯!§°µ±¡¯`²:³!§3®¬9¯!¡¨®®½6²9¯D±µ¯!È¯¡½ËF³!½µ³z§ḠÀK·m¬µ½¡µ¯!©¡¡·|»Tµ¯!±µ®¯!9³z§®9Ê§ö®¨°¡±¡¨³¥¨¯!ª¨°¡¹2³z§¨§¯°[¼A¼¼Ì²9¨°µ¹³z§½µÀK¨¯ÀKµ¨#Ço¯³!§®¨±µ¯!È¯¡½¡©kèé©ê;ëKì½¬µ½,¡¯°¡¹³!§¨§¯qÀK¨#Ço¯!§wÄ·§ºl±µ¸±¡¯!ÈoÆ!gÊ3¼°µ±¡½µÀK¯!±µ¹Ê°µ¨³z±µ¯`²:³!§®¨2ÀA°¡±¡½9ÀK¯!¡¯!Á¡½Ëµ#Âµ¬µºµ¨ª!ẅ¦§#°µl³z§36²:±µ§¡¨ª¨;³z½µÀK°¡¬µ¯!¼³`ÁÀK¯!§¨G²: §3¼9¸!·®2ḠÀK¨ª!¨›zÍÎ ³z¡¨®¨©±µ¸!®½µ®ḠÀK¨©o¡½Ço¯§¯!¨±µ½¡½,±µ¸±¡¯!Èo½µÀK¨³!§½,½¡¡§¯±¡®¬µº¡¨©¬9½Á¸¾¡½ËT²:¨°¡¹2³z§½µÀK¨ª!¨,ÀK¡¨#ÇoḠ³z§®2±µ¯!È¯¡½¡©gÍ¡¥ÒÚ*ÛÍ4Ü3ÄÀK·ä¹2Ç¯Ý®2³z§±µ¯!Âµ¬µ½µ³zº,³¡¯!©µ¡¨©«¸3²:#Â¡½b¨j²9¨°¡¹2³z¼3»ÃË¡®¬¡Ë¡¯!§³ÏËb¡¨®Ëb93¬µ½¡§½¡Â¡¯³!¼3ËÑ»6±µ¼¡§¯±¡½ÁM°¡±¡½µµ#²µ¬9¯zÇ¡¨³!§ºx ∈ Ξ»2±µ¼¡§¯!±¡½¡Ģ¾¡½µ¯!©oè,Íb±K¨9tol (A, b)| mid A · x − mid b | ≤ rad b − rad A · |x|.¦!¼¡®½¡®¬µ¯!µ§µ‡L6Ã8jþri@c£Wr_ºob]lV/hÇU=V/WsX—Yn[^]wR:_EV/]wT¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76am |g[^Xev@Wr`ŸY m Y%]E[l`badt?]^ceWsUU=V/WsX—Yn[^]£R:_@V=h ]S[lh:Y?]bq X£V=U/Xgfb“¬w¨,¨§°¡±9®µ¨©F§¨Â¡¼6¨©8µÈ½»F±9³!³ÀK¨§±µ¯!¡½¡©8Ë¡®¬¡Ë¡¯!§³ÏË²4³w´(x) = ²4³w´ (x; A, b) =min1≤i≤m⎧⎨⎩ rad b i −∣ mid b i −∣⎫ n∑ ∣∣∣∣ ⎬a ij x j⎭j=1

Á2ó §¡óÍ õ ï¤è õé ë ÷!÷ ï éâõ ñ2ð&è¤ë é êgð ƒ ê é ù ë þ õ ÷ø ëx ∈ Ξ;5,=@2%A?+@B0C:DEA%F:,G6D¥|³z°¡¨2ÀK¡½µÀ|Ê©Â¡§¨tol (A, b)Ax ⊆ b Íµåw¨2³`¬µ¯z²9¡¯¯Ý®¼¬µØqÂ¡¯!¡½µ¯wÀK¨#Ç¡¨,°¡¯±¡¯!°µ½µ³!§º®³`¬µ¯z²9¹6ØDÅḠÀÑ®½6²9¯ ¯³z¬µ½m½m§¨¬µº¼6¨Ã¯³z¬µ½¡·lÊ€¦!§Ä½9³!³`¬9¯`²9¹¯ÀE3ËÃ§¨Â¡¼x∈ R n ÊKA³²9±¡¹6ª!¨© ¦x ∈ÜEŠÛm: R n → Rñ?` kob] m Y:U/V/]ôkRnXeqV=`lc£Wr_` YnXwobXgR%TnXeW¡cŸT©öÈi@V=v>¢/U=`bV/]wRµ²4³w´Ξ tol (A, b) (x; A, b)W†ÕXwõu`0“m i@c£WrU”f`lXf¨V=ùq XwceW¡[l`≥ Y%]E[lV=`lceU”R:_EV:]|²%³/Áx = bYnX£~sX£V=U=aKU/V=W¡XZYn[E]£R:_EV/ùa R:U/V/X£a=V=`ba ceU:ceW¡XgfhXwceWs_ fV/ẁqxX£c£Ws[lìkYn`b[^V-T{ x∈R n | ²%³/´ (x; A, b)≥0 }öÈiEV/v>¢/U=`bV/]£Rn]²%³/´@˜mid b i −n∑a ij x j ⊆ [−rad b i , rad b i ], i = 1, 2, . . . , m,j=1Â¡§¨¦¼¡®½¡®23¬µ¯¡§¡¨∣∣ mid b i −∣n∑ ∣∣∣∣a ij x j ≤ rad b i , i = 1, 2, . . . , m.j=1é¬µ¯`²:¨®§¯z¬9º¡¨9Êx ²9¯!©9³z§®½¡§¯z¬9º¡¨¥°µ±¡½¡µ3²µ¬µ¯zÇ½¡§ Ξ tol (A, b) §¨ªÏ²:|½l§¨¬µº¼6¨§¨ªÏ²´¡Ê¼6¨ªÏ²:(x; A, b) = min²%³/´1≤i≤m⎧⎨⎩ rad b i −∣ mid b i −∣⎫ n∑ ∣∣∣∣ ⎬a ij x j⎭ ≥ 0.j=1§ÀK¯!§½9Àk³z±9¸!¹ÝÇ¯ÊGÂ¡§ẅÐw¹2µ¼¡¾¡½¡¨µ3¬²4³w´(x; A, b) ¡¯!°µ±¡¯!±¡·®¯!;°µ¨D®³!¯À³z®¨½µÀb±¡ª¹ÀK¯!¡§3À¿¼¼V®,¨Ä·Â¡¡¨Àb³!ÀK·¥³z¬µ¯ ¦§¨F³z¬µ¯`²9¹¯!§,½¡¸;µ¯!°¡±¡¯±¡·®µ¨ÁÎ³!ÀK·¥³z¬µ¯l°¡¨,êl½¡°¡Èo½¡¾¡¹Í³z§½%½¡µ§¯!±¡®23¬µº¡·|»j±¡½µÐlÀK¯!§½¡Â¡¯³!¼¡½»k¨°¡¯!±9¾¡½¡©:›`Êg§3¼Q½%®VÄ¨¬9¯!¯,³z½¬µº¡¨ÀÄ¹²9¯À¤µ¸!·®§ºmÐw¹6¡¼¡¾¡½¡¨µ3¬h²4³w´(x; A, b) Y?]^c m è¡^V/]^ŽrU”flÊ°¡¨Á³z¼6¨¬9º¼¡¹o¸!µ¼¯ª!¨l¸µ#Â¡¯!µ½¡©°¡¨¸®¨¬¡Ë¡¯!§SI#±µ³z°¡¨¸!µ§ºýÞ·§¨Â¡¼¡½½¡¸Ξ J tol (A, b) Íë¥±¡ª¹ÀK¯!¡§·×¦!§¨ª!öÐw¹6¡¼µ¾¡½¡¨93¬9lµ¯D®°µ¨¬µ¡¯D±µ®¡¨°¡±9®µ·M³¥¨G²:¡¨©F³!§¨±µ¨Á²:µ¡·¯T¸3²:#Â¡½gÊe§#Í¯ÍÀE§±µ½¡¾µA ½k®¯!¼¡§¨±A°¡±9®¨©AÂµ³z§½ b Í]åw¨T¦!§¨©k°µ±¡½¡Â¡½¡µ¯ÀK·s½j¨§²9¯!¬¡Ë¡¯À½»;²:±¡¹6ª¨§Ý²9±¡¹6ªzÝ§¨Â¡¼6¨©³¸°Ë¡§¨©gÍÙw¨ªÏ²´D®§¨±¡½µÂ¡¡·¯¥±µª!¹ÀK¯!¡§·b±9³z°µ¨Á¦ A b ¦ µ¯³z¹6Åo¯³z§®¯!µ¡·lÊ2ÀK·mÄ¹²9¯À%¨°¡¹³!¼6§º½»]Êª!¨®¨±2ËF°µ±¡¨³!§¨¨Ðw¹2µ¼¡¾¡½¡¨µ3¬µŌ²4³w´¸!µØDÅo¯!ª!öÐw¹2µ¼¡¾¡½¡¨µ3¬9(x) ½ Í

Ü^Š@ªé ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýî6ï9ðñ2ðÆÁ2ó¨‡L6 N›·iEV/v>¢=U/`bV/]£Ra²4³w´(x)[l` pV=iEWsh a:˜¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76(x) ¯³z§ºµ½ÇËË8¨ª!½¡Ä2ØqÅo3ËFÐw¹6¡¼¡Á;5,=@2%A?+@B0C:DEA%F:,G6„l¹2µ¼¡¾¡½¡¨µ3¬a²%³/´∣¾¡½¡¨µ3¬µ¨®ς i (x) = rad b i −∣ mid b i −∣n∑ ∣∣∣∣a ij x j , i = 1, 2, . . . , m,j=1ς½VµÀÃ²9¨³!§#§¨Âµ¡¨¬µ½¡Èº,¹2³z§3¡¨®½¡§ºo®¨ª!µ¹2§¨³z§ºo¼3Ç²9¨ª!¨,½¡¸i (x)åw¹³z§ºx, y ∈ R n Êλ ∈ [0, 1] Íwé©¹2ÄG²:½µ³z§±µ½¡Ä¹6§½¡®¡¨³!§ºm½¡¡§¯!±¡®23¬µº¡¨©±¡½9ÐlÀK¯!§½¡¼µ½8®¬µ¯!Âµ¯!§§¨ªÏ²:Ímid b i −n∑a ij (λx j + (1−λ)y j )j=1() (n∑n∑⊆ λ mid b i − a ij x j + (1−λ) mid b i − a ij y j).j=1j=1¦ Ðw¹6¡¼¡¾¡½¡Ë| · | ¦ ÀK¨µ¨Á§¨µµ°¡¨,®¼¬µØqÂ¡¯!µ½¡Ø ³À|Í¥2¬9®¹ šb›`Ê¡³z¬µ¯`²9¨®§¯z¬µº¡¨èw¸!®¯³z§¡¨µÊÂ¡§¨,Ä³!¨¬µØq§µËT®¯z¬9½¡Â¡½¡µ½¡¡§¯±¡®¬9∣∣n∑∣ mid b i− a ij (λx j + (1−λ)y j )∣j=1(≤∣ λ mid b i −≤ λ∣ mid b i −) ()∣n∑n∑ ∣∣∣∣a ij x j + (1−λ) mid b i − a ij y jj=1j=1∣ n∑ ∣∣∣∣ a ij x j + (1−λ)∣ mid b i −j=1∣n∑ ∣∣∣∣a ij x j ,j=1Â¡§¨½F²9¨¼¸!·®¯!§åw±¡¯`²µ¬9¨#Ço¯!µ½¡¯Íèw§#¼gÊ°¡¨G²9ª±µÐw½µ¼hyp = { (x, t) ∈ R n+1 | x ∈ R n , t ∈ R, t ≤ (x) } ²4³w´ ²4³w´: R n → R Ë¡®¬¡Ë¡¯§³ÏËQ®·°¡¹6¼¬µ·¥ÀãÀK¡¨#Ç¯³z§®¨À|ÍÌýÞ¨#Ç¡¨²:3Ç¯|°¡¨¼Ģ§º9Ê#Â¡§¨¨§¨Ä±µÇo¯¡½Ëi²4³w´hyp³z§±9µ³!§®Ì®R n+1 Ê´§GÍ¯Í´Ë¡®¬¡Ë¡¯!§³ÏËA®·°¡¹6¼¬µ·¥À×ÀK¡¨ª¨ª!±9¡¡·¥À×ÀK¡¨#ÇoḠ³z§®¨À|Í²4³w´|¯³!§ºw°¡¯!±µ¯³z¯Â¡¯!µ½¡¯¼6¨¡¯Â¡¡¨ª¨qÂµ½µ³`¬9w°µ¨¬µ¹6°¡±¡¨Á

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R n˜î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ý(x) ≤ minåw¨³!¼2¨¬µº¼µ¹6²%³/´1≤i≤m rad bÊ#ÀK¨#ÇoḠÀ8¸¼¡¬µØDÂ¡½µ§ºµÊGÂ¡§¨q®³z¯i γ k ≤ 0k = p + 1, . . . , q Ê´§#¼k¼¼j®Ì°µ±¡¨§½µ®¡¨2ÀÕ³`¬µ¹6ÂµōÐw¹6¡¼¡¾µ½¡¨µ¬i²4³wóÄ·|¬jÄ·¡¯!¨ª!±µ¡½¡Â¡¯8³!®¯±»¹Í6åw¨,¦§¨©V°¡±¡½µÂ¡½¡¡¯Ê°[¼¡½µÀV¨Ä±µ¸¨À|Êmax x∈R n²%³/´Ü^Šíou`lceWsU£ k]EXeWp`5f£]lvE| vE`bV/X£t%V=`¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76 ‡L6à€h·iEV/v>¢=U/`bV/]£R;²%³/´(x)ceU”fiuf£]sV/]S[^cwXgfhyp¬µ·¥ÀÉÀK¡¨ª!¨ª!±µ¡¡·¥ÀÃÀK¡¨#Ç¯³z§®¨À|ÊË¡®¬µË¡¯!§³ÏËF®·°¡¹6¼¬µ¨©T¨Ä¨¬µ¨Â¡¼6¨©¼6¨¡¯ÂÁ;5,=@2%A?+@B0C:DEA%F:,G6¥åw¨G²9ª!±9Ðw½¡¼bÐw¹2µ¼¡¾¡½¡¨µ3¬9¡¨ª¨kÂ¡½9³`¬9Q§¨Â¡¯!¼(c k , γ k ) k = 1, 2, . . . , p (c ²%³/´ÊYÄ¹²9¹6Â¡½«®·°¡¹6¼Ák , γ k ) k =p + 1, . . . , q R n+1 Ê® ¨}šßl«1Íåw±¡½8¦!§¨2ÀÉÀK·m²:¨¬¡Ç¡·ä½µ³z¼¬9ØDÂ¡½¡§º9°¡±µ®¬µ¯¡½¡¯(0, . . . , 0, 1) (x) Ê§#¼Ì¼¼a²4³w´ ®³zØ²:¹T¨°¡±µ¯`²9¯!¬µÆ!gÍ?¥¨¬µ¯!¯w§¨Â¡¡¨µÊÊE½bµ°¡±9®¬9¯!¡½¡© Ê Êhyp ²%³/´ ={ q∑ λ k (c k , γ k ) ∣ c k ∈R n , γ k , λ k ∈R, λ k ≥ 0,k=1}p∑λ k = 1 .k=1maxx∈R n{ q∑ ∣ ∣∣= max λ k γ k λk ≥ 0,k=1}p∑λ k = 1²4³w´(x) = max { t | x ∈ R n , t ∈ R(x, t) ∈ hyp ²4³w´ }{ p∑ ∣ ∣∣= max λ k γ k λk ≥ 0,k=1k=1}p∑λ k = 1k=1= max1≤k≤p γ k.(x) ³z¨®°µ3²:¯§³ÀE¼³z½µÀK¹ÀK¨2ÀF°¡¨D¡¯!¼6¨§¨±¡¨2ÀK¹¦ °¡¨®³zḠÀγ¼6¨¡¯Â¡¡¨ÀK¹,ÀK¡¨#Ç¯³z§®¹¸µ#Â¡¯¡½¡©TÐw¹6¡¼¡¾¡½µ¨µ¬9 Ê ¦ ½,²:¨³z§½Ákªz¯§³ÏË9w¸!µ#Â¡¯¡½¡½T±¡ª!¹ÀK¯!µ§#µÊ³z¨¨§®¯!§³!§®¹6Åo¯ÀQÀE¼³z½µÀE¬µºµ¨ÀK¹½µ¸¦!§½»γÊk k = 1, 2, . . . , p Í⎛⎞ ⎛ ⎞[2, 3] [−1, 2] [0, 60]⎜⎟ ⎜ ⎟⎝ [1, 2] [1, 3] ⎠ x = ⎝ [10, 72] ⎠[−1, 1] [0, 1] [−10, 36] ªµÍÜß@›¿.Gò013+-.·H;¬¡Ë8½¡¡§¯±¡®¬µº¡¨©F¬9½¡¡¯!©µ¡¨©3 × 2 ÁN³!½µ³z§ḠÀK·

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Á2ó §¡óÍ õ ï¤è õé ë ÷!÷ ï éâõ ñ2ð&è¤ë é êgð ƒ ê é ù ë þ õ ÷ø ëaÊg¬µ¯®¨ÀK¹j½¬µ½Ã°¡±9®¨2ÀK¹Ê[ÀK¨#Ç¯!§ÞÄ·§ºµÊe®¸zË¡§¨ÀK¹k¯ÅÆF³T°¡±¡¨§½¡®¨°¡¨¬µ¨#Ç,Ákjk ÁN¨©%³z§±¡¨¼¡½ÃÀE§±µ½¡¾¡·Ÿ½µÀK¯!¯§V¡¹G¬µºÞ¼¼¡½µÀ€ÁN§¨Þ½¡Ţ³!®¨½»k¼6¨¡¾µ¨®µÍ\¬w¨Ì²:Ço¯§¯!±µ®3¬9º¡·|»¬µ½µ¡¯!©¡µ·|»8³z½9³z§ḠÀMß@§?šx ∈ int (Ξ²µ¬¡ËV®³!¯z»tol ∩ O k )¥äÂ9³z§¡¨³!§½gÊ:°¡¨³z¬µ¯`²9µ¯!ō³!¨¨§¡¨Èo¯!¡½µ¯;¨Ģ#Âµ¯§GÊ9Â¡§¨ÌµT¡¯¼6¨§¨±¡¨©Þ¨§3Á¼¡±¡·§¨©V¨Ä¬:³z§½Ì®R n ¬µ½¡Ä¨Írad b k − mid b k + ∑ nj=1 a kjx j = 0 = Ú9â>´ ß , ªµÍÜÛE›rad b k − mid b k + ∑ nj=1 a kjx j = 0 = Ú9â>´ ß , ªµÍÜEª@›¬µ½¡Ä¨rad b k + mid b k − ∑ nj=1 a kjx j = 0 = Ú9â>´ ß , ªµÍÜ2æ^›¬µ½¡Ä¨rad b k + mid b k − ∑ nj=1 a kjx j = 0 = Ú9â>´ ß , ªµÍÜí@›¬µ½¡Ä¨§¯!±µ®3¬:¡Í¡¥¨ĢÀK¨#Ç,µ·m¬µ½V°¡¨G²9¨Ä¡·¯Ý±µ®¯!µ³!§®&¥®AÂ¡ÆGÀÒÀK¨#Ç,µ¨k¹6Ä¯z²9½¡§º2³ÏËQ°¡¨2³`¬µ¯Ì±µ³!°¡½µ³!·®µ½ËQ¨°¡±¡¯z²9¯z¬µ¯¡½ËÑÀK¨G²9¹G¬¡ËÉ½µÁH½µÐlÐw¯!±¡¯¡¾¡½¡±µ¹Ëw®·±µÇo¯!µ½Ë ª¡Í4ÜÛ@›à ª¡Í4Üí-›:°¡¨x jÊj = 1, 2, . . . , n Ê!¬µ¯!ª¼6¨¹6®½6²9¯§ºµÊÂ¡§¨l½»°¡±¡¨½µ¸!®¨G²:¡·¯|±µ®¡·¿¼¼2¨2ÀK¹2Á¬µ½µÄ¨Ý½¡¸¼6¨¡¾¡¨®w½¡¡§¯±¡®¬µ¨®¡·¥ÀQ¸!µ¼6¨À|Íåw¨¬µ¹6Âµ¯§³ÏËgÊ3Â¡§¨;¦§½°¡±¡¨½¡¸!®¨G²9¡·¯¥ÀK¨ª!¹6§w¨G²9¡¨®±¡¯ÀK¯¡¡¨lĢ3Á¡¹G¬µ½¡§º2³ÏËm¬µ½¡ÈoºÉ®¿³z¬µ¹6Âµ¯Ê¼6¨ªÏ²:%¼Ço²:·©Õ½¡ķ½µ¡§¯!±µ®3¬9º¡·|»«¦z¬µḠÀK¯!¡§¨® ª¡Í4ÜÛ@›à ª¡Í4Üí-›|¡¯;±µ®¡·ã¡¹G¬µØoÊ9Ḡ³`¬µ½rad b§¨ªÏ²´®·±µÇo¯¡½Ëk ≠ ±mid bÊkb§#Í¯Í[¯³z¬µ½É¼6¨µ¾¡¨À ¡¯Ë¡®¬¡Ë¡¯!§³ÏË%¡¹G¬µº9Í Ý §¨k½Q²9¨¼¸·®¯!§8¹2§®¯!±2Ç²9¯¡½¡¯iåw±¡¯`²9¬µ¨#Ço¯¡½ËgÍ¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76 ‡L6 4ÇœcgR:Up²4³w´(y; A, b) > 0“y ∈ int Ξ tol (A, b) ≠ ∅˜Ws`: R n →ÀK¡¨#ÇoḠ³z§®¨;5,=@2%A?+@B0C:DEA%F:,G6ˆ°[¼m¼6¼«¨§¨Ä±µÇo¯!µ½¡¯²%³/´RX = { x ∈ R n | (x) > 0Ë¡®¬¡Ë¡¯§³ÏË8¨§¼¡±¡·§·¥À|Í Î ¡¨,§3¼Ç¯¡¯!°µ¹³z§¨µÊ°¡¨³!¼2¨¬µº¼µ¹}y ∈ X ⊆ Ξ tol (A, b) X ⊆ int Ξ ¡¯°¡±¡¯!±µ·®¡¨9Êtol (A, b) Ê|°¡±¡½µÂ¡ÆÀ Í²4³w´é¬µ¯`²:¨®§¯z¬9º¡¨9Êy ∈ int Ξ tol (A, b) ≠ ∅ ÍÀK¯!§¨G²:½¡¼6¯½µ³!³z¬µ¯`²9¨®¡½Ë±µ¸!±¡¯È½µÀK¨2³z§½¬µ½µ¡¯!©¡µ¨©,¸3²:#Â¡½¨D²:¨°¡¹2³z¼3»]Ê#§#Í¯Íåw¨G²9·§¨#Ç½¡®ËQ°µ±¡¨®¯`²9Æ¡¡·¯±µ³!³!ÀK¨§±¡¯¡½ËgÊ€°¡±¡½»2¨G²9½µÀâ¼Ñ³`¬9¯`²9¹6ØDÅo¯!©¼Ã¼¡±¡½¡§¯!±¡½¡Ø‰°¡¹2³z§¨§·á#¡¯°¡¹³!§¨§· ²:¨°¡¹2³z§½µÀK¨ª!¨kÀKµ¨#Ço¯³!§®V±µ¯!È¯¡½¡©É½µÁ

é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýî6ï9ðñ2ðÆÁ2ó¨±K¯!ÈḠÀjĢ#²:#Âµ¹Ä¯¸!¹2³`¬µ¨®¡¨©ÀE¼6³!½µÀK½¡Ģ¾¡½¡½®¨ª¡¹6§¨ªẅÐw¹6¡¼µ¾¡½¡¨93ÁT =åw¹2³z§º ²4³w´max x∈R nτ ∈ R n Í=°[¨ªÏ²:¾¡½µ¨µ¬µ¨ÀÑ®µ¯!¼6¨§¨±¡¨©8§¨Â¡¼6¯• T ≥ 0 τ ∈ Ξ tol (A, b) ≠ ∅ Ê§GÍ¯ÍY¬µ½¡¡¯©¡µ3Ëb¸3²:#Âµj¨²9¨°µ¹³z¼»V²µ¬¡Ë%½¡¡§¯±¡®¬µº¡¨©k¬9½¡¡¯!©µ¡¨©Ã³z½9³z§ḠÀK·Ax = b ³z¨®ÁÊE§¨ ¯³z¬µ½τÀK¯³!§µ,½V§¨Âµ¼6• T > 0 τ ∈ int Ξ tol (A, b) ≠ ∅ Ê½V°¡±¡½¡µ3²µ¬µ¯!Ç,µ¨³z§º§¨ÂÁ¬µ¯zÇ½¡§®o²9¨°¡¹³!§½µÀK¨2ÀÑÀK¡¨#Ç¯³z§®¯w±¡¯!Èo¯!¡½µ©BÕτ ²9¨°¡¹2³z§½9ÀK¨ÀK¹ÃÀK¡¨#ÇoḠ³z§®¹%±¡¯!Èo¯!¡½¡©¿¹³!§¨©¡Âµ½¡®Þ¼¿ÀE3¬µ·¥ÀÊ¡§¨ ¼¡½ ¯³z¬µ½¦ ®¨¸ÀK¹6Å¯¡½ËµÀ%²´¡¡·|»• T < 0 Ξ tol (A, b) = ∅ ÊG§#Í¯Í!¬9½¡¡¯!©µµ3Ë;Ģ#²:#Âµ|¨²9¨°¡¹2³z¼3»Ê!§¨ ¯³z¬µ½ßE§2Ü²4³w´(x; A, b) =min1≤i≤m⎧⎨⎩ rad b i −∣ mid b i −∣⎫ n∑ ∣∣∣∣ ⎬a ij x j⎭ .j=1¬9(x; A, b) ½¦!§¨;¸!µ#Âµ¯!¡½¡¯²:¨³z§½¡ªz¯§³ÏË,Ðw¹6¡¼¡ÁÀE§±¡½µ¾¡·ä½V°¡±9®¨©FÂµ³z§½Þ³!½µ³z§ḠÀK·&Õ²µ¬¡ËV½¡¡§¯±¡®¬µºµ¨©Ì¬µ½¡¡¯©¡¡¨©V³z½9³z§ḠÀK·Ax = b µ¯³z¨®ÀKḠ³z§µµÍÎ §ÀK¯!§½9À|Ê©Â¡§¨Ñåw±¡¯`²9¬µ¨#Ço¯¡½ËjªµÍá9Í¤šzà”ªµÍá9ÍÛ¡ÊK%§3¼Ço¯Þ°¡±µ½¡®¯z²9Æ!¡93Ë«®·|Á¨³!§#¡¹6§³ÏËm³z°¡±µ®¯z²µ¬µ½¡®·¥ÀK½gÊY¯³z¬µ½m¨°µ±¡¯`²9¯!¬µ½¡§ºÑ±µ³!°¡¨¸µØqÅ½¡©mÐw¹6¡¼¡¾¡½µ¨ÁÈ¯lÀK¯!§¨G²9½¡¼½µ³!³z¬µ¯`²9¨®¡½ËF±µ¸!±¡¯È½µÀK¨2³z§½Ì¬µ½¡µ¯!©¡¡¨©V¸#²´#Â¡½Fö²9¨°¡¹³!¼6»µ3¬²4³wẃÄ¨¬µ¯!¯w¨ÄÅ½µÀÑ®·±µÇo¯¡½¡¯À|Ê¼¼²4³w´(x; A, b) :=min1≤i≤m⎧ ⎛⎨⎩ s ⎝ i rad b i −∣ mid b i −∣⎞⎫n∑ ∣∣∣∣ ⎬a ij x j⎠⎭ ,j=1ªÏ²9¯s iÊi = 1, 2, . . . , m Ê ¦ ¼¼¡½¡¯zÁN§¨;°µ¨¬µ¨#Ç½¡§¯!¬µºµ·¯DÂ¡½µ³z¬9¡Í”¬w½Ç¯qÀK·Õ¹6®½¡Á²9½µÀÞ°µ±¡½µÀK¯!±µ·É§¨ª¨µÊ#¼¼;°µ¨G²9¨Ä¡¨ª¨®½6²:qÐw¹6¡¼¡¾¡½¡¨µ3¬9·Ã°¬µ¨G²9¨§®¨±¡¡¨¥°¡±µ½ÁÀK¯!Ë¡Øq§³ÏËV®±9¸z¬9½¡Â¡¡·|»V³z½¡§¹2¾¡½¡Ë»´Í¥mµ³!§¨#Ë¡Å¯¯q®±¡¯À€Ë8ÀE¼³z½µÀK½µ¸¾µ½ËÌ¡¯ª*¬9#²:¼¡½»T®¨ª!¡¹6§·|»FÐw¹6¡¼µ¾¡½¡¨93Á¬µ¨®,Ë¡®¬¡Ë¡¯!§³ÏË8¡¯°¬µ¨#»2¨±9¸!±9Ä¨§3¡¡·¥ÀÑ®¨°¡±¡¨³!¨ÀÑ®·Â¡½µ³z¬µ½¡§¯!¬µºµ¨©V¨°¡§½ÁÀK½¡¸¾¡½¡½gÍlÞ]°µ¨³`¬9¯`²9¡½µĒ²9Ḡ³ÏË¡§½¬µ¯§½Ë°¡±µ¨È¯z²9È¯ª!¨¥®¯!¼Ä·|¬µ¨¥°¡±¡¯z²µ¬µ¨#Ç¯!¡¨¡¯ÀE¬µ¨¥¦GÐlÐw¯!¼¡§½µ®¡·|»ÝÂ¡½µ³`¬9¯!¡¡·|»;ÀK¯§¨G²9¨®±µ¯!È¯¡½Ë;¦§¨©lĢ#²:#Â¡½ ³!À|ÍÊµÁ°¡±¡½µÀK¯±gÊÀK¨¡¨ª!±µÐw½¡½º¨áµÊnšw§Ê:šGÜÊ:šuŠÊÜ^§Ê¡ÜÛb«´½F¹2¼¸µ·¯¥§3ÀÉ³³z·|¬µ¼¡½:›`Í Ý §¨±¡½¡©A±9¸!±µ¯!È½9ÀK¨³z§½A¬µ½¡µ¯!©¡¡¨©A¸3²:#Â¡½Þ¨T²9¨°¡¹2³z¼3»F²9¯!©9³z§®½¡§¯z¬9º¡¨Ì®°¡¨¬µ¡¯²:Æ§,¨³zµ¨®¡½¡¯Ý°¡¨¬:ªz§ºµÊ6Â¡§¨±µ¸!®½¡§·©VµÀK½V®,¦!§¨À¿°µ±µª!±µÐw¯q¼µ±¡½¡§¯!Á§¨Âµ¡¨¯¸µ#Â¡¯!µ½¡¯¥ÀE¼³z½9ÀK¹6ÀE;®¨ª¡¹6§·|»Ðw¹6¡¼µ¾¡½¡¨93¬µ¨®w³ÀKµ¨ª!¨ª!±µ¡¡·¥ÀK½°¡±µ¼¡§½¡Â¡¯gÍµ¥mÂ9³z§¡¨³!§½gÊ®¨ÜÛu«[¨°¡½µ³!¡·Õ¡¯¼6¨§¨±¡·¯lÀK¯!§¨G²9·lÊ93»¨G²9Ë¡Å½¡¯

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¨§®¯!§·äµ,¦!§½8½Ì²:Ço¯wÄ¨¬µ¯!¯w³`¬9¨#Ç,µ·¯Ý®¨°¡±µ¨³z·lÍÅÇËÈÍÍ{Ø;_`{Õ6a"ÆÈØe_{Ê0ËÈÍ{Ä b{ÍÄÌÆµËÈÏ Ü{Ä ÙnÊ0Õ×ÁÃÀ0ÄÁ2ó§ò2óÖ õ ê¤ê é ‚?„¤ë ® ï¤ë¤è é ú è õ úƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýß@§Û§¯z¬9¯³zµ¨ª!¨VÄ±¡¹³Ì®¡¹6§±¡¯¡¡¯!©%¨¾¡¯!¡¼µ½k²:¨°¡¹2³z§½µÀK¨ª!¨ÞÀK¡¨#Ç¯³z§®8±¡¯!Èo¯!¡½µ©gÍÙw±¡¨ÀK¯Ý§¨ª!¨µÊ6µÈooÀK¯!§¨G²9½µ¼6I#±µ³z°¡¨¸!µØDÅo¯!ª!¨Ðw¹6¡¼¡¾¡½¡¨µ3¬:J²9¨°¡¹³!¼¯!§²:3¬9º¡¯©¡È¯¯T±µ¸®½¡§½¡¯Ê€°¡¨¸®¨¬¡Ë¡ØDÅo¯!¯T¨¾¡¯!¡½µ®§ºk³z§¯°¡¯!µºk±µ¸!±¡¯È½µÀK¨2³z§½¯²:¡¡·¯½¬µ½b¡¯!±9¸!±µ¯!È½9ÀK¨³z§½Ñ¸3²:#Â¡½¿Ä²9¨°¡¹³!¼3»Ã½Ñ¼6¨±µ±¡¯!¼¡§½¡±¡¨®2§ºQȳ®Ìµ¹Ç¡¨2ÀmµÀ×³!ÀK·¥³z¬µ¯Í]åw¨V³z¹6§½gÊ]Âµ½µ³`¬µ¨®¨¯¸!9#Â¡¯!¡½µ¯ÀE¼6³!½µÀK¹ÀEF±9³z°µ¨Á¸!µØDÅo¯!ª!¨ÉÐw¹6¡¼µ¾¡½¡¨93¬9jË¡®¬µË¡¯!§³ÏËm®Ç¡¨©¿»2±µ¼¡§¯!±¡½µ³!§½¡¼6¨©«¬9½¡¡¯!©µ¡¨©¸3²:#Â¡½m¨Ã²:¨°¡¹2³z¼3»]Ê°µ¨¸!®¨¬¡ËËÕ±µ¯!Èo§º¿ÀKµ¨ª!½µ¯k§¨¡¼¡½¡¯Q®¨°¡±¡¨³!·lÍ°e¼¡½»®¨¸ÀK¨#Ç¡¨³!§¯!©Ì°¡¨G²µ»2¨G²Vè,Í%±K¨µ§3¼¡Ço¯w¡¯l°¡±¡¯z²9¨³!§3®¬µË¡¯!§#ÍÏÿ Ó §©§(©©º'¸()()¾¥Ð ¥0¢©¾ Ð ©¿¨µ†¥3¼ÃSåw±¡¯`²´³z§3®½µÀÑ³!¯!Ä¯w±¡Ḡ3¬µº¡·©8°¡±¡¨¾µ¯³!³Ý±¡¯È¯¡½ËV¼6¼6¨©ÁN¡½¡Ä¹!²:ºo¼6¨µ¼¡±¡¯!§¡¨©°¡±µ¼¡§½¡Â¡Ḡ³z¼6¨©¸#²´#Â¡½gÍåw¨Â¡§½;®2³z¯ªÏ²:|¨;µ¯¸¼¡Â¡½µ®¯§³ÏËw°¡¨³z¬µ¯§¨ª!¨9Ê!¼¼ÀK·s°¡±¡½µÈ;¬µ½j¼k®·®¨G²9¹8¨F§¨2À|Ê:Âµ§¨Ì¦§3T¸3²:#Âµµō½µÀK¯¯!§Ì±¡¯!Èo¯!¡½¡© ¡¯±µ¸!Á±¡¯!Èo½µÀE@›zÍ7Þ]¼¸Â¡½¡¼¡¹Êµ¼¼k°¡±µ®½¬µ¨9Ê9½¡µ§¯!±¡Ḡ³z¡¨Ì°¡¨¬9¹2Âµ½¡§º8½V²:3¬9º¡¯©¡È¹6Ø½¡µÐw¨±µÀE¾¡½µØâ¨,§¨2À|Ê®¯!¬µ¯!¡½¡Ë»Ã²:µ¡·|»Ñ¸3²:#Â¡½b½gÊE®¨Á1®§¨±µ·|»´ÊK¡¯Þª¨G²9½¡§³ÏËb®%¼6#Âµ¯³z§®¯8¾¡¯!¡§±µµ³z¼6¨¬9º¼6¨µÊ2®,¼2¨¬µ½¡Â¡Ḡ³z§®¯!¡µ¨ÀÉ½µ¸ÀK¯!±µ¯!¡½¡½[Êµ¡¯±µ¸±¡¯!Èo½µÀE±9³!³ÀE§±µ½¡®¯ÀE3ËÌ¸#²´#Âµ¡Ê¼¼V³`¬9¯`²9¹¯!§½¡¸ÀK¯¡½¡§ºT®»¨G²9µ·¯¥²:¡¡·¯w¸3²:#Â¡½gÊÂµ§¨Ä·¨µ,³z§33¬:o±9¸!±µ¯!È½9ÀK¨©gÊ½V§#Í²€ÍÄ¹¯!§³ÏËw¹6¼6¸§ºª!±µ¡½¡¾¡·%®2±¡½µ¾¡½¡©l®»¨G²9¡·|»¥²:¡µ·|»´ÊG®¥°¡±µ¯`²9¯!¬93»w¼6¨§¨±¡·|»¬l°¡±µ¨§½¡®9Êz¯³z¬µ½l½µ³Ï»2¨G²9µ3ËÝ¸3²:#Âµ|¨¼¸3¬:³zº±µ¸!±¡¯È½µÀK¨©gÊ!§¨D¡¯!±¡¯z²9¼6¨¥§±¡¯!Á¸3²:#Âµw®³!ÆD¯ÅÆD¨2³z§3µ¯!§³ÏË±µ¸!±¡¯È½µÀK¨©gÍ-ýA·Õ³z°¡¨2³z¨Ä¡·¿²:§º;±µ¸®¯±¡¡¹6§·¯í³`¬9½¿ÀE§±¡½¡¾µA ½Ñ³!¯!±¡¯z²9½¡µk°¡±µ®¨©ÃÂ9³z§½ mid b °¡¨j¼¼¡½µÀ€ÁN§Ä°µ±¡½¡Â¡½ÁµÀÉ¸Ðw½µ¼6³!½¡±¡¨®µ·b½Ì¨³!§#ØD§³ÏËTµ¯!½¡ĢÀK¯!¡¡·¥ÀK½[Ê¡§¨¹2®¯z¬µ½µÂ¡¯!¡½µ¯D±µ3²9½¡¹³!¨®®³!¯z»Þ¼6¨ÀK°¡¨¡¯!¡§F®¯!¼µ§¨±µbµT¨G²9µ¹8½j§¹VÇo¯®¯!¬µ½¡Â¡½µ¡¹ K °¡±¡½µ®¨G²9½¡§#Ê:¼¼K (x; A, b) Íé¬µ¯`²:¨®§¯z¬9º¡¨9Ê¬µ¯!ª¼6¨®½6²9¯§ºµÊµ¼8²9¨Ä2®¬9¯!¡½¡Øã¼6¨9³z§3¡§·max¼AÐw¹6¡¼¡¾µ½¡¨µ¬µ¹W²4³w´ ²4³w´(x; A, b + Ke) = K + maxx∈R nx∈R n²4³w´(x; A, b),

î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýßE§@ªe = ([−1, 1], . . . , [−1, 1]) ⊤ Íí³`¬µ½V¬9½¡¡¯!©µµ3Ë8¸#²´#Âµö²9¨°¡¹³!¼3»T¡¯±µ¸±¡¯!Èo½µÀE½ªÏ²9¯maxx∈R n²4³w´(x; A, b) = T < 0,ÀK· ÀK¨#Ç¯Àu³ ²9¯!¬9§º¿¯ÆQ±9¸!±µ¯!È½9ÀK¨©ä³Q§¨©×Ç¯Ã³!ÀK¨©äÀE§±¡½¡¾¡¯©A °¡¨Á³z±¡¯z²:³z§®¨Àb±µ³zÈo½¡±¡¯!µ½Ë8®¯!¼¡§¨±µ,°¡±9®¨©VÂµ³!§½ÞµKe Ê K ≥ 0 Í9åw±¡½V¦!§¨ÀÀK¡¨#ÇoḠ³z§®¨¸!µ#Âµ¯!¡½¡©j±¡ª¹6ÀK¯¡§3¡Ê¡²9¨³!§#®¬¡Ë¡ØqÅ½»VÀE¼³z½9ÀK¹6ÀmÐw¹6¡¼µ¾¡½¡¨93Áτ ∈ Arg max (x; A, b) Ä¹²9¹6§q9®¯±¡ËÁ¼Ì°µ±¡½¡µ3²µ¬µ¯zÇ,§ºÞµ¯!°¡¹2³z§¨2ÀK¹V²9¨°µ¹³z§½µÀK¨ÀK¹QÀK¡¨#ÇoḠ³z§®¹A±¡¯!Èo¯!¡½µ©Q½µ¡§¯!±¡Á²4³w´ ¬µ¹²4³w´¡¯|½¡¸ÀK¯¡½¡§³ÏËgÊ3µ¨Ý§¨Â¡¼¡½®¬µº¡¨©T¬µ½µ¡¯!©¡µ¨©Þ³z½µ³!§¯ÀK·Ax = b + Ke.¬l¨Ä¨±µ¨§#ÊḠ³`¬µ½maxx∈R n²4³w´(x; A, b) = T ≥ 0,§3¼Â¡§¨;¬µ½¡¡¯©¡µ3ËĢ#²:#ÂµÝ¨Ý²9¨°¡¹³!¼3»o±9¸!±µ¯!È½9ÀE¡Ê§¨¨µw¨2³z§3¡¯§³ÏË,§#¼6¨Á¹ÀK¯!µºÈ½9Àm±µ#²:½¡¹³!·s®³!¯z»Þ¼6¨ÀK°¡¨¡¯!¡§8®¯¼Á§¨±9o°µ±µ®¨©8Âµ³!§½Ì9,®¯z¬9½¡Â¡½¡¡¹®¨©V²:3Çō°µ¨³`¬9ō§¨ª!¨9Ê:¼¼kÀK·K K ≤ T Í¬w¯!±¡¯z²9¼6¨l°¡¨G²9¨Ä¡¨¯|±µ®¡¨2ÀK¯!±¡µ¨¯¹6È½¡±µ¯!¡½¡¯D®³z¯!»o¼6¨ÀK°µ¨¡¯¡§l®¯¼¡§¨±9bÊ°¡±¡¯z²9°¡¨¬:ªz§ºµÊ6Â¡§¨µÀÉĢ#²:8ÀE³zÈ§3Ä½¡±¡¹6ØqÅ½¡©8®¯¼¡§¨±ÀK¨#Ço¯§o¨¼6¸§º³ÏËT¡¯°¡±¡½¡ḠÀ€¬µ¯ÀK·¥À¿µ°¡±µ¼¡§½¡¼6¯Í¡åw¨¦§¨©8°¡±¡½µÂ¡½¡¡¯;³!§#¡¯Àv = (v 1 , v 2 , . . . , v m ), v i ≥ 0,b§3¼6¨©Â¡§ö¹6®¯!¬µ½¡Â¡¯¡½¡¯qÈ½¡±¡½µ¡· ®o°¡±¡¨¾µ¯³!³!¯D¼6¨±µ±¡¯!¼¡¾µ½¡½T¸3²:#Â¡½²9¨¬¡Ç¡ïvÄ·§º,°¡±µ¨°¡¨±¡¾¡½¡¨µ3¬µº¡·¥À Íµ¥·Â¡½µ³z¬µ½µÀ¿§¯!°¡¯±¡ºiT v = maxx∈R n²%³/´v(x; A, b),ªÏ²9¯v(x) =²4³w´min1≤i≤m⎧ ⎛⎨⎩ v i −1 ⎝ rad b i −∣ mid b i −∣⎞⎫n∑ ∣∣∣∣ ⎬a ij x j⎠⎭ .j=1 ªµÍÜEŠ@›í³`¬9½m¬µ½¡¡¯©¡µË«Ģ#²:#Â9j¨Ã²9¨°¡¹2³z¼3»b³AÀE§±µ½¡¾¡¯!©A ½m®¯¼¡§¨±µ¨À£°¡±µ®¨©³z§½8³z§3µ¨®½µ§³ÏË±µ¸±¡¯!Èo½µÀK¨©T°µ±¡½K ≥ | T v | Íåw¨G²9Ä½¡±µ3ËTÀE³zÈ§3Ä½¡±¡¹6ØqÅ½¡¯Âµ³!§½b °µ¯!±¡®¨µ#Âµ¬µºµÄ¡¯8½µÀK¯z¬9j±¡¯È¯!µ½¡©gÊE§¨Q¡¨®23ËÉĢ#²:#Âµj³F§¨©¿Ço¯A (bÀE§±µ½¡¾¡¯!©i + Kv i [−1, 1]) i=1 m½Ã¹6Èo½¡±¡¯!µ¡·¥À×®¯!¼¡§¨±¡¨2À ®Þ°¡±µ®¨©jÂµÁ

¬l½¡Ä¨¬µ¯¯T®Ç,µ·©ÑÂµ³z§¡·©b³`¬µ¹6Âµ©Ñ±µ³!³ÀK¨§±¡¯¡¡¨©Ñ¼6¨µ³!§±¡¹6¼¡¾¡½µ½ ¦ÀK¯³!§µJU“UÁ2ó§ò2óÖ õ ê¤ê é ‚?„¤ë ® ï¤ë¤è é ú è õ úƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýß@§2æÀK¡¨#Ç½¡§¯!¬µ½v iµ¹Ç¡·|»%µÀä¼6¨2ÀK°¡¨¡¯¡§Þ°¡±µ®¨©QÂ9³z§½Ã¨Â¡¯¡ºAÀE3¬µ¯¡º¼µ½ÁÀK½gÊ[ÀK¨#Ç¡¨Ì²9¨Ä½¡§º2³ÏËk§¨ª!¨µÊ]Âµ§¨Ä· ¨µ½j°¡±9¼¡§½µÂ¡¯³!¼¡½%¡¯½¡¸ÀK¯Ë¬µ½µ³!ºV°¡±¡½¼6¨±¡±¡¯¼¡¾¡½¡½[Ê¡o¹6Èo½¡±2Ë¬µ½µ³zºÄ·ä§¨¬9º¼6ö§¯l¼2¨2ÀK°¡¨µ¯!¡§·ä®¯¼¡§¨±9b Ê¼6¨§¨±¡·¯½µÀK¯!Øq§²9¨°µ¨¬µ¡½µ§¯z¬µº¡·¯w¼Ìµ½µÀb¡¨ÀK¯±µ¡Í²4³w´(x) =min1≤i≤m⎧ ⎛⎨⎩ | b i| −1 ⎝ rad b i −∣ mid b i −∣⎞⎫n∑ ∣∣∣∣ ⎬a ij x j⎠⎭j=1¦!§¨8¨Ä¯³!°¡¯!Â¡¯¡½¡¯;¨G²:½¡µ¼6¨®·|»V¨§¡¨2³z½¡§¯z¬µº¡·|» °¡±¡¨°¡¨±µ¾¡½¡¨93¬µº¡·|»AÄ³!¨Á¬µØD§¡·¥À%¸!µ#Âµ¯!¡½ËµÀÈ›€¹6®¯z¬µ½¡Âµ¯!¡½¡©,±µ3²9½¡¹2³z¨®w¼6¨ÀK°µ¨¡¯¡§l°¡±µ®¨©Âµ³z§½gÊ¼6¨ÁªÏ²:v i = | b i | ²µ¬µË8¡¯¡¹G¬µ¯!®·|» b iÊi = 1, 2, . . . , m Í Î Ä¨¸µ#Â¡½µÀ½V°¡¹³!§ºT = maxx∈R n²%³/´0(x; A, b).¥¯z¬µ½¡Âµ½¡µT Ë¡®¬¡Ë¡¯!§³ÏË,³z¹6ÅḠ³z§®¯!¡µ¨qÄ¨¬µ¯¯©§¨¡¼6¨©o¼2¨¬µ½¡Â¡Ḡ³z§®¯!¡µ¨©;»6±µ¼Á®8Ú!Ú*Û¡ÍÛ½ºª¡Íß§¯!±µ½µ³z§½µ¼2¨©Ã³z¨®2ÀK¯³!§¡¨³!§½j¬µ½¡¡¯©¡¡¨©Q¸3²:#Â¡½Q¨Ì²9¨°¡¹2³z¼3»]Ê:Â¡¯Àä®®¯`²9Æ¡µËM =min1≤i≤m⎧⎨⎩ χ(b i) − max1≤j≤na ij≠0⎫⎬χ(a ij )⎭ ≥ 0.é©¹²µËT°¡öÀK¨G²:¹¬9ØT Ê6ÀK¨#Ç¡¨®¯³zº2ÀEl§¨Â¡¡¨¨°¡±¡¯z²9¯z¬µ½µ§ºoÀK¯±¡¹,¡¯³!¨®ÀKḠ³z§µ¨Á³z§½ÃĢ#²:#Âµ½Q®A³z¬µ¹6Âµ¯T < 0 ½¬µ½gÊeµ¨Ä¨±¡¨§#Ê]§¨9ÊI#µ³z¼6¨¬9º¼6¨V¸#²´#ÂµV³z¨®Á¸°µ³;¹³z§¨©¡Â¡½µ®¨³!§½Þ±µ¸!±¡¯È½µÀK¨ª!¨F³!¨³z§¨#Ë¡¡½Ë/›|®F³z¬µ¹6Âµ¯T ≥ 0 Íí³z§¯³z§®¯!µ¡¨µÊ¡®2³zÆw¦!§¨²:¨³z§½¡ªz¯§³ÏËÌ¾¡¯¡¨©F²9¨°¡¨¬µµ½¡§¯z¬9º¡·|»8®·Â¡½µ³z¬µ¯!µ½¡©gÍÅËÈÍ{ÍØ;_6`Õ0a¦ÓºÄÌÊ0ÍÕ6`{Î ÙnÕÙnÊ0Ø7ÓºÎ×ÁÃÀ]\ýÞ·u°¡±µ¨G²9¯ÀK¨µ³z§±µ½¡±¡¨®23¬µ½¿®¨¸ÀK¨#Ç¡¨³!§½É¼6¨±µ±¡¯!¼¡¾µ½¡½É¬µ½µ¡¯!©¡µ¨©b¸#²´#Â¡½É¨b Í[é©¯!©¡Â9³ÀK·Ã¨Ä2³z¹²9½µÀ|Ê¼6¼l¸3²:#Â¡¹D¨²9¨°¡¹2³z¼3»wÀK¨#Ç¡¨¥¼6¨±¡±µ¯!¼¡§½µ±¡¨®2§º°¡¨2³z±¡¯z²:³z§®¨À²9¨°µ¹³z¼»8°¡¹6§ÆGÀäÀK¨G²:½µÐw½¡¼¾¡½µ½A¨G²9¡¨ª!¨F®¯!¼¡§¨±µÌ°¡±µ®¨©kÂµ³!§½s, x ∈ IR“A Íåw±¡¯`²´³z§33¬¡Ë¡ḠÀE3Ëb¡½Ço¯ÞÀK¯§¨G²9½¡¼½µÀK¯!¯§³z®¨¯!©8¨³!¡¨®¨©®±¡º½¡±µ¨®¡½ËÉ¦!¬µ¯ÀK¯¡§¨®ÃÀE§±¡½¡¾¡·4\6Ã8jþri@c£Wr_ YnU=t=‹—f ikY?]^[lV/`b[EXe~¡XeV=V/h a¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76 mrad x ≥ rad (x ⊖ s) Tn[wR%TnXeW¡cŸT m Y?]^[lU”R:_EV=hLfúU=V=W¡XZYn|kob] s˜!n=`¤|x ⊖ s| = |x| − |s|˜s ‘U=V/WsX—Yn[^]£R[^]£R:`pf

î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýx + s ≤ 0 x ≤ 0 Ê³z¨¬µØD§µ¨ª!¨F¸!µ#Â¡¯¡½ËÞ¨§F¨Ä¯½»VÂµ³!§¯!©gÍ´èwµ#Â¡¯Ê¡¯³z¬µ½|x| = −x Ê:§¨ Ê½8ÀK·Õ³z¡¨®l½9ÀK¯!¯ÀÉ®o³z½¡¬µ¹½¡¸!®¯³!§¡¨ª¨;¡¯±µ®¯!µ³!§®Ý²9¬¡ËÌÀK¨G²9¹G¬¡Ë²%³/´(x; A, b)åw±¡½µÀK¯Àb³`¬9¯`²9¹6ØDÅo½¡¯l¯³!§¯³!§®¯¡¡·¯l°¡±µ¯`²9°¡¨¬µ¨#Ço¯¡½ËMßE§íÎ Ä¨¸!µ#Âµ½µÀ%²µ¬µËV¼µ±µ§¼6¨2³z§½s := |s| = rad s Í9åw±¡¯z²9°¡¨Á¬µ¨#Ç½µÀÉ§3¼Ço¯Ý²µ¬µËÌ¨°µ±¡¯`²9¯!¬µ¯!µ¡¨³!§½gÊ¡Â¡§¨;5,=@2%A?+@B0C:DEA%F:,G6|x| = max {|x|, |x|} = |x|,|x| ≥ |x| x ≥ 0 Íåw±¡½Þ³ ²9¯z¬9¡¡·|»F9ÀK½8°¡±¡¯`²:°¡¨¬µ¨#Ç¯!¡½¡Ë»F¨¼¸·®¯!§³ÏËgÊ2Â¡§¨§#Í¯Í|x − s| ≥ |x + Íµ¥×Âµ³!§¡¨³!§½gÊ¦§¨¨¸µ#Âµ¯§ s|. ª¡ÍßE§-›rad x = x − x2≥ s,H¯!©µ³!§®½µ§¯z¬µº¡¨µÊ¹6§®¯!±2Ç²9¯!µ½¡¯låw±¡¯`²µ¬9¨#Ço¯!µ½ËÌ®¬µ¯!Â¡Æ§í³`¬9½x + s ≥ 0x − s ≥ x + s.§3¼8Â¡§¨Ê´§¨Þ¡¯±µ®¯!µ³!§®¨ ªµÍß@§@›D°¡¨¬µ¹6Âµ¯!§³ÏËQ¨§³!Ø²:Ì®¸zË¡§½¡¯ÀâÄÁ±µ¸¡¨³!§½|x − s| ≥ |x| − |s| ≥ |x| − s = −x − s = |x + s|.¬l¼6¨¡¯¾gÊ¡³Ý¹6Â¡Æ!§¨2À ª¡ÍßE§-›|x ⊖ s| = max {|x − s|, |x + s|} = |x − s| = x − s = |x| − |s|,|x| = |x| ²:¨¼Ģ§¯!¬µº³!§®ö°¡±¡¨®¨G²9½¡§³ÏËFµ3¬9¨ª!½µÂÁ¡·¥À¿¨Ä±9¸!¨2À|Í¼¼F½8§±¡¯Ä¨®3¬9¨³zº9ÍlH;¬¡Ë°¡¹³!¼3»Þ³;½¡¡§¯!±¡®23¬µº¡¨©AÀE§±¡½¡¾¡¯©åw±¡¯`²9°µ¨¬µ¨#Ç½µÀ|ÊKÂ¡§¨%µÀâ²:µkµ¯³z¨®ÀKḠ³z§µËÉ¬µ½¡µ¯!©¡µËÑ¸3²:#ÂµkÄ²9¨ÁA ½A½¡¡§¯±¡®¬µºµ·¥À«®¯!¼µ§¨±¡¨2À«°¡±µ®¨©Âµ³!§½b Í{°[¨ªÏ²:bÄ¯!¸!¹2³`¬µ¨®¡·©äÀE¼³z½µÀK¹À¤±µ³!°¡¨¸µØqÅ¯ª!¨¿Ðw¹6¡¼µ¾¡½¡¨93¬9τ ∈ R n Ê6Ë¡®¬¡Ë¡¯§³ÏË8¨§±¡½¡¾9§¯!¬µº¡·¥ÀM®,§¨Â¡¼6¯max (x; A, b) = (τ; A, b) = T < 0.²4³w´ ²4³w´x∈R nÊK¨k¼6¨§¨±¡¨2ÀãÄ¹²9¯À ³zÂ¡½¡§3§ºj½µ¸!®Ḡ³z§µ·¥À|ÊeÂ¡§¨j¨Ã²9¨2³z§½¡ªz¯!§³ÏËýÞ¨#Ç¡¨;¬9½Ì¹ÀK¯!µºÈ½µ§ºoÀK¯±¡¹T¡¯±µ¸±¡¯!Èo½µÀK¨³!§½T¬µ½µ¡¯!©¡µ¨©FĢ#²:#Âµ½T¨;²:¨°¡¹2³`ÁA½V¼¼8¼2¨¡¼¡±¡¯§¡¨,¦!§¨T³ ²9¯z¬9§º&¥¼3»F°¡¹6§ÆGÀ¿³!¹2Ço¯!µ½Ë8ÀE§±¡½¡¾¡·

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ß=šßÁ2óÁ2ójØ õ ê þ ¯ ï üâï ® êgð ƒþ é ê õ ñ ã ê ¯ ÷ ð.ê é ù é è¤ë ®? 9@&¡b?ªR GRaV@&TH*4=&2¤,¾hj*7,e?/7#14&?0$@&¦ 3CoÊ&&)@#)'=¦h4=:H?4&#150#)'=¦Ž@&&'&#)7&'&*7N’H?/$?1Ê& *$*8=:1“>?9g4?’),e#)4?Œ²:¡µ¨ª!¨t ∈ ΞH;¬¡Ëtol (A,®·Â¡½9³`¬¡Ë¡ḠÀ¢µ½¡Ä¨¬µºÈo¯!¯¡¯!¨§±¡½¡¾9§¯!¬µº¡¨¯b)ηη · Ae ⊆ b ⊖ At.Êµ§#¼6¨¯ÝÂ¡§¨(t + ηe) e = ( [−1, 1], . . . , [−1, 1] ) ⊤ Ê¯³!§ºl®¡¹6§±µ¯!¡ËË,¨¾µ¯!¡¼D²9¨°µ¹³z§½µÀK¨ª¨;ÀK¡¨#ÇoḠ³z§®2q±µ¯!È¯¡½¡©èw¡§¯±¡®¬µº¡·©Ì®¯¼¡§¨±Ξ tol (A, b) Ê ÍΞ tol (A, b)˜U = (t + re)U=V/WsX—Yn[^]£R:_@V=h aQ[^X£vEWs`ŸYW¡]lv^qxXÆ¢:XgR:U=vE`pf RnXkq U=W [^`†f¨V=`b|q XwceW¡[^XYnX£~sX£V=U=a¡¯!©µ¡¨©Ģ#²:#Â¡¯|ẅ²:¨°¡¹2³z¼3»oÀE§±¡½¡¾9;5,=@2%A?+@B0C:DEA%F:,G6€åw±µ¯`²9°¡¨¬µ¨#Ç½µÀ«³zµ#Â93¬9µÊµÂ¡§¨8®F±9³!³ÀE§±µ½¡®¯ÀK¨©V¬9½ÁA ½µÀK¯¯!§l¡¹G¬µ¯®¹6ØÕÈ½µ±¡½¡¡¹Ê§#Í¯Í®¯!Åo¯zÁr A =min1≤i≤m⎧⎨rad b i − ∣ mid b i − ∑ ∣n ∣∣j=1 a ijt j∑⎩nj=1 | a ij|⎫⎬⎭ , ªµÍßá-›³z§®¯!¡93Ë ¦ A = A ¦ ½8°¡¨§¨2ÀK¹vert A = A Í¡åw±¡¯`²´³z§3®½µÀÃ¼3Ç²9·© x ∈ U®,®½6²9¯x = t + y ÊªÏ²9¯ max 1≤j≤n | y j | ≤ r A½| (Ay) i | =∣∣n∑ ∣∣∣∣a ij y j ≤§3¼Âµ§¨¥²9¬¡Ëi = 1, 2, . . . , m ³!°¡±µ®¯`²µ¬9½¡®¥³z¬µ¯`²:¹2ØqÅoË¾¡¯!°µ¨Â¡¼¥µ¯!±µ®¯!9³z§® Mj=1n∑| a ij | | y j |j=1n∑≤ r A · | a ij |j=1≤ rad b i −∣ mid b i −∣n∑ ∣∣∣∣a ij t j .j=1

î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ý§3¼Ç¯F®¼¡¬µØDÂ¡¯Ñ®Ξ tol (A, b) ÍK¥£Âµ³z§µ¨³z§½gÊK¯³`¬9½Ñ®³!¯ U Vß?šáAx = At + Ay Ê¡ÀK·ä°µ¨¬µ¹6Âµ¯Àé¬µ¯`²:¨®§¯z¬9º¡¨9Ê2°µ¨³z¼6¨¬µº¼¡¹(At) i − rad b i + | mid b i − (At) i |≤ (Ax) i≤(At) i + rad b i − | mid b i − (At) i |½¬µ½gÊµÂ¡§¨±µ®¡¨2³z½¬µº¡¨µÊb i − ( mid b i − (At) i ) + | mid b i − (At) i |≤ (Ax) ib i − ( mid b i − (At) i ) − | mid b i − (At) i |.≤ ªµÍß2ÛE›¥Õ³z½¡¬µ¹F§¨ª!¨9ÊÂ¡§¨−z + |z| ≥ 0½− z − |z| ≤ 0b i ≤ (Ax) i ≤ b i ,zÊ¡¯!±µ®¯µ³z§®¨ ª¡ÍßÛ@›®¬µ¯!Âµ¯!§²µ¬¡ËV®³ÏË¡¼6¨ª¨®¯!ÅḠ³z§®¯!¡µ¨ª!Äx ∈ b Êµ¼6¼Ì½V¨#Ç½6²:¬µ¨³!ºµÍ åw¹³z§ºm§¯!°¡¯±¡º×ÀE§±¡½¡¾µ§¨,¯³!§ºA t ∈Ξ tol (A, b) ≠ ∅ ÍÈ±E³!³!ÀK¨§±¡½µÀŸ³!¨®¨¼¡¹6°¡¡¨2³z§ºQ®³!¯z»Ñ¬9½¡¡¯!©µ¡·|»¿Ģ#²:#Âb¨Q²9¨Á¸3²:#Â¡½ã½µÀK¯¯!§m¡¯¡¹G¬µ¯!¹6Ø£Èo½¡±¡½¡µ¹ä½V x = b V ∈ vert A Í:é©¨ª*¬:³zµ¨°µ±¡¯`²:³!§3®Á¬µ¯!µ½¡Ø ªµÍ¤šu§@›⋂³ °¡¹³!¼3»²µ¬µËÌ¬µ½µ¡¯!©¡µ·|»V³!½µ³z§ḠÀΞ tol (A, b) =V ∈vert AΞ tol (V, b),½¯³z¬µ½o²µ¬¡Ë¼3Ç²9¨ª¨V ∈ vert A ½µ¡§¯!±µ®3¬9º¡·©®¯!¼¡§¨±,±¡¯!Èo¯!¡½ËT³!¨¨§®¯!§ÁU³z§®¹6ØDÅ¯©kĢ#²:#Â¡½k¨T²9¨°¡¹³!¼3»Þ¯³!§º ÊV U V ⊆ Ξ tol (V, b)¡·©V®¯¼¡§¨±U Ê¡§3¼2¨©FÂ¡§¨ U =⋂Ê´§¨V½¡µ§¯!±¡®23¬µºÁV ∈vert AU V ªµÍßá-›*Ê¡§¨½µÀK¯!ḠÀ¾¡¯!µ§±V½8½¡»Ì±µ3²9½¡¹³!·Õ¨°µ±¡¯`²9¯!¬¡Ë¡ØD§³ÏËVÐw¨±µÀK¹G¬µ¨©U = t + re,½9ÀK¯!ØD§Q¨ÄÅ½¡©

ª!¹6±¡¡·|»%³z¼6¨Ä¼3» ªµÍß@ª@›*Íeèw¡·¥ÀK½Ã³`¬µ¨®ÀK½[ÊgÀK·£ĢÀK¯Ë¡¯Àä°¡¯!±µ¯ÀK¯!µ¡·¯a ijÁ2óÁ2ójØ õ ê þ ¯ ï üâï ® êgð ƒþ é ê õ ñ ã ê ¯ ÷ ð.ê é ù é è¤ë ®ªÏ²9¯r =min r V =V ∈vert Amin1≤i≤mminV ∈vert A⎧⎨rad b i − ∣ mid b i − ∑ ∣n ∣∣j=1 a ijt j∑⎩nj=1 | a ij|⎫⎬⎭ .ß=šGÛ°g¯¨±¡ḠÀE¡Ê¡§ḠÀ¿³ÀK·¥À|Ê6²:¨¼Ģµ°¡¨¬µµ¨³z§ºØÍ±E¡¯¯|¥ÝÍ¥ÝÍRpr =min1≤i≤mminA∈A⎧⎪⎨⎪⎩rad b i −∣ mid b i −¬w¯³!ÀK¨§±2ËÌµ®¡¯!Èo¡ØDØÒ°¡±µ¨³z§¨§¹,²:¨¼Ģ§¯!¬µº³!§®µÊ¹6§®¯!±2Ç²9¯!¡½µ¯¨°g¯!¨Á∣n∑ ∣∣∣∣a ij t j±¡¯ÀK·úªµÍ ªµÍ¤šwË¡®¬¡Ë¡¯!§³ÏËÞµ½¡Ä¨¬µ¯!¯l§¨¡¼¡½9À«½¡¸;±¡¯¸!¹G¬µº3§3§¨®°¡¨G²9¨Ä¡¨ª!¨,³z¨±µ§#µÍ©6²9¹6±¡¨®w¹2³z§3¡¨®½¬®¡¨Û¡Ê?š#æw«NÊÂ¡§¨ÝḠ³`¬µ½t ∈ Ξ tol (A, b) Ê§¨Ý²µ¬¡Ën∑| a ij |i=1j=1⎫⎪⎬⎪⎭ ªµÍß@ª@›½¡¡§¯±¡®¬µºµ·©Õ®¯¼¡§¨±(t + re) ®¼¬9ØDÂ¡Æ×® Ξ tol (A, b) Í¥Ùl¼Õ®½6²9½µÀ|Ê|¯!ª¨®¡¹6§±µ¯!¡¡½µ©8ÀK½¡µ½µÀK¹ÀÑÄ¯!±¡Æ§³ÏËÌ°¡¨[^cwXgfŸÀE§±¡½µ¾µÀÐw¨±µÀK¹G¬9¿°¡±µ¼¡§½¡Âµ¯³z¼µ½â³!¨®°9#²:¯!§¿³jµÈ¯!©ä¸b½µ³!¼¬µØDÂµ¯!¡½¡ḠÀu§¨ª!¨9ÊÂ¡§Ä ∈ A Ê¡oµ¯Ý°µ¨(v@ùV:Xet%|V=`pfiÞÀK¡¨#ÇoḠ³z§®¹V¼¡±µ©µ¡½»AÀE§±µ½¡¾gÍ Ý §¨µÊ:§ḠÀm¡¯,ÀK¯!¡¯¯Ê:¡½¡¼6¨½µÀ«¨Ä±µ¸!¨2À¡¯8®¬µ½Ë¡¯§kµj¨¼6¨¡Âµ§¯z¬µº¡·©Ã±¡¯!¸¹G¬µº3§3§#ÊK¼2¨§¨±µ·©¿³z¨®¯±¡È¯¡¡Ä½6²:¯!¡§½¡ÁÂ¡¯!8²9¬¡ËA¨Ä¯!½»VÐw¨±µÀK¹G¬[Í?H¯!©µ³!§®½¡§¯z¬µº¡¨µÊ:ÀK¨#Ç,µ¨T°¡¨¼¸§ºµÊÂ¡§¨8Ðw¹6¡¼¡¾¡½Ëi = 1, 2, . . . , m ª¡ÍßEª@›lË¡®¬¡Ë¡¯§³ÏË¿¼µ®¸½¡®¨ª!¡¹6§¨© ³À|ÍÚwš@šÍ-æ#®-›zÍÌýA½Á¡½µÀK¹ÀÒ¼¡®2¸½¡®¨ª!¡¹6§¨©%Ðw¹6¡¼µ¾¡½¡½Ã²:¨³z§½¡ªz¯§³ÏËgÊ[¼¼Ñ½¡¸!®¯³z§¡¨µÊ€®k¼¡±9©¡¡½¡»®QÐw½µª!¹6±¡¡·|»Ñ³!¼2¨Ä¼3»Ê9®·±µ3Ç¯!¡½¡ËA®FÐw½¡ª!¹6±¡µ·|»Þ³!¼6¨Ä¼» ªµÍß@ª@›|°µ±¡½¡¡½µÀEØD§V³z®¨½§¨Âµ¼6»b®·°¡¹6¼¬µ¨©m¨Ä¬:³z§½m¨°¡±¡¯z²9¯z¬9¯!¡½ËgÍ°[¼¡½µÀ

“i = 1, 2, . . . , m Ê¡½V¸§¯À¿°¡¨¬9ªzḠÀî6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ý“t ∈ Ξ tol (A, b) YnXp¡libR:_E|ß?šuª? 9@&¡b?ªR f>aV@&TH*4=&2¤,ÜE?/7=$&:=4=*3¤?6$@&¦Ž3CoÊ&&)@#)'=¦h4=:H?4&#150#)'=¦Ž@&&'&#)7&'&*7N’H?/$?1Ê& *$*8=:1“>?9g4?’),e#)4?Œt ∈ ΞH;¬¡ËT²´¡¡¨ª!¨tol (A, b)∣®·Â¡½µ³z¬¡Ë¡¯À¿½¡¡§¯!±¡®23¬µ·∣ ∣∣∣∣∣n∑rad b i −∣ mid b i − a ij t jj=1r i =,n∑| a ij |j=1ϱ :=min r i .1≤i≤m(t + ϱe) e = ( [−1, 1], . . . , [−1, 1] ) ⊤ Ê¯³!§ºl®¡¹6§±µ¯!¡ËË,¨¾µ¯!¡¼D²9¨°µ¹³z§½µÀK¨ª¨;ÀK¡¨#ÇoḠ³z§®2q±µ¯!È¯¡½¡©èw¡§¯±¡®¬µº¡·©Ì®¯¼¡§¨±Ξ tol (A, b) Ê Íåw¨³z¼6¨¬µº¼¡¹Q½ÑÂµ½µ³`¬µ½µ§¯z¬µºQ½Ñ¸µÀK¯µ§¯z¬µºQÀK½¡¡½µÀK½¡¸½¡±¡¹¯ÀK¨ª¨k®·±µ3Ç¯zÁ¡½ËT³z¨G²:¯!±2Ç§w¬µ½¡Èº°¡¨;¨G²9¡¨2ÀK¹®»¨#Ç²9¯!µ½¡ØÕ¼3Ç²9¨©½µḐ°¡¯±¡¯ÀK¯¡¡·|»a ij®°¡¯!±µ®¨©Þ³!§¯!°¡¯¡½gÊ´§¨FÀK¯§¨G²µp£©6²9¹6±¡¨®µÊµ®,²9¯©µ³z§®½¡§¯!¬µºµ¨³z§½gÊµ¦¼¡®½¡®23¬µ¯Á§¯!¨¾µ¯!¡½¡®2¡½µØ«ÀK½µ¡½µÀK¹ÀED²9±¡¨Ä½o¼¼Âµ³z§¡¨ª!¨Ý¨§wÀK½¡¡½µÀK¹ÀEwÂ¡½9³`¬µ½¡§¯z¬¡Ë¼¼;¡¯§±¡¹²9¡¨Ý°¡¨¼6¸§º£¨4Û6Ê”š#æw«NÊ#§¯Àk®·Èo¯ÊGÂ¡¯ÀAÀK¯¡ºÈo¯!©Ë¡®¬¡Ë¡¯§³ÏË;È½¡±µ½¡µµÀE¼³!½µÀK¹ÀV¸!µÀK¯!µ§¯z¬¡ËgÍ Î §¡¨³!½¡§¯!¬µºµËl§¨Â¡µ¨³z§º§#¼6¨ª!¨¨¾¡¯!µ½¡®¡½ËgÊÀE§±µ½¡¾¡·A Í¿.Ì+Eð0B0,À5+@ñÌò0+ËÇ76àÇ76Ã8 PSR%TºR:žb`ba Wsùt%v@U=| ¢:XeV=W{Y?]Ws]^Wrhm `’R:iEt?]EXgfhÈXº]£R¡ p`ŸY:U/W3f¨`’f¦r`ba-f£]la/X—Y%]c£`l[ m ]ôb]lWc¡Y\X’¡îuR:_@Ws]l|Ws]wfUd]£R¡ p`ŸY:U/W3f£]AÝ ]la/ouieY:`l[^]E˜;5,=@2%A?+@B0C:DEA%F:,G6:H¯!©µ³!§®½µ§¯z¬µº¡¨µÊ¹2³`¬µ¨®½¡¯η · Ae ⊆ b ⊖ At¨¸µ#Âµ¯§½η · (Ae) i≥ (b ⊖ At) iη · (Ae ) i ≤ ( b ⊖ At ) ,ii = 1, 2, . . . , m,

A ÍÁ2óÁ2ójØ õ ê þ ¯ ï üâï ® êgð ƒþ é ê õ ñ ã ê ¯ ÷ ð.ê é ù é è¤ë ®ªÏ²9¯Ae ¦¹6±µ®¡¨®¯!È¯¡¡·©Ì½¡¡§¯±¡®¬µº¡·©8®¯¼¡§¨±gÊ¡®,¼6¨§¨±¡¨À− (Ae) i= ( Ae ) i = | (Ae) i|, i = 1, 2, . . . , m.¥¨¬µ¯¯Ý§¨ª!¨9Ê2²µ¬¡Ët ∈ Ξ tol (A, b)ß=š#æ(b ⊖ At) i≤ 0 ≤ ( b ⊖ At ) ii¡½¡©½gÊg§3¼µ½µÀ×¨Ä±µ¸!¨2À|Ê:²µ¬¡Ë%®³ÏË¡¼6¨ª¨M{ ( b ⊖ At ®¯!±µµV³`¬µ¯z²9¹6ØDÅoËj¾¡¯°¡¨Âµ¼6V³!¨¨§¡¨Èo¯zÁ)iη ≤ min, ( b ⊖ At ) }i( Ae ) i ( Ae ) i= min { −( mid b i − rad b i ) + (At) i , ( mid b i + rad b i ) − (At) i}| (Ae) i |= min { rad b i − ( mid b i − (At) i ), rad b i − ((At) i − mid b i ) }| (Ae) i |= rad b i − max { mid b i − (At) i , (At) i − mid b i}| (Ae) i |= rad b i − | mid b i − (At) i |.| (Ae) i |i = 1, 2, . . . ,³z¨¨§®¯!§³!§®¹6ØDÅo¯!ª¨Q½µ¡§¯!±µ®3¬:H;¬¡ËmÊ°¡¨¬µ¹6ÂµḠÀK¨ª!¨Ã®É¬µª!¨±¡½¡§ÀK¯5°[Ä¬[Íª¡Íß¡Êr ii η = ϱ Í °¡±¡½¡®¨G²9½¡§¼8±9®¯µ³z§®¹°¡¨³z¬µ¯`²9µ¯!¯;®·±µ3Ç¯!¡½¡¯l³!¨®°9#²:¯!§³w¡½Ç¡¯©Aª!±µ¡½¡¾¡¯©§3¼8Â¡§¨®¸!Ë¡§½¡¯;ÀK½¡µ½µÀK¹ÀE,°¡¨,®³zḠÀÙl¼ÞÀK¯!§¨G²µp£©²9¹6±¡¨®2¡Ê§3¼Þ½AÀK¯§¨G² ¬w¨©µÀE©¡¯!±µ,°µ±¡¨³!§·ä½V¬9¯!ª!¼6¨±¡¯!Á3¬µ½µ¸!¹ḠÀK·M¯³z¬µ½m¹2Ço¯Vµ©²9¯!µ%§¨Â¡¼t∈ Ξ tol (A, b) Ê§¨Ã°¡¨³!§±¡¨¯¡½¡¯Þ½¡¡Á§¯!±µ®3¬9º¡¨ª!ẅ±¡¯!Èo¯!¡½¡Ë®¨¼¡±¡¹6ª¡¯Æ§±¡¯Ä¹¯§l®³!¯!ª!Ö(mn) ±¡½µÐlÀK¯§½¡Â¡Ḡ³z¼¡½»¨°¡¯±µ¾µ½¡©gÍ ¬w¨k¦§Ä²:¨³z§½¡ªz¯§³ÏËÉ¾¡¯!µ¨©Ñ¸µ#Â¡½¡§¯z¬µº¡¨ª¨k¨ª±¡¹6Ä¬µ¯¡½ËÉ¨¼2¨ÁÂµ§¯z¬µº¡¨ª¨j±¡¯¸!¹G¬µº3§3§3µÊ€¨2³z¨Ä¯!µ¡Ä²9¬¡Ë¿Èo½¡±¡¨¼µ½»Ñ½¡¡§¯!±¡®23¬µº¡·|»¿ÀE§±¡½¡¾åw¨2³z§±¡¨½µÀäÄ±¡¹2³o®¡¹6§±¡¯¡¡¯!©Ã¨¾¡¯!µ¼¡½j²9¨°¡¹2³z§½9ÀK¨ª!¨ÞÀK¡¨#Ç¯³!§®¿.Gò013+-. ªµÍá”›`Í[¥ Ú’ªµÍáÞÀK·ŸµÈ;¬µ½Ã§¨Â¡¼¡¹(0, 0) ⊤ ½¡¸®¡¹6§±µ¯!¡¡¨Á³z§½«¯Æ8²9¨°¡¹2³z§½9ÀK¨ª!¨ÃÀK¡¨#Ç¯³!§®j±¡¯!Èo¯!¡½µ©gÊ½«¦!§¹Ñ§¨Â¡¼¡¹¿ÀK¨#Ç¡¨%®¸zË¡§º%®±¡¯!Èo¯!¡½µ©É³z½µ³!§¯ÀK·

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î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýßÜík¡¨³!§½gÊí³`¬µ½ ¦X k ≠›`Ê¡§¨°¡¨¬µ¨#Ç½µÀX kX ′ := ( )X 1 , . . . , X k , . . . , X n ,ýÞ·ä³!§3¡ḠÀÑµ¸!·®2§º°¡¨G²9¨Ä¡·¯Ý¼6¨ÀK°¡¨¡¯!¡§·äÄ±¡¹³X U=V£¢=U/obX£V=WsV=hLfU9Í¦!§¨Ì½¡²9¯!¼³o½¡¡¾¡½6²:¯!¡§µ¨©k¼6¨2ÀK°¡¨¡¯¡§· ½j°µ¨¦!§¨ÀK¹Ê:®8Âµ³!§3ÁX ′′ := ( X 1 , . . . , X k , . . . , X n).X ′ ½X ′′ Ë¡®¬¡Ë¡Øq§³ÏË m `bWr`pfv-]£fUâ¨§X ½µ¸·®§º³!ÀK¹Ì°¡±¡¨¾¡¯`²9¹6±¡¹Ì½»Ì°µ¨±¡¨#Ç²9¯¡½Ë ogYn`bž’RnXeV/U/XgfŸ½9³Ï»¨G²9¡¨ª!¨Ä±¡¹2³!¥¹!²:¯ÀŸª!¨®¨±¡½¡§º9ÊYÂ¡§¨ÉÄ±¡¹³!·X Í=ýÞ¨¡¨§¨¡¡¨2³z§º½¡¡§¯±¡®¬µºµ¨©V±¡½µÐlÀK¯§½¡¼¡½V°¡¨,®¼¬µØDÂµ¯!¡½¡ØÒ®¬µ¯!ÂµÆ!§½V°¡¨¦§¨ÀK¹Φ(X) ≤ Φ(X ′ )½Φ(X) ≤ Φ(X ′′ ),{}Φ(X) ≤ min Φ(X ′ ), Φ(X ′′ ) .¥¿²9¯!©µ³!§®½µ§¯z¬µº¡¨³!§½gÊµo°¡±µ¼¡§½¡¼6¯w¦!§ö¡¯±µ®¯!µ³!§®öÂµÅo¯D®2³z¯!ª¨Ä¹!²:¯!§Arg Ψ(X) Arg Θ(X) Ä¹²9¹6§%°¡±2ËµÀK·¥ÀK½°¡±¡¨½¡¸!®¯`²9¯¡½ËµÀK½gÍgèÝ»k°µ¨°µ3²:¡½µō®V±µ¸!¡·¯,Ä±µ¹³z·|ÁN°¡¨§¨ÀK¼¡½j®V±¡¯!¸¹¬9º3§#§¯½ ³z§±µ¨ª!½9À|Ê§3¼«¼¼«ÀK¡¨#ÇoḠ³z§®2X Êµ°¡±µ½µÀK¯!±gÊµ§#¼6¨ª!¨9ÊÂ¡§¨±µ¸Ä½¡¯¡½ËArg Ψ(X) ⊆ X ′ Arg Θ(X) ⊆ X ′′ ,½½µÀK¯!¯§³`¬µ¯z²:³z§®½¡¯ÀΨ(X) = Ψ(X ′ ½) Θ(X) = Θ(X ′′ ),Ψ(X ′′ ) > Ψ(X)½Θ(X ′ ) < Θ(X).§¨ªÏ²ó¼¼Ψ(X) / Θ(X) < Ψ(X ′ ) / Θ(X ′ )é¬µ¯`²:¨®§¯z¬9º¡¨9Ê½Ψ(X) / Θ(X) < Ψ(X ′′ ) / Θ(X ′′ ),§#Í¯Í¡®o²9¯!©µ³!§®½µ§¯z¬µº¡¨³!§½{}Φ(X) < min Φ(X ′ ), Φ(X ′′ ) .

X ′ ½X ′′ §3¼Ço¯ÀK¨ª¹6§ÌÄ·§ºÌ±9¸!Ä½¡§· °¡¨°¡¨¬9À«²µ¬¡Ëj²:3¬µº¡¯!©µÈ¯!ª¨F¹6§¨ÂÁJÊ3¹2Ço¯½µ³!°¡¨¬µº¸!¨®µ¡¨©l®2¬9®¯ßM¨ÄÅ¯!©o³Ï»¯ÀK¨©©I3ÀK¯§¨G²:Ý®¯§®¯©½ª!±µ¡½¡¾Φ(P ) P ⊆ X Ê ¦ ÊP ∈ IR n P ⊆ X L Ģ°¡½µ³!ºÃ³z°¡½µ³!¼ Ê½9ÀK¯!ØDÅo¹6Øµ½9ÀK¯!¡ºÈ¯!¯¸µ#Â¡¯¡½¡¯¨¾µ¯!¡¼¡½ÊΦ(P )ÀE¡Ê:ÀK·âÄ¹²9¯À¿µ¸·®§ºd[^X£oui@ŽsX£aº¡E] m U:ce_@Íµ¥×9#Âµ3¬µ¯l9È¯ª!¨3¬9ª!¨±µ½¡§ÀEÍ©Ùl¼b¨Ä·Â¡µ¨µÊE°¡¯±¡®¹6Ø ¼Ãµ#Âµ3¬9¹k§¯¼¡¹6Å¯!ªÄÈoªzV3¬µª¨±¡½¡§Áß2Ü^ŠÁ2óLîóï¥ï ë3õ ê

Q ← XΘ(Q) Arg Θ(Q) Ψ(Q) Arg Ψ(Q) ½ Õ Ê ®·Âµ½µ³`¬¡Ë¡ḠÀ ½ Õ(Q, Φ(Q), Ψ(Q), Θ(Q), Arg Ψ(Q), Arg Θ(Q))Õ°µ±¡½µ³z®2½¡®2ḠÀΦ(Q ′ ) ← Ψ(Q) / Θ(Q ′ ) Φ(Q ′′ ) ← Ψ(Q ′′ ) / Θ(Q) Õ ½(Q, Φ(Q), Ψ(Q), Θ(Q), Arg Ψ(Q), Arg Θ(Q))Õßß@§î6ï9ðñ2ðÆÁ2ó¨ é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ý³z§±µ¹2¼µ¾¡½ËµÀK½º°[Ä¬[Í=ªµÍ ªµÍ,¸§¯À«µ,¼Ço²:¨ÀbÈoª!¯;³z°µ½µ³z¨¼AÀK¨G²9½µÐw½¡¾¡½µ±¡¹¯§³ÏËÞ®T³z¨¨§®¯§³z§®½¡½A³l½µÁ?9 @=¡b?6ª> ªRaU@=TH*+4>2¤,ƒ:>2¤*9Ê&'#%'&¦4&?’1,;#14=*3h*$*8&:¤1“>?9g+4>:H?4=#150#)'&&¦N@=&'#%7&'*97B’1?/$?1Ê=Φ(X) = Ψ(X)/Θ(X)®·Âµ½µ³`¬¡Ë¡ḠÀLM½¡¡½µ¾¡½µ3¬9½¡¸!½¡±µ¹¯À¿±µÄ¨Âµ½¡©8³!°¡½µ³!¨¼L ← { ( X, Φ(X), Ψ(X), Θ(X), Arg Ψ(X), Arg Θ(X) ) }ÕÕä/ö_å ( ß'àµø )õ Arg Ψ(Q) ∩ Arg Θ(Q) = ∅QÕ®·Ä½¡±9¯ÀÑ®,®¯`²:¹2Åo¯ÀÑÄ±¡¹2³z¯w½¡¡¾¡½²9¯!¡§¡¹6ØÒ¼2¨2ÀK°¡¨µ¯!¡§¹ιQ Q ′ ½Q ′′ §3¼gÊ¡Â¡§¨Ä·Q ′ ι½Q ′′ιË¡®¬¡Ë¬9½µ³zº,°¡±¡¨§½¡®¨°¡¨¬µ¨#Ç¡·¥ÀK½Ì¼6¨¡¾µÀK½8½¡¡§¯!±¡®23¬9±9¸!Ä½¡®¯ÀQ 9o°µ¨§¨2ÀK¼¡½ ι½Arg Ψ(Q) ⊆ Q ′ ÊArg Θ(Q) ⊆ Q ′′ Õ¹²:¬¡Ë¡¯À¿½¡ļ±µÄ¨Â¡¯!ªö³z°µ½µ³z¼L Ä·®È¹6Øã®¯`²:¹2Åo¹6ØâĢ°¡½9³zº®·Â¡½µ³z¬¡Ë¡¯ÀΘ(Q ′ ) ½ Arg Θ(Q ′ ) Ê Ψ(Q ′′ ) ½ Arg Ψ(Q ′′ ) Õ(Q ′ , Φ(Q ′ ), Ψ(Q ′ ), Θ(Q ′ ), Arg Ψ(Q ′ ), Arg Θ(Q ′ ) ) (½Q ′′ , Φ(Q ′′ ), Ψ(Q ′′ ), Θ(Q ′′ ), Arg Ψ(Q ′′ ), Arg Θ(Q ′′ ) )°µ¨ÀK¯!ÅḠÀÉ¸°¡½µ³!½®T³z°¡½9³z¨¼L®,°¡¨G²µ»2¨G²µË¡ÅḠÀÃ°¡¨±2Ë6²9¼6¯ °µ¨®¨¸±µ³!§#¡½¡Ø®§¨±¡¨ª¨,°¡¨¬¡Ë/›fÕ¨Ä¨¸!µ#Âµ¯À%¡¨®¹6Øã®¯z²9¹6Å¹6Øã¸°µ½µ³zº³z°µ½µ³z¼L Â¡¯!±µ¯!¸å-æ-ßkß'à¥%±¡¯!¸¹¬9º3§#§¯®·°¡¨¬9¡¯!¡½¡Ë3¬µª¨±¡½¡§ÀE¥°¡¨±¡¨#Ç²:¯§³ÏË;¡¯!¹6Ä·®ØqÅoËÝ°¡¨Á

Φ(y) = min{ Φ(x) | x ∈ vert X } ÍÀK· °¡¨2³z§3±93¬µ½µ³!ºÌÀE¼6³!½µÀE3¬9º¡¨Ì¹2Âµ¯³z§ºÌ¨2³z¨Ä¯!¡µ¨³z§½A±¡¯!ÈḠÀK¨©kĢ#²:#Âµ½ ¦ I#§¯³!§D³z±µ¯`²9¡¯©;§¨Â¡¼¡½J¡¨§¨¡¡¨2³z§º±¡¯¬µ½¡¸¹¯ÀÌ§¨¬µº¼6¨°¡¯±¡®·©JÁ2óLîóï¥ï ë3õ ê

• °¡¨G²µ²9¯!±2Ç½¡®§º,®¯Å¯³!§®¯¡¡·©Ì°µ±µÀK¯!§± φ Ê¡±µ®¡·©é ù é è¤ë é ï¤ë¤è é ú è õ úºƒ ðâð è ë õâõ æ ¯ ÷ ‚ð ýî6ï9ðñ2ðÆÁ2ó¨• Φ(mid P ) ½®·Â¡½µ³`¬µË¡§º®¯!¬µ½¡Â¡½¡µ¹ßß2Üµ½µÀK¯!µºÈḠÀK¹T½¡ļ¸µ#Â¡¯!µ½¡©Φ(mid P ) ²µ¬¡ËV®³z¯!»FÄ±µ¹³z¨® P Êmin{ Φ(x) | x ∈ X } ≤ φ,¼6¨ªÏ²´3Á1¬µ½¡Äö°¡¨±µ¨#Ço²:Æ!¡¡·|»83¬µª¨±¡½¡§ÀK¨ÀÉ²9¨,§¯!¼¡¹6Åo¯!ª!¨,Èªz¡Í°g¨ªÏ²:½V®³z¯wĢ°¡½µ³!½( )P , Φ(P ), Ψ(P ), Θ(P ), Arg Ψ(P ), Arg Θ(P ) ,φ < Φ(P ) ª¡Íá2Û@›¼6¨§¨±¡·¯w¹!²:¨®¬9¯!§®¨±2Ë¡ØD§ÀK¨ª¹2§lÄ·§º;¹²:¬µ¯!¡·m½µḐ³z°¡½9³z¼L Ä¯!¸¼¼6¨ª!¨Á1¬µ½¡Äẅ®¬µ½Ë¡µ½Ëµl±¡¯¸!¹G¬µº3§3§¬w¨kµj°¡¹6§½Ñ±µ¯3¬9½¡¸¾¡½¡½É¦§¨©¿½6²9¯!½b³z§¨#Ë¡§j¸!µ#Âµ½¡§¯z¬9º¡·¯8§±¡¹²9¡¨2³z§½[Í±µÄ¨§·â3¬µª¨±¡½µ§ÀE¡ÍªµÍ ªÞ²9½µÀK¯§±¡·¤®¯`²:¹2Åo½»ÃÄ±¡¹2³z¨®A¡¯¨Ä#Ë¡Ģ§¯!¬µº¡¨T³!§±¡ḠÀ€Ë¡§³ÏËk¼A¡¹G¬µØoÊ´§#¼A¼¼AÄ±¡¹³!·ä²9±¡¨Ä#Ë¡§³ÏËA§¨¬µº¼6¨T°¡¨F½µÁH¯z¬µ¨j®k§¨2À|ÊEÂ¡§¨Q®Q¬µª!¨±¡½¡§ÀK¯°[Ä¬[Íª¡Ê¡¯³!¨ÀK¡¯¡¡¨9Ê!Ë¡®¬¡Ë¡¯!§³ÏËl®¯³zº2ÀE®·ª¨G²9¡¨©gÊ!§3¼¼6¼;¨Ä¯³!°¡¯!Â¡½µ®¯§|¦!¼6¨¡¨2ÀK½¡Ø¾¡½6²9¯¡§¡·¥Àã¼6¨2ÀK°¡¨¡¯¡§3À|Í[åw¨G²:¨Ä93Ëj¨2³z¨Ä¯!µ¡¨³!§ºÞ3¬µª¨±¡½¡§ÀEº°[Ä¬[Í0ª¡Í§¨ÀK¹Ê¡Â¡§¨±9¸!µ¨³z§º½Ì¾µ¯z¬µ¯µ°¡±9®¬9¯!¡¡¨2³z§º®·Âµ½µ³`¬µ½µ§¯z¬µº¡·|»F¹2³z½¬µ½µ©gÍ?¬w¨,¨µÇo¯w°¡±¡½µ®¨G²9½¡§,¼(Φ(mid P ) − Φ(P ))®¯z²9¹6Å½»QÄ±¡¹2³z¨®ÀK¨#Ço¯§¥¨³!§#®§º³ÏËlÄ¨¬µºÈo¯!©l¡¯¼6¨§¨±¡¨ª¨¥°¡¨¬9¨#Ç,½µ§¯z¬µº¡¨ª¨¥Â¡½µ³z¬9|²´3Ço¯²µ¬¡ËQ ÍJw½9³`¬µ¯¡¡·¯T¦¼6³!°¡¯!±µ½µÀK¯!¡§·

Q ← Xφ ← Φ(mid X)Θ(Q) ‘ Arg Θ(Q) ‘ Ψ(Q) ‘ Arg Ψ(Q) ‘ Φ(X) = Ψ(X)/Θ(X)Q ′ $&@¦Ž3¤1#%gj ∈ {1, 2, . . . , n} 2+?9“=&g‘Ê&2¤* wid Q ′ j ≠ 0 Q ′ jΘ(Q ′ ) Arg Θ(Q ′ ) ‘ Ψ(Q ′′ ) Arg Ψ(Q ′′ )Φ(Q ′ ) ← Ψ(Q) ‹ Θ(Q ′ ) Φ(Q ′′ ) ← Ψ(Q ′′ ) ‹ Θ(Q)Φ(Q ′ ≤ φ ‘=2¤*8&*+,;#+Ó6?#),²’H?98=)A `Q′‘Φ(Q ′′ ) ≤ φ ‘&2¤*68*,e#\Ó?9#1,²’H?/8&&)A `Q′′‘`Q, Φ(Q), Ψ(Q), Θ(Q), Arg Ψ(Q), Arg Θ(Q)´Q ′′Φ(Q ′ ) ‘ Ψ(Q) ‘ Θ(Q ′ ) ‘Φ(Q ′′ ) ‘ Ψ(Q ′′ ) ‘ Θ(Q) ‘ßßßÁ2óLîóï¥ï ë3õ ê2¤*Ê='#)'=¦h ? @=¡b?Œª> ¼4&?’1,;#14=*3Ž 4=:H?64&#150#)'=¦Ž@&='#)7='*7N’H?H$?1Ê& *6$*8=:1“>?9g3+CoÊ=)@&¦=#),Q ιÊ2+*+ CQ ′ ιQ ′′ιQ ιQ Q ′ Q ′′ 2?“¥‘¦R3+@¦R@&&)AŒ8&4&*2+&3+*8*9@*/.'=CV, “R*'¡b?,'?68&*+2¤*,;“=Arg Ψ(Q) ⊆ Q ′ ‘Arg Θ(Q) ⊆ Q ′′L ← ˘ ` X, Φ(X), Ψ(X), Θ(X), Arg Ψ(X), Arg Θ(X) ´` ´ú£ûüEý_þfÿ¡ Arg Ψ(Q) ∩ Arg Θ(Q) = ∅ & (φ − Φ(Q)) ≤ ɛ ¯3CV &4?#),µ33+#%$:£Ó0#1,Ë +4=:¤1#!&'b&>$#)'&2+'&:q¨“R*+,8*9'#)'&2+:'?/ *@&A+50#)7 $@=&'&C¨B4?911#1“>?#),&'&2¤#14>3?9@?'?9g*1$&,¾#112¤#)12+3¤#%'&'&CV# ‰+ª¤I]'?Q ′ jQ ′ j#1)@= 4&#)’):@=A92+?92¤*+,ƒ¦R3+@&¦=#12+%¦'#)*+2¤4>¡b?92¤#)@=A+'&Co7 &'&2¤#14>3?9@£‘R2¤*’H?9,e#%'¦>#1,'?Œ#)TH*@#)3Co7 “R*'#b‘&?Œ#1)@=B'&#)8*9@*/.&2¤#)@&A'&Co7£‘2¤*Œ'?68&4?93Co7 “R*'&#(b3Co8*9@&'¦>#1,¾&'&12¤4=:>“b&&8&4&#%$&C]$:£Ó0#)TH*Œ8=:R'“&2?!$@&¦N 4=:H?3CoÊ&&)@¦>#1,8&4=&)3?/&3?9#1,Arg Ψ(Q) ‘ Arg Θ(Q ′ ) ´ 3)8&&1*+“L 3'=:=.6'*,Ë8&*+4R¦R$“R##))@&Arg Ψ(Q ′′ ) ‘ Arg Θ(Q) ´ 3)8=1*“L 3Œ'&:>.'&*+,¾8*+4R¦$“R#EΦ(Q) ‘ Ψ(Q) ‘Dà8*3¤*+’)4?)2+?/'&qi3¤2¤*4&*9TH*ŒÊ&@&#)'?+I¤#))@&Co3+5j:q ’H?/8&&)A ` 3+#%$:£Ój:q :/$?/@¦>#1,ƒ Q‘Θ(Q) Arg Ψ(Q) Arg Θ(Q) ´ &’-)8=1“>?‘ ‘ω ← min{ Φ(mid Q ′ ), Φ(mid Q ′′ ) }3CoÊ&&)@¦>#1,φ > ω φ ← ω LΦ(P ) Ψ(P ) Θ(P ) ‘ ‘ ‘Ê&&12+,ƒ%8&)*+“ ‘>2¤*Œ8&*@?/T1?9#1,LArg Ψ(P ) ‘ Arg Θ(P ) ´‘=Ê&2¤*Φ(P ) > φ#))@&‘&29 #+:/$?9@&¦=#),¾&’!'#%TH*Œ3¤)#j2+?9“=#!’H?98=) `P‘* +*’)'?1Ê?9#1,µ'&*3:qi3¤#%$&:£Ó0:qL’H?98=)A%8&)“=?0Ê#14=#1’¡¢wúZú£û

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ÀaÁÜ;.ÈL³L´#³rÇ¤ÀeÂ..¼¡ØŽÀ¹¤º9¾ÆÂ´#Õa³à·¼¥Á;Ö¼¡Ø}¹.¼½e³¥ÈàÁ;´¥ÖÀ¹¤¾¤³#ÈêÇ¤ÀÅ,À¹¤º.À)·¥¹¤ÀaÅ3Ê¤Ax = b ¹.´¹¤ÀÙëÁa´¡Â6´¡Ã¤ºçÂ.¤¾ëº!½JÛ¼;Â!¹¤¼ÙFº¤¹¤³#ÀaÇ¤·´#.È¥¹¤¼Ù.º¤¹.ÀÙ¤¹¤¼Ùë½eº!½e³#À;ÖÚÇ¤Àa¸¤ÜÇ¤Ç¤À¹.³#¹¤¼¥µa¼ë·ºÂóË‚~Ï¨ÍÑBÏ f ¼Ç¤ÀÅ,À¹¤º.ÀÄ·¥¹¤ÀaÅÆ¹¤ÀaÙoÁaĹÂ´¡Ã¤º‹Â.¤¾O½º.½e³¥ÀaÖÚ½B.ÀeÂ!Üèá¿,´#¾â³#¼³}·¥¼¥».Ç¤¼½ŠÂ´#Õa³9£i¢pË%¯?°6š?°Ö± ËÎ³¥À¼¥Ç.ÀaÖ§´ ï ´#Ù.ÀÇ.´#Ê‚M´#Ç.¹¤¸À;Ñ;ÕQ£i¬é´¥ærº¤¸9½eº.Ç¤Ü¥À;Ö˜Cºò·¥Àa¸¤³#¼Ç ºÉ%¬¨Ê?ËÌ£Ë‹Í£«9ÎnÏÍ‹Ðn¬z° Ö§´#³#Ç¤º.É¤ÜÇ.´¥½a½Ö¼³#Ç¤º.Ö÷¼¥³#¼Ó¥Ç.´#Ø,Àa¹¤º¤À ∈ C ˜d ∈ dR n → R n ».Ç.´#·º9.Ü Þ9Â!ÀÙ.½³#·¥Üèà¿ÆÀÀr»¤¼ù(jijŸž oysu¥¨§©õ|@ noJýgFoJý ø §£~ßàÃ¤À·ºÂ!¹¤¼!ÞÃ¤³#¼É%¬¨Ê?ËÌ£Ë‹Í£«9ÎnÏÍ‹Ðn¬z°Ξ(A, b) = { x ∈ R n | (∃A ∈ A)(∃b ∈ b)( Ax = b )}= { x ∈ R n | (∃A ∈ A)(∃b ∈ b)( x = (I − A) x + b )}.¼A ∈ A (I − A) ∈ (I − A) Þ!º»¤¼¥³¥¼ÖÜuÖÚfº¤³#È}·Ú)¸9!´¡Â¸¤ºç½e.ÀBÂÜèà¿Æº.Ö÷¼¥ÓÇ.´#Áa¼ÖÖ¼¡Ø,À;Öo»¤Ç¤¼;Â¼#¤ØΞ(A, b) = { x ∈ R n | (∃(I − A) ∈ (I − A))(∃b ∈ b)( x = (I − A) x + b ³#¼µJÂ´¤Þ¤¸¼µJÂ´ ºF³#¼¥.È¥¸¼ ³#¼µJÂ´ )}= { x ∈ R n | (∃C ∈ (I − A))(∃b ∈ b)( x = Cx + b )}= { x ∈ R n | (∃C ∈ C)(∃d ∈ d)( x = Cx + d )}C = I − A d = bÞ9Ã.³#¼ ºF³#Ç¤ÀÓ¼¥·Ĺ.¼½eÈ.ÏÞ Â.¤¾¸9´#¸O¹!ĹÛ¼;Âº¤³#Èë¼É¤À¹.¸¤ºÖ¹.¼¡Ø,Àa½³#·×Ç¤ÀÅ,À¹¤º¤Ù»¤¼;Â!¼Ó¥¹¤ÚMÛO½º.½e³¥ÀaÖ¡+ëßà³#·À³×¹.Ć ∈ IR n×n Äd ∈ IR n ÄÂ£B DµΞ = { x ∈ R n | (∃C ∈ C)(∃d ∈ d)( x = Cx + d ) }¡&¡¹h"($+nº{++¨ (@»BDCh (IÇń$¨+n I+¨ ¡I½B$BD-o + pEGHD+¨$Ïl$+¨)‚EGC#ńµ+x = Cx + d x ∗ ∈ IR n o – òEi3#ńµ+ ¡EGHD+¨$¢=!¡I¿AÂ{E#hC+n +$Ï)¾¿ *{"¡#Ä‘# $nB BÈ¤£x = Cx + d B C ∈ C $ d ∈ dË¦¥RÑw"{´‹0e´:9õ&¡ ¡Ień$+Ï +¨ ¡ISB$B )¾¿x ∗§Ξ ⊆ x ∗ $1(I − C) + ¡B ¡&¡+¨+n# ÈÃ¨ )·{E#I+ IÒ3‚Es "+ wºZEs H) +$B "¡‚En@â$DB–0Cw&òEGÂ£BË¦¥ ¥RÑwC·(´:9õ/+¨$nB D !D *{"¡#Ø$SD –ńµ£· wD ¡*{"¡#¡Ä· *{"¡#Ø$+¨)‚EGC#ńµ+n#0Ò3#‘EG$¨Æn#x ↦→ ˜Cx + ˜d.[´¥¸ç¸9´¥¸{ ˜Cx + ˜d | x ∈ x ∗ } ⊆ ˜Cx ∗ + ˜d ⊆ Cx ∗ + d = x ∗ ,

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Ž”‚¡€”Ž”ƒ)|¡ŒŽª”Š¡‚¡˜€Ÿ¤•”†„ ˜•”|&ƒ)Š‚¤>€K}¤„œ§•”ƒ.Œ)„‡}¡|¡Š¤Ž•S‚¡}¤‘=„‡}'e•”}¡ƒ¨•>€K˜š“”˜ƒ$•ƒ)ª”Š¡‚¡˜€KŸ¡•”†œ›e˜”•”|¡ƒ)Š”‚7>€}¤‚•a c ˜ij•|>&”PƒS’4¡)|‚¡•”ƒ=–…•”µ¨µ ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”¡µÅ}¡—ƒ)•””¡¦„‡•”}'Pƒ.Œ-|&‚7 Šƒ)–…ƒ=•”˜”~Ξ αβ (A, b)¢|&}¤“”•”¡µ{‚¡•”ƒ=–…•”µ¨µ{˜•”|&ƒ)Š‚¤>€KŸ¡•”¡µ{}¡—”ƒ=•””¡e„‡•”}'Pƒ.Œ-|&‚7…Šƒ)–…ƒ=•”˜”~&‰¤}&€ƒ=ƒΞ αβ b)¢E¦Q¡7}¤•”ƒ)—³§-“”|&}¤‰¡†ØŒ-Š¡‘)¡¨PƒÓ‚¤}¤Œ)ª”}&€Ÿ¤‘)}¤‚¤&|¡Ÿ¤ŒŽ ª”}¡€¡“&ƒ.„‡}¡~Q‚›7}.’Kƒ(A,ÙÚ Ìx i⎛⎞∈ ( a c ii )−1 ⎝ opp ∑ a c ij x j + b c ⎠ij≠iÞ7ß¨îß•ï©äVî'ãïcœ-u>ÝDv¡ä”éñ ë¤ã.î'åy×ÜuRð4í)í²uñóòVãWñy+ã!Ý³´8}¤’4”§8ƒ.Œ!€˜x ∈ Ξ αβ (A, b)§³|&}[’K}&€”e• ‰¤†|&ŸcŒ-ªŠ&‚¤ƒ!’€˜‚¡}¡~×Œ!€Kƒ!’K¡µ ‹S>Ž—”ƒ)ª}¡“”SŒ-}¡}¤|&•”}¤–…ƒ)•˜”~⎛⎞= ( a c ii )−1 ⎝ ∑ opp ( a c ij x )j + bci⎠j≠i⎛⎞= ( a c ii) −1 ⎝ ∑ ( opp a c ij) x j + b c ⎠ij≠iª”}¤Œ)}&€Ÿ¤”¡e‚¤Œ)ƒx j⎛⎞⊆ ( a c ii) −1 ⎝ ∑ ( opp a c ij) x j + b c ⎠ij≠i∈ x j˜& |>&xjKR}¤ª”ƒ)Š¡—”˜”˜‚Œ-¡|&Ÿ«|&}¤“”ƒ)“”•†ƒ¡Š”˜ÕQ„‡ƒ)|¡˜”“”ƒ=Œ)”˜”ƒ‚¤€µ “ƒ)•”˜”µP¢ „‡}¤•”}¡|&}¤•”•”†{ª”}˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”†~‚¡ƒ)|&}¡Š ˜xª”}7Œ-Š”ƒ-’4Œ-|¡‚¡}¤„´³¡”˜„}¤‰¡Š¡‘)}¤„œ§ƒ=Œ!€K˜c}¡ª”Šƒ!’Kƒ)€˜”|&ŸŠ&‚¤ƒ)•Œ)|&‚ ª”}¡}7„‡ª”}¡•”ƒ=•”|&•”†œ›⎛⎞˜x i := ( a c ii )−1 ⎝ ∑ ( opp a c ij ) x j + b c ⎠i , i = 1, . . . ËÍÌ ¢ÏÎ>±&Ð, n,j≠i|&}«‚eŠ¡Œ=Œ)„ ¡|&Š”˜”‚7&ƒ=„‡†œ›¯•K¡„‡˜ ¡¤Œ-€}¡‚¤˜Ž›¯}¡•b c i§‚¤‘-Ž”|&˜”ƒ$ª”Š”}¤|&˜”‚¤}¡ª”}¡€}'e•”†œ›z)€ƒ=„‡ƒ=•”|&}¡‚e˜ |'¢ª8¢”‚«‚¡†Š&Pƒ=•”˜”˜ ËÍÌ ¢Î'±¡ÐW§‚¡‘-Ž”|¡ŸSªƒ)Š”ƒ.Œ-ƒ)“ƒ)•”˜”ƒŠ¤Œ-|&ƒ.Œ-|&‚¤ƒ)•”•}eª”}¡z=|&}¤„‡¡x ∩ ˜x ⊇ Ξ αβ (A, b),

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=KþDCI?ø@H¡û)ü4ý¡þK÷ESYXZOÍú¦û£C ø=/e/?RSXgCP÷d@H)üd@?RCLGd| x ⋆ | ≤ 〈A〉 −1 | b ËÍÌ ¢ÏÎ Ì Ð|.ü=Kþ/?¥SKþ÷!S G¥=G¥CI?úQÿA x = b XoCP÷K”MPOøc?¥QI?-þ/=DSr=þ/CL?Iø@H&û)ü4ý¤þ÷!S〈A〉 | x ⋆ | ≤ | b |.“+Gü= AÈ?AÙ7Û ÎÞ7ß W”ß•ïSí)íùÝ®ãóy+å£œ-u>äD4ã¨å¤æ¡å¤æ'ç«è!ä”ä4å£±'åQë¤ã'î'åyuSÜuRðKí=í²uñóò®ã!ñy+ã!Ý³ËÍÌ ¢ÏÎEŒ¤ÐŠ¤Œ)Œ)„ ¡|&Š”˜”‚7&ƒ.„ž|¡}&€Ÿ¤7}«•”ƒ=}¤Œ)}¡‰¡ƒ=•”•”†ƒ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†ƒ „ &|¡Š”˜”—”†m,D‚ƒ2„2…*†0‡ˆ…‰,p6§©}&€KŸ¯Œ-}¡Š”}e„‡†}¤‰¡‹…•}¤Œ-|¡˜ „‡}'«•}¦Œ-“˜šA§+|¡}¦‰¤ƒ)‘}¤)Š¡•”˜”“”ƒ=•”˜Ž|>&|&ŸK§“”|&}0 ∉ a ii§i = 1, 2, . . . , n¢}¤ª”Š”ƒ!’4ƒ-€”Ž”µ¨‹…˜›ò}¡‰¡}¤‰&š°'¡7Œ)Œ=>š ƒ)~’4ƒ-€”Ž8§>ẽ¡“”˜”|¡†‚¤>Ž8§¤“”|&} Ê©ƒ)Š”ƒ)›¤}.’ŽÔª”Šƒ!’Kƒ)€¡‚ØŠ¤Œ-“”=|&•”†œ›žÕ©}¤Š„‡¡.€K>›V§lim'x =‹…=•”•”†~e˜”•”|¡ƒ)Š”‚7>€Ÿ¤•”†~¯„‡ƒ)|&}.’lim ˜x = x( ( ∑) )x ⋆ i = x⋆ i ∩ ( a c ii )−1 ( opp a c ij ) x⋆ j + bc i , i = 1, 2, . . . , n.j≠i⋆§”ª”}¡€¡“”˜„ó¤€ƒ!’4}¡‚¤¡|&ƒ-€KŸ¡•”}K§7ª}SŠ&~”•ƒ)~[„‡ƒ)Š”ƒx ⋆ i ⊆ ( ac ii )−1 ( ∑j≠i( opp a c ij ) x⋆ j + bc i), ËÍÌ ¢ § ª Ð) ∣ ∣∣∣| x ⋆ i | ≤ ∣ ∣∣∣( a c ii )−1 ( ∑j≠i( opp a c ij ) x⋆ j + bc i˜ ª”}¡|&}7„‡¡( ) ∣ ∑ ∣∣∣∣ ∣ ( ac ii )−1 ( opp a c ij ) x⋆ j + bc i = ∣ ( a c ii )−1∣ ∣ ∑·( opp a c ij∣) x⋆ j + bc i∣j≠ij≠ii = 1, 2, . . . , n§8|>&¬¡¦‚ }¤‰¡ƒ)˜”›“¡Œ)|.Ž›‚¡€Kµ “”ƒ=•”˜Ž Ë:Ì¢¨§ ª Ð¨Œ)|&}&šŽ”|cª”Š¡‚¡˜€KŸ¡•”†ƒ¯˜•”|&ƒ)Š‚¤>€K†Q¢0 ¨—”ƒ=•”˜„ÖŒ-‚¤ƒ)Š7›¡¦ªŠ&‚¤†ƒe“¡Œ)|&˜ ª”}¡€¡“”ƒ)•”•†œ›’€”Ž×‚¤Œ)ƒ-›•”ƒ)ŠK&‚¡ƒ=•Œ-|¡‚D%( )∑≤ | a −1ii | | ( opp a c ij ) x⋆ j | + |bc i |j≠i≤ 〈 a ii 〉 −1 ( ∑j≠i| a ij || x ⋆ j | + |b i |)

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|( D c ) −1 | = 〈D〉 −1§˜ ª”}¤z)|&}7„‡¡|&ƒ-€KŸ¡•”†œ›z)€ƒ=„‡ƒ=•”|&}¤‚§|( D c ) −1 | | opp E c | = 〈D〉 −1 ËÍÌ ¢ §§¡Ð|E|.ÜÝ/u£œkueÞß¬àâá®ãWäœku>ä®ã¨å¤æ¡å¡æ>ç«è!ä”ä”éœêë‡ä4å)ì¯ã.í)îœQï+ãWð«ã!ä”ñAƒ.Œ-|&ŸLÒ š„ ¡|&Š”˜”—K”§7|&}…ƒ=œ’K˜¡)}¡•K>€Ÿ…Œ)}¤Œ)|&}¡˜”|…˜”‘ ª”}¡€}'e˜š&P>€ƒ=ƒ¡§¡ƒ.Œ!€˜Ù¤Û¡ÙD|>&”Pƒ©ƒ=Œ)|&ŸÀÒ šI„ &|&Š”˜—”¢¦©}[˜ „ &|&Š”˜—cŒ-ŠK&‚¡•ƒ)•”˜Ž §©Š”}¤„‡ƒ©|&}¤)}K§ Ž”‚&€Ž”ƒ)|¡ŒŽbÒ šI„ &|&Š˜”—”ƒ)~³§³‰ ¡=’K¡“”˜Ø}.’4•”}¡~˜”‘¯|&}¤“”ƒ)“•”†œ› „ ‚ª”Šƒ!’Kƒ)€K>› A¢Óó¤€ƒ!’4}¡‚¤¡|&ƒ-€KŸ¡•”}K§8ª”}¤Œ)}&€Ÿ¤”¡〈A〉〈A〉 =〈D〉 |E|§˜”‘QŠ”ƒ)‘=¡.€Ÿ>|>&|>a¦©}¤~„ &~ƒ)Š Ë´³ƒ)}¡Šƒ=„ #§”¢Ì ¢ÏÎ'ÐÓ‚¤†|&ƒ)&ƒ=|−ρ ( 〈D〉 −1 |E| ) < 1,Ë:Ì ¢¨§§¤Ð¤ªŠ”˜”‚¡}.’4˜”|P •”ƒ=Š&‚¤ƒ)•Œ)|&‚ ¡“”|&}e‚7„‡ƒ=Œ-|¡ƒ©Œρ ( |( D c ) −1 | | opp E c | ) < 1.“˜”|&Ÿ§K“”|&} ˜”|&ƒ)ŠK&—”˜”}¤•”•”†~[ª”Š}¡—”ƒ.Œ)Œ‚ Ü[†ˆ„‡}'Pƒ.„œ§|>&”˜K„ž}¤‰¡Š¡‘)}7„œ§”‘=¡€µKRx (0) ← x ⋆ ,n§}¡ª”Š”ƒ-’Kƒ-€Ž”ƒ=„‡†~[Õ©}¡Š„‡¡.€K¤„‡˜x (k+1) ← ( D c ) −1( ( opp E c ) x (k) + b c ) , k = 0, 1, . . . ,x ∗ }¡|&}¤‰¡Š&Pƒ)•˜Ž Œ›¤}.’K˜|¡ŒŽc¦ƒ-’K˜”•Œ)|&‚¡ƒ=•”•”}¤~•”ƒ=ª”}.’K‚¡˜”«•}¡~|¡}¡“”ƒKRx ↦→ ( D c ) −1( ( opp E c ) x + b c ) .n§7’Kƒ)~KŒ-|&‚¤¡7µ¨‹…ƒ=)}«ª”}«ª”Š¡‚¡˜€¡KR n →x ∗Ž”‚&€”Ž”ƒ=|¡ŒŽ Õ©}¤Š„ >€Ÿ¤•”†„ØŠ”ƒ=–…ƒ=•”˜”ƒ=„˜”•”|&ƒ=Š”‚¤&€Ÿ¡•}¡~¯€K˜”•”ƒ)~•”}¡~Œ-˜Œ)|&ƒ=„‡† Ê©Š”˜z=|&}¤„x = ( D c ) −1( ( opp E c ) x + b c ) ËÍÌ ¢ §$¤Ð.“ƒ)•”˜Ž ËÍÌ ¢ §”Î.Ð¤ª”}e˜”•’4¡7—”˜”˜[„‡}'e•”}«‚¡†‚¡ƒ.Œ-|&˜³§“”|&}&P>€ƒ=ƒ¡§˜”‘Q‚¤€µx ⋆ ⊆ x ∗ .¢ § Ù Ð&…ƒ)~Œ)|&‚¡˜|&ƒ-€Ÿ¤•”}§ËÍÌx ⋆ ⊆ xƒ=Œ-€˜x ⋆ ⊆ x (k)§”|¡}§”ª”Š˜”•”˜„ &Ž‚¤}S‚¤•”˜„ ¡•”˜”ƒŒ-‚¤}¡~Œ)|&‚¡}…„‡}¤•”}¡|¡}¡•”•”}7Œ-|&˜˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”†œ›e&Š˜ÕQ„‡ƒ)|&˜“”ƒ=Œ)”˜›¯}¡ªƒ)Š¡—”˜”~e‚KR(0)§”˜“”ƒ=•”˜”µ…§•”ƒ)|&Š¡)’4•”}e‘=¡€µ “˜”|&Ÿª”}«‚¡€Kµx ⋆ ⊆ ( D c ) −1( ( opp E c ) x ⋆ + b c )⊆ ( D c ) −1( ( opp E c ) x (k) + b c ) = x (k+1) .

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Š¡‚¡• •”¡.€µ ˜’K}¤Œ)|&˜”-&ƒ=|¡ŒŽ•‚¤ƒ)Š”–P˜”•”ƒ}¡|&}¤Š>Ž[’K€”Ž¦˜Œ›7}.’K•”}¤~¦Œ-˜KŒ-|&ƒ.„‡†( 1 2 , 0) •¡˜”‚¡•}¤„Øª”Šƒ!’K}¤‰ ¡¤Œ-€K&‚¡€˜”‚7&•”˜”˜¡7‚¤ƒ-€˜“”˜”‚¤¡ƒ)|¡ŒŽƒß§”ª”Š”˜. ËÍÌ ¢ § Û ÐÙ7Û Þ7ß_^7ßa`ï³ãóy+å¤æð4íùÝ/u>œ¡Ýœku>ä®ã ±«šŠ”ƒ)–Pƒ)•”˜”~ «„∃∃ ∃∃∃∀∀∃∃∃«„∃∃«šJŠƒ)–…ƒ=•”˜”~ž’€”ŽÔ˜Œ›7}.’K•”}¡~Â(…óY¡n¢0£(©‘”¨Ó˜Œ¡¢ Ì ¢ÏÎ¨•”ƒ)|¡Š”¡=’K•”}…‚¤˜’Kƒ=|&Ÿ§7“”|&}S„‡•”}'Pƒ.Œ-|&‚¤} „Ä•&˜‚&š•”} ª”Š”ƒ-’K}¡‰¤¡¤Œ!€K}¡‚&€Kƒ)•”•”}¤~ƒÅŒ)˜Œ-|&ƒ.„‡† Ë:Ì¢¨§ Û Ð ‚ ª”ƒ=Š”‚¡}7„¨}¤Š”|>&•”|&ƒ«•”ƒ¯Œ)}.’Kƒ)Š7e˜”|( 4 3 , 5 3 ) ˜¬ª”Š”˜”€ƒ)-¡µ ‹…¡µ ¦•ƒ)~¬“¡Œ)|&Ÿ Ë•&ªŠ”˜„‡ƒ)Š³§V|&}¡“”¡(1, 1) ⊤Ð‚¡ƒ=Š”–…˜”•¡ ¢ § Ú Ð-¢oÃâ}&€ƒ=ƒ|¡}¡)}K§ ËÍÌ•”˜e•ŽŽÔ}¡—”ƒ=•”¬‚¡|&}¤Š”}¡~ò}¡}¡Š¤’K˜•&|&†j|&}¡“ƒ)Ôz)|&}¤)}Ø„‡•”}'Sƒ=Œ-|¡‚¤¬Š”ƒ)–Pƒ)•”˜~8§„‡•”}'Pƒ.Œ-|&‚7 „Ór.p×013*-.( ) ( )2 [−3, −1]E[1, 3]E−1 [2, 5] A x =[1, 10] E¨ ¡Œ=Œ)„‡}¡|¡Š”˜„ž˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”¡µ{€˜”•ƒ)~”•”¡µˆŒ)˜Œ-|&ƒ.„‡¡§ËÍÌ ¢ § Ú Ð}¡|&}¤Š”}¡~ ’€”Ž( 2 −2mid A =−1 3.5)(, (mid A) −1 0.7 0.4=0.2 0.4),Œ-Šƒ!’K•”ƒ=~8§”ƒ.Œ-|&Ÿ •”}¡~237([11, 26] [−10, 10][−10, 10] [11, 26])x = 237([7, 14][4, 11])œ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•K>Ž$Œ)˜Œ-|¡ƒ=„ ”§Ä•¡˜”‚¡•”}œª”Šƒ!’K}¤‰ ¡¤Œ-€}¡‚¡€ƒ)••>ŽƒÅâŒ‡ª”}¤„‡}¤‹…Ÿ¡µØ}¤‰¡Š¡|>šÄ•¡˜”‚¡•”}QªŠ”ƒ!’K}¤‰ ¡7Œ!€}¤‚&€ƒ=•”•”}¡~Å3(…óY¡n¢0£õ•”ƒ}¡‰'Ž”‘.&|&ƒ)€Ÿ¡•”} Œ-}.’Kƒ=Š¤e˜”|„‡•”}'Pƒ.Œ-|&‚¤}Š”ƒ=–…ƒ=•”˜”~¬˜Œ›¤}.’K•}¡~W(…óY¡n¢0£§8 ‚¡•”ƒ=–¦š(©|'¡8§„‡•”}'Pƒ.Œ-|&‚¤}QŠ”ƒ)–Pƒ)•”˜”~•ŽŽ[}¡—”ƒ=•”«„‡•”}'Pƒ.Œ-|&‚7«Š”ƒ)–Pƒ)•”˜”~Ä•&˜”‚¤•”}eª”Š”ƒ-’K}¡‰¤¡¤Œ-€}¡‚¡€ƒ)•”•}¡~ƒÅSŒ-˜Œ)|&ƒ=„‡†„‡}'Pƒ=|¦˜ •ƒ ‰¡†|¡ŸP‚¤•”ƒ)–P•”ƒ)~¯}¤—”ƒ)•7}¤~Œ-}¤}¡|&‚¤ƒ)|¡Œ)|&‚ ¡µ¨‹…ƒ)=}P„‡•”}'Sƒ=Œ)|&‚¤QŠƒ)–…ƒ)š•”˜”~ ˜KŒ›¤}.’K•”}¤~ Œ-˜Œ)|&ƒ=„‡†Q¢ ´³ƒ=„Ö•”ƒ„‡ƒ)•”ƒ=ƒ¡§‡‚¡†œ›7}.’ ˜”‘ Œ)}¡‘!’®&‚¡–Pƒ)=}¤ŒŽ×‘=&|¡Š”¡=’š;?RGRG¥CL=§aŒ-}¡}¤|>š|&}¤„œ§4“”|&} „‡†{’K}¡€”e•”†Öª”Š”ƒ-’K}¡‰¤¡¤Œ!€4&‚&€K˜”‚¤¡|&Ÿe•”ƒ˜Œ›¤}.’4•”¡µ ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”¡µ €K˜”•”ƒ)~•”¡µjŒ-˜KŒ-|&ƒ.„‡¡¬‚¤}¡}¤‰¡‹…ƒ¤§³m¤4ûø”û£šy¥•}¡‚¡ŸØ}¤‰¡Š¡|&˜„Œ-˜KŒ-|&ƒ.„+% •”ƒ)~•”†œ›x ∈ Ξ αβ (A, b) ⇐⇒ A c · x ⊆ b c .

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Ù ±§ÜÝ/u£œkueÞß¬àâá®ãWäœku>ä®ã¨å¤æ¡å¡æ>ç«è!ä”ä”éœêë‡ä4å)ì¯ã.í)îœQï+ãWð«ã!ä”ñ¡¤Œ-€}¡‚¤˜”~[z)”‚¤˜”‚¤&€ƒ)•|&•”}¤Œ)|&˜ªŠ”ƒ)}¤‰¡Š¡‘)}¡‚7&•”˜~˜‘ô.Î¡¢¨§&¯z=|&} „‡•}'Pƒ=Œ)|&‚¡}Œ-}¤‚&šª'’®&ƒ)|PŒ’K}¤ª”¡¤Œ)|&˜„‡†„ž„‡•”}'Pƒ.Œ-|&‚¤}¤„×Š”ƒ=–…ƒ)•˜”~˜”•|&ƒ)Š”‚7>€Ÿ¤•”}¡~ Œ-˜Œ)|&ƒ=„‡† Ë §¡Ð-§⎧⎪⎨⎪⎩[1.5, 2.5] x 1 + [2.5, 3.5] x 2 +[0.8, 1.4]1 + x 2 1= [5, 15],[0.9, 1.1] x 1 − [1.8, 2.2] x 2 + [1.6, 2] cos x 2 = [8, 20].¦ª”}¤|&}¤„‡¡P’4&€ƒ)ƒ$„‡† „‡}'Pƒ.„×Š¡Œ=Œ)„ ¡|&Š”˜”‚7&|&Ÿ¦‘.'’4'“¡S‚¤•”ƒ)–P•”ƒ)=}¦}¡—ƒ)•”˜”‚7&•”˜”Ž¡7Pƒ¨’€”Ž ËÍÌ ¢ $Î.ÐW¢ÿ '“”ƒ=Œ)|&‚¡ƒ$ª”Š”˜K„‡ƒ)Š«Š¤Œ)Œ=„‡}¡|&Š”˜K„˜”•|&ƒ)Š”‚7>€Ÿ¤•”¡µÖŒ-˜Œ)|&ƒ=„‡¡ ¡7ŠK&‚¡•ƒ)•”˜”~⎧⎪⎨⎪⎩2x 1 + [−0.5, 0.5] x 1 + [2.5, 3.5] x 2 +[0.8, 1.4]1 + x 2 1= [5, 15],[0.9, 1.1] x 1 − 2x 2 + [−0.2, 0.2] x 2 + [1.6, 2] cos x 2 = [8, 20],ËÍÌ ¢ $-§¡Ðÿ Œ-˜”€¡Š”ƒ=‘)¡.€Ÿ>|>&|¡}¡‚ ô=Î¡¢¨§&P}¡•KPz=”‚¡˜‚¤>€Kƒ)•”|&•KPŒ)˜Œ-|&ƒ.„‡ƒ⎧⎪⎨⎪⎩2x 1 = [−0.5, 0.5] x 1 − [2.5, 3.5] x 2 −[0.8, 1.4]1 + x 2 1+ [5, 15],2x 2 = [0.9, 1.1] x 1 + [−0.2, 0.2] x 2 + [1.6, 2] cos x 2 − [8, 20].}¡|&}¤Š>Ž8§4‚Œ)‚¡}¤µ²}¤“”ƒ)Š”ƒ-’KŸ§®ª”}7Œ!€ƒSª”ƒ=Š”ƒ)•”ƒ.Œ-ƒ=•”˜Ž“€Kƒ)•”}¤‚„‡}'Sƒ)|‰¤†|&Ÿª”ƒ)Š”ƒ)šª”˜Œ=&•S&¦Q&}¡•ƒ)—8§ª”Š”˜”›¤}.’K˜„⎧⎪⎨⎪⎩x 1 = [−0.25, 0.25] x 1 − [1.25, 1.75] x 2 −[0.4, 0.7]1 + x 2 1+ [2.5, 7.5],x 2 = [0.45, 0.55] x 1 + [−0.1, 0.1] x 2 + [0.8, 1] cos x 2 − [4, 10].⎧⎪⎨⎪⎩[1.5, 2.5] x 1 + 3x 2 + [−0.5, 0.5] x 2 +[0.8, 1.4]1 + x 2 1= [5, 15],x 1 + [−0.1, 0.1] x 1 − [1.8, 2.2] x 2 + [1.6, 2] cos x 2 = [8, 20].y¥}¤‘=„‡}'Sƒ)•˜¦’KŠ¡7=}¡~¬Œ-ª”}7Œ-}¡‰z)‚¡˜”‚7>€ƒ=•”|&•”}¤)}ª”Š”˜”‚¤ƒ!’Kƒ=•”˜Ž¦˜•”|&ƒ)Š‚¤>€KŸ&š•”}¡~[Œ)˜Œ-|&ƒ.„‡† Ë:Ì ¢ $§¤Ðâ Sƒ-€K¡ƒ=„‡}¤~ Š”ƒ=”¡Š”Š”ƒ=•”|&•”}¤~cÕ©}¡Š„‡ƒ¤¢/ ¨| Ë:Ì ¢ $§¤Ðœ„‡†õª”ƒ)šŠ”ƒ-›7}.’K˜„

yâ†Š>«>Žx 1˜”‘Q‚¤|&}¡Š”}¤)}S¡Š&‚¤•”ƒ)•”˜”Žã 1Xa 2X@]E]]¥Xa lX=7ún?R}hza=¤a=þ/CL?Iø@H&û)ü4ý¤þDJ} þ/?)÷E &ø@?ROx ∗ ∈ IR n X3€˜”|¡˜”“”ƒ=Œ)}¡ƒe‚¡†ŠK>Pƒ=•”˜”ƒS’€Žx i§>ªK&Š¤„‡ƒ)|&Š”}¤‚a j§>“”ƒ=|&†Š”)›P&Š”˜KÕQ„‡ƒ)|&˜”“ƒ=Œ-˜›…}¡ªƒ-š˜³§”‚¡}¤‘=„‡}'e•”}§7ƒ)‹PQz-€ƒ.„‡ƒ)•”|>¡Š”•”†œ› Õ©¡•””—˜”~ |¡˜”ª Š&—˜”~ § sin exp§log§‚¡}¤‘)‚¡ƒ-’Kƒ)•˜Ž‚«—”ƒ-€K¡7µˆŒ)|&ƒ)ªƒ)•”ŸK§”‚¡‘)Ž”|&˜Ž }¤Š”•Ž ˜ |'¢ª8¢cos§)+*-,/.0*-132546…„687–àJGRCPýxHV=þ/CL?Iø@H&û)ü4ý¤þ÷!SWG¥=G¥CI?ún?ÝJ!ø”û>H>þ?-þ/=DSx = G(a, x) + bÿ4ø”û!AF?)þD=/S÷ECS÷!~ùøû!AÈ?-þ/=-d?RGRCL?G¥CIH>?-þKþ÷£TW÷¦=þ/CL?Iø@H&û)ü4ý¤þ÷£TW÷ø”û>GRQa=&ø@?-þ/=-dx ↦→ G( dual a, x ),P KR n û£CL=/?N >ø4÷£GRCø”û>þ/GRCLH¡û XgCP÷a=þ/CL?Iø@H&û)ü4ý¤þKûRdrGR=/GRCL?ú…ûnJ!ø”û>H£Oþ/?)þD=DS?RGRCPýx = G( dual a, x ) + O GUA $$7Ð ¢ Ë:Ì bCP÷ùø®÷!?\d@H=üd@?¥CIGÎdxH>þ/?RQPþ?¥Sx=þ/CL?Iø@H&û)ü4ý¤þ÷!S÷Ee?-þ/?)þþK÷!?• >ø”û>H£=ü4ý¡þK÷!? öÓ÷ùø7ú¦û=ü4ý¡þ÷£?Pøc?¥QI?-þD=?Œ-|&ƒ.„‡†õ¡Š¡‚¡•”ƒ=•”˜”~m,D‚ƒ2„2…*†0‡ˆ…‰,p6¤Ê©Š”ƒ)P’4ƒ¯‚¤Œ)ƒ))}[ª}¡>Sƒ=„œ§8“”|¡}[’€”ŽØ˜”•|&ƒ)Š”‚7>€Ÿ¤•”}¡~ØŒ)˜šx = G(a, x) + Ë $¤Ðb

i§4˜cz=|&˜c‚&›7}'S’Kƒ)•”˜”Ž‚¯ŠK&‘)•†œ›‚¤†Š>Sƒ)•”˜Ž›( G i (a, x) + b i )•”ƒPª”ƒ)Šƒ=Œ-ƒ=>šÜÝ/u£œkueÞß¬àâá®ãWäœku>ä®ã¨å¤æ¡å¡æ>ç«è!ä”ä”éœêë‡ä4å)ì¯ã.í)îœQï+ãWð«ã!ä”ñÙ ± Ù⎧⎪⎨⎪⎩x i ≥ maxa∈a G i(a, x) + b i ,x i ≤ mina∈a G i(a, x) + b i ,i = 1, 2, . . . , n.ËÍÌ ¢ $ Ù Ðª”Š”˜”•K'’€ƒ)e•”}¤Œ)|&Ÿ|&}¤“””˜ x’K}¡ª¡¤Œ-|¡˜„‡}¤„‡¡Q„‡•”}'Sƒ=Œ-|¡‚ ¡$Š”ƒ)–Pƒ)•”˜”~Ξ tol (G, a, b)‚&€ƒ=“”)|«Œ-ªŠ&‚¤ƒ!’€˜‚¡}¤Œ)|&Ÿ«Œ!€ƒ-’K¡µ ‹…ƒ=~[Œ-˜Œ)|&ƒ=„‡†{•”ƒ)ŠK&‚¡ƒ=•Œ-|¡‚’K}¤&‘=¡|&ƒ-€KŸ¤Œ-|¡‚¡ƒ ´³ƒ)}¡Šƒ=„‡† Ú ¢Î¡¢¨§§„‡† }¤‰¡}¤Œ)•”¡ƒ.„ ËÍÌ ¢ $ Ù Ð¤ª”¡|&=„z)”‚¤˜”‚¤&€ƒ)•|&•”†œ›«ª”Š”ƒ)}¤‰¡Š¡‘)}¤‚¤&•˜”~eŒ‚¡†â’4ƒ-€”Ž”µ¨‹…˜„×ª”Šƒ!’K˜”¡|&}¤„Ø„‡•”}'Sƒ=Œ)|&‚¤§Q&˜ª”Š”˜Š”ƒ)–Pƒ)•”˜~8¢D(Q„‡ƒ)ƒ.„Ξ tol (G, a, b) = { x ∈ R n | (∀a ∈ a)(∃b ∈ b)( x = G(a, x) + b ) }= { x ∈ R n | (∀a ∈ a)(∃b ∈ b)( x 1 = G 1 (a, x) + b 1 &· · · &x n = G n (a, x) + b n ) }= { x ∈ R n | (∀a ∈ a)((∃b1 ∈ b 1 )(x 1 = G 1 (a, x) + b 1 ) &· · · &(∃b n ∈ b n )(x n = G n (a, x) + b n ) ) } ,’Kƒ[ª”}7Œ!€ƒ-’K•”ƒ)ƒ Š¡‚¡ƒ=•Œ-|&‚¤}¦’Kƒ)~Œ)|&‚¡˜|&ƒ-€Ÿ¤•”}×Ž”‚&€Ž”ƒ)|¡ŒŽÔŠ&‚¤ƒ)•Œ)|&‚¡}7„ˆª”}¤|&}¤„‡¡¤§Å« }¡|=’Kƒ)€Ÿ¡•”†„ž“€ƒ=•¡„}¡•3O8µ¨•””—”˜~€ $ ®¢µ |¡ŒŽ¯’KŠ”¡$Œ’KŠ”¡=}¤„œ¢(©‘)‚¤ƒ=Œ-|¡•”}§7“”|&}«|&}¤’4…„‡†õ˜„‡ƒ)ƒ.„Øª”Š¡‚¡}Äª”Š”}¡•}¤Œ-˜|&ŸDÅi ( G“”|&}’€”Ž&P’4}¡)} ‚‚¤†Š>Sƒ)•”˜”ƒi (a, x) +‚¡›¤}.’K˜”|c€˜”–…Ÿ¬}.’K•˜”‘b i )∃ Ä”‚¤¡•”|&}¤Š”†õŒ)¡7‹Pƒ=Œ)|&‚¡}¤‚¤&•˜ŽËÚ ¢¨§¡ÐŸžËÚ ¢ $7ÐW§¤„‡† „‡}'Pƒ=„²ª”Š”}¡š’K}&€«˜|&Ÿ•¡–…˜ ‚¤†€K>’K”˜Œ‚¡†â’Kƒ)€”Ž”µ ‹P˜„Öª”Š”ƒ-’K˜”&|¡}¤„ÖŒ!€Kƒ!’K¡µ ‹P˜„Ö}¤‰&šy¥}7Œ-ª”}¡€Ÿ¡‘=}¡‚7&‚¡–P˜Œ-Ÿz)”‚¤˜”‚¤&€ƒ)•”|¡•”}¤Œ)|=Ž„‡˜

‚¡ƒ=Š7›•˜„‡˜×˜Ø•”˜e•”˜„‡˜ }¤•”—¡„‡˜Øƒ.Œ-|&ƒ.Œ-|&‚¤ƒ)•”•†œ›¦˜”•”|¡ƒ)Š”‚7>€Ÿ¤•”†œ›¬ŠK¡Œ-–P˜šŠ”ƒ)•˜”~"G i x)¢yŒ)‚¡}¤µ¨}¡“”ƒ=Š”ƒ!’4Ÿ§&}7Œ-•”}¤‚¡†‚¤&ŽŒ-Ÿ$•QŒ-‚¤}¡~Œ)|&‚¤&›P&Š˜ÕQ„‡ƒ)|&˜”˜(a,Ξ αβ (F, a, b) ={x ∈ R n ∣ ∣ (∀a ∈ a)( (x1 ≥ G 1 (a, x) + b 1)&Þ7ß_€7ßa¥ä®ãWð…ä®ã)ã©å'á4ã!äœku>ä4ãYyVÝ³ä4ã!Ý”ä4ã!ñ”ä”éœê¦íÎVí=î'ã!ëÙ ± ÚŠ&‘=}¤„(x1 ≤ G 1 (a, x) + b 1)&· · · &(xn ≥ G n (a, x) + b n)&(xn ≤ G n (a, x) + b n) )}=={x ∈ R ∣ ( ) n (∀a ∈ a) x1 ≥ G 1 (a, x) + b 1 &{(∀a ∈ a) ( x 1 ≤ G 1 (a, x) + b 1)&· · · &(∀a ∈ a) ( x n ≥ G n (a, x) + b n)&(∀a ∈ a) ( x n ≤ G n (a, x) + b n) }x ∈ R n ∣ ∣ ( x 1 ≥ max a∈a G 1 (a, x) + b 1)&(x1 ≤ min a∈a G 1 (a, x) + b 1)&· · · &(xn ≥ max a∈a G n (a, x) + b n)&(xn ≤ min a∈a G n (a, x) + b n) } ,•”ƒ=Š&‚¤ƒ)•Œ)|&‚ ËÚ ¢Î.Ð!¢Š¤Œ!€˜ &S’K†~Ô˜”‘cª¡Š¡„‡ƒ=|&Š”}¡‚“”|&}¯Œ)}¡‚¡ªK'’4¡ƒ)|PŒ©Œ)˜Œ-|&ƒ.„‡}¡~a 1§a¢=¢)¢=§ 2§a‹P˜›ò˜•”|&ƒ)Š‚¤>€KŸ&š•”¡µ •”ƒ)}¤ª”Š”ƒ-’Kƒ-€=•”•”}7Œ-|&ŸK§V‚&›¤}.’4˜”| •ƒS‰¤}&€ƒ=ƒ«}.’K•”}¡=} ŠK&‘=‚ ª”ƒ=Š”‚¡}¤~¬Œ)|&ƒ)ª”ƒ=•”˜l§¤˜„‡ƒ=µG 1§G 2§¢)¢=¢)§G n§¤|&}¦’€”ŽÔ‚¤Œ-ƒ)›‚×&S’K}¡ƒ˜”‘[}¤„‡ª”}¤•”ƒ)•|&•”†œ›‚¤†Š>Sƒ)•”˜”~i = 1, 2, . . . , nmax G i(a,˜x) min G i(a, x)a∈a a∈aG i (a, x)˜G i (a, x)Œ-}¤‚¡ª>’4&µ¨|PŒ

ªŠ”ƒ!’4Œ)|>&‚¡˜|&Ÿ«z)|&˜[‘=•'“”ƒ)•˜Ž ‚«‚¡˜’Kƒ§Q'¡7›¤ƒ)ŠK”§„‡†{„‡}'Sƒ=„G i (dual a,˜x) G i (dual a, x).Ù ± ÛÜÝ/u£œkueÞß¬àâá®ãWäœku>ä®ã¨å¤æ¡å¡æ>ç«è!ä”ä”éœêë‡ä4å)ì¯ã.í)îœQï+ãWð«ã!ä”ñ|&}¡“” (©|>&8§ƒ.Œ!€˜ ªŠ”˜”•>’€ƒ-e˜”|’K}¡ª”¡7Œ-|&˜K„‡}¤„‡¡ „‡•”}'Pƒ.Œ-|&‚¤¡õŠ”ƒ=–…ƒ=•”˜”~˜xΞ tol (G, b)˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”}¤~ Œ)˜Œ-|&ƒ.„‡† ¡7ŠK&‚¡•ƒ)•”˜”~a,x = G(a, x) + Ë $¤Ðb' &ª”¡7Œ-|&˜K„‚KR n ˜|&ƒ)Š¡—”˜”}¤•”•”†~˜x ∈ G( dual a, ˜x ) + b.˜c‚¡†ª}&€•”ƒ=•”†Ö¡¤Œ-€}¡‚¤˜Ž’K}¡¡‘)†‚¤¡ƒ=„‡}¤~|¡ƒ)}¡Šƒ=„‡†Q§4¤Œ)¡µ ‹…˜ƒ=ŒŽ[‚&›7}'P’4ƒ)•”˜”~‚ Ë $7Ðâª&Š¤„‡ƒ)|&Š}¡‚¯ŒQ˜”•|&ƒ)Š”‚7>€Ÿ¤•”}¡~•”ƒ=}¡ª”Š”ƒ-’Kƒ-€K)•”•”}7Œ-|&Ÿ¤µ…§”|¡}§‚¯Œ)˜€¡ ËÍÌ ¢ $ Ù ÐŒ-ª”ŠK&‚¡ƒ-’€˜”‚¤}P‚¤€µ¨“”ƒ)•”˜ƒª”Š”}¤—”ƒ=Œ=Œ¨ª}«Õ©}¤Š„‡¡.€K¡„x (0) ← ËÍÌ ¢ $ Ú Ð˜x,x (k+1) ← G( dual a, x (k) ) + b, k = 0, 1, 2, . . . Ë:Ì ¢ $ Û Ð.Œ-}7„‚¡ƒ=”|&}¡Š† Œ-}.’Kƒ=Š7S&||&}¡“”¡ ¦©ƒ)|&Š¡)’4•”}×ª”}¡&‘.&|&Ÿ¦ª”}¬˜”•’K¡”—”˜˜8§Ó“”|&}×‚7Œ-ƒª”}¡Š”}'S’4¡ƒ=„‡†ƒz)|&˜K„ˆª”Š”}¤—”ƒ=Œ-šx (0) z)|¡} ‚¤ƒ)Š”•”} ª”}¤Œ)|&Š”}¤ƒ)•”˜”µP§V˜×ƒ=Œ!€K˜˜x ∈ x (k) ’K€”Ž¦•”ƒ=}¡|&}¤Š”}¡=} •”}7„‡ƒ)Š ˜x¢&…ƒ)~Œ)|&‚¡˜|&ƒ-€Ÿ¤•”}§’€”Ž §V|¡}[˜”‘-š‘= „‡}¡•}&šk‚ KRª”}«‚¡€µ¨“”ƒ)•˜”µ|&}¡••”}¤Œ)|&˜[&Š”˜ÕQ„‡ƒ=|&˜”“”ƒ.Œ-”˜”›}¡ª”ƒ=Š&—˜”~˜x ∈ G( dual a, ˜x ) + b ⊆ G( dual a, x (k) ) + b = x (k+1) .ó¤€ƒ!’4}¡‚¤¡|&ƒ-€KŸ¡•”}K§˜x• Œ)¤„‡}¤„’Kƒ-€ƒeªŠ”˜”•>’€ƒ-e˜”| ‚7Œ-ƒ=„x (k)§³7}¤|&}¡Š†ƒPª”}¡š‰¡†|&Ÿ¦ª”Š&‚¤˜€Ÿ¤•”†„‡˜˜”•|&ƒ)Š”‚7>€Ÿ¤•”†„‡˜ ‚¤ƒ)”|&}¤Š¡„‡˜³§a•ƒ)‚¡‘=˜šŠ>Žò•¬‚¡}¤‘=„‡}'e•”}¡ƒ•&€˜”“”˜”ƒ[‚×‚¤†Š>Sƒ)•”˜”˜ z)|&}7„‡¡¬’K}&€«•†G(dual a,•”ƒ)ª”ŠK&‚¡˜”€Ÿ¡•”†œ›x)¦Q&}¡•”ƒ=—8§â|&ƒ)}¤Š”ƒ=„ Â‹ Š!’Kƒ=Š}ž•”ƒ)ª}.’K‚¡˜e•”}¤~¨|&}¤“”ƒ ˜”•”|&ƒ=Š”‚¤&€}¡‚K¢ÂŒ”¢±¢Î.Ðª”Š”˜”‚¤}.’K˜”|¬•¡ŒÔ‚¡†‚¤}.’K¡Ø}×|&}¤„œ§¤“”|&}×ª”Š”˜Ô•&€}'Pƒ=•”•”†œ›•Ë|&ƒ)}¤Š”ƒ=„ G(dual a, x){ x (k) }Œ›7}.’K˜”|¡ŒŽ¨•ƒ)}¡|&}¤Š”}¤„‡¡òª”Š”ƒ-’Kƒ-€K¡>€KŸ¡•”}7„‡¡¦Š”ƒ)–Pƒ)•”˜µ¨Œ-˜Œ)|&ƒ=„‡†ò¡Š&‚¤•”ƒ)•”˜~ ¡¤Œ-€}¡‚¤˜Ž›žª”}7Œ!€ƒ-’K}¡‚7&|&ƒ)€Ÿ¡•}¤Œ-|¡Ÿ ¢ $$7ÐW¢ Ë:Ìx ∗ Ê©Š”˜z)|&}7„ª”Š˜”•'’K€ƒ-e•”}7Œ-|&Ÿ "Öƒ!’K˜•Œ-|&‚¤ƒ)•”•}¤„‡¡PÕ©}¤Š„˜x ∈ x (k)˜x ∈ limk→∞ x(k) = x ∗ .‚¡€ƒ)“)|€µ ‰¤}¡~|&}¡“”˜ ˜x˜”‘Q„‡•”}¡šPƒ.Œ-|&‚7PŠƒ)–…ƒ=•”˜”~ Ê©}¤Œ)7}¡€Ÿ¡¡P•¡–…ƒ©Š¤Œ)Œ-¡7S’Kƒ=•”˜”ƒ©Œ-ª”Š¡‚¡ƒ-’€˜”‚¤}Q’€”ŽΞ tol (G, b)§”|&}S’4ƒ)~Œ)|&‚¡˜”|¡ƒ-€Ÿ¤•”}a, Ξ tol (G, a, b) ⊆ x ∗¢

x ∗ ∈ IR n X3ø®÷£G¥C øû>þG¥CIH&û IR n ]Y9”÷£TU¢'û¦=KþDCI?ø@H¡û)ü4ý¡þ4ûRd#GR=/GRCL?ROx ∗ ∈ IR n X¦¢WüdÙ ±¡±Þ7ß_€7ßa¥ä®ãWð…ä®ã)ã©å'á4ã!äœku>ä4ãYyVÝ³ä4ã!Ý”ä4ã!ñ”ä”éœê¦íÎVí=î'ã!ë„‡•”}'Pƒ.Œ-|&‚7 Šƒ)–…ƒ)š•”˜”~ ó¥}¡‚¡ƒ=Š”–…ƒ=•”•”}[Œ›¤}.’4•”†„¨}¤‰¡Š¡‘)}¤„Ô’€”Ž×¡7ªŠ&‚¡€”Ž”ƒ=„‡}¤)}Ξ ctrl (G, a, b) := { x ∈ R n | (∀b ∈ b)(∃a ∈ a)( x = G(a, x) + b ) },Œ-˜Œ)|&ƒ=„‡† Ë $7Ðâ}¡‰¡}7Œ-•”}¤‚¡†‚7&ƒ)|¤ŒŽ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•}¡~$7Ð×þ/?R DJ-G¥CS÷XI=XIø”û>H)üd@?ú¦÷!? ú¦þ÷EAF?G¥CIH>÷Óøc?¥QI?-þ/=DS)+*-,/.0*-132546…„6_^BàJGRCPýnJ=þ/CL?Iø@H&û)ü4ý¤þ÷!SÈG¥=/GRCL?ú¦ÿ˜J-øû>H&þ/?-þ/=DS?RGRCL?G¥CIH>?-þKþ÷£TW÷¦=þ/CL?Iø@H&û)ü4ý¤þ÷£TW÷ø”û>GRQa=&ø@?-þ/=-dG : KR n → KR n K÷I >ø”û>H£=7üJXo¢!?¥SG¥CIH£J}hzI?R?¦Ξ ctrl (G, a, b)G(a, x) H>ÿ4øû£AF?-þ/=-d G(a, x)÷!CP÷!~ùøû£AF?-þ/=/?x ↦→ G( a, x ),P KR n &ø4÷£G¥C ø”û>þ/GRCLH&û ]\9÷£TU¢'ûÀG¥=G¥CI?ú¦ûLJ!ø”û>H>þ?-þD=/S?RGRCPýx = G( a, x ) + dual G¥A{û£CL=/?3 O b=7úZ?R?¥C?¢=Kþ/GRCLH>?)þþK÷!?• >ø”û>H£=ü4ý¡þK÷!? öÓ÷ùø7ú¦û=ü4ý¡þ÷£?Pøc?¥QI?-þD=?CP÷ùø®÷!?\d@H=üd@?¥CIGÎdxH>þ/?RQPþ?¥Sx=þ/CL?Iø@H&û)ü4ý¤þ÷!S÷Ee?-þ/€KŒŽS}¡•S’€Ž¯€K¡Œ=Œ-˜”“ƒ=Œ-}¤~S˜•”|&ƒ)Š‚¤>€KŸ¡•”}¤~¯&Š”˜ÕQ„‡ƒ=|&˜””˜³§¡‚…Œ)}¡‚¡ƒ=Š”–…ƒ=•”•”})+*-,/.0*-132546…„6G(a, x)>ø=þ/?¥ÿ4ø”û£OG : IR n → IR n X’KŠ”¡)˜”›S|&ƒ=Š„‡˜”•&›V§¤ẽ‰¤ƒ)‘˜Œ)ª”}&€KŸ¡‘)}¤‚¤¡•”˜ŽPª}¡•Ž”|&˜”ŽÕ©}¡Š„ &€Ÿ¡•”}¤)}©Š”ƒ=–…ƒ)•˜Žƒ%Ë•¡–P ª”ƒ)Š”ƒ.Õ©}¡Š„‡¡.€˜Š”}¡‚¤¯Š”ƒ)‘=¡=€KŸ>|'¡|&}¡‚ ´8ƒ=}¡Š”ƒ.„‡† Ù ˜ó¤€ƒ!’®Œ-|&‚¤˜Ž Û ˜”‘¦°7€K&‚¤† Î¤Îˆ”•”˜”=ĩÏÎ ®ÐàJG¥CSý–¢!üdŽ?RGRCL?G¥CIH>?-þKþ÷£TW÷i=þDCI?øcH&û)ü4ý¤þ÷£TW÷øûG¥QL=>ø@?)þD=kdAF?)þD=-d÷!CP÷!~ùøû£AF?-þ/=/?¢!?¥S/GRCLH>J}hzI?R?¦ K÷L &øûH£=7üJx ↦→ G(a, x),d@H)üd@?¥CIGÎdú¦ûIJ-øû>H&þ/?-þ/=DSx = G(a, x)ËÍÌ ¢ $7±¡Ð=7úZ?R?¥C?¢=Kþ/GRCLH>?)þþK÷!?# >ø”û>H£=ü4ý¡þ÷£? öÓ÷ùø7ú¦û)ü4ý¤þ÷£?Pø@?RQL?)þD=?&øû>H?R¢!ü=/H>÷LG-÷'÷!CPþK÷!QL?)þD=/?

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mid A · Ù Œ$˜€˜⎧⎨⎩mid A · x − mid b ≤ ( rad A ∃ − rad A ∀ )·|x| + ( rad b ∃ − rad b ∀ ),−mid A · x + mid b ≤ ( rad A ∃ − rad A ∀ )·|x| + ( rad b ∃ − rad b ∀ ),s,Çß s¤ßZǹuí=íÎœ¤ä”éñ#w®ãï³ã=æ¡å-ïVä”éñru&Ý±'å)ï@Vî=ë⎧N‡£É{ sgn x 1 , . . . , sgn x n } · | x | − ( rad A ∃ − rad A ∀ ) · | x |“”|&}ez=”‚¡˜”‚7>€ƒ=•”|&•”}«Œ-˜KŒ-|&ƒ.„‡ƒ⎪⎨≤ mid b + ( rad b ∃ − rad b ∀ ),⎪⎩−mid A · ≤ −mid b + ( rad b ∃ − rad b ∀ ).N‡£É{ sgn x 1 , . . . , sgn x n } · | x | − ( rad A ∃ − rad A ∀ ) · | x |“¤Œ-|&˜ ª”}&€K¡7“ƒ)•”•”†œ›¯•ƒ)Š¡‚¡ƒ)•KŒ-|&‚«„‡}'e•”}P¡ª”Š}¤Œ-|¡˜”|&ŸS˜¯’4&€ƒ)ƒ¤§‚¤Œ-ª}&š„‡•”˜”‚e}¤ª”Š”ƒ!’4ƒ-€ƒ=•”˜”ƒ$›7¡Š&”|¡ƒ)Š”˜Œ)|&˜”“”ƒ.Œ-}¡=}P‚¤ƒ)”|¡}¡Š Ê©Š&‚¤†ƒbc%mid b + ( rad b ∃ − rad b ∀ ) = b c ,−mid b + ( rad b ∃ − rad b ∀ ) = (−b c ).yƒ.Œ-|&Ÿ‚¡ƒ)|&}¡Š¡‰¤Œ)}&€µ¨|&•”†œ›×‘)•'“”ƒ=•”˜”~ }7„‡ª”}¡•”ƒ=•”| x§‡|'¢ƒ¤¢yÊ©¡¤Œ-|¡Ÿi =| x i i = 1, 2, . . . , n§”˜|§S = diag { s 1 , s 2 , . . . , s n }, s i = sgn x i = ±1,x ∈ Ξ αβ (A, b) ∩ O¡|&Š”˜”—K”§‡}¡‰¤Š&‘=}¡‚¤¡•”•&Ž¦‘)•K&¡„‡˜ }7„‡ª”}¡•”ƒ=•”|‚¤•”¡|&Š”ƒ=•š•”˜›Ø|&}¤“”ƒ) ŠK¡Œ)Œ=„ &|&Š˜”‚¤¡ƒ=„‡}¡=}[}¡Š”|>¡•”|> "’K˜K&)}¤•>€KŸ¡•&ŽØ„ §a|'¢ƒ¤¢ O x =’K€”ŽSy x O¢‡Ê©Š”˜z)|&}7„¡7Œ!€}¤‚¡˜”ƒ∈¡ƒ=|y ∈ R n§K¡)’4}¡‚&€Kƒ)|&‚¤}&šŠ7Ž”µ ‹…˜~•ƒ)Š¡‚¡ƒ)•KŒ-|&‚7¡„ ‚¡†ª”}¡€•”ƒ=•”}e|&}¡’®e˜c|&}¡€Ÿ¡}e|&}¤’4”§K7}¤’4eŒ-¡‹Pƒ=Œ-|¡‚

$&%('¡)Ax = b¢(©•”|&ƒ=Š”‚¤&€Ÿ¡•&Ž¯€˜”•ƒ)~”•&ŽŒ)˜Œ-|&ƒ.„$,+%¡'¡)}¤—”ƒ)• ¨ª”|&˜„ Œ)•”˜”‘=¡¯ª”} 7}¤}¡Š¤’K˜”•K&|&ƒ&€Ÿ¡•K>Ž y νšJ}¤~y ← + ∞ 8:ª”Š”}¡=Š¡„ „‡˜Š”}¡‚7&•”˜ŽŠ”ƒ=–…˜”|&Ÿe‘.'’4'“¡«€˜”•”ƒ=~”•”}¡=}ª ¢ §¤ÐÙžËÎ ª ¢ $¤ÐW§‚¡†“”˜Œ-€”ŽŽz ← min{ xËÎν | x ∈ Ξ αβ ∩ O i 8 }ÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ãÙ Œ Ùüû*Ê\õ(¥ø¨(¤¨§© p ¨²öÿ¡¤¨§©Võ¥÷ËUÿ(ø¨ôcõRö÷-øùõnüãûp EøõR÷¦¡¤-ÿËUÿL¦i£¥¤-ø¨#|(¤iu¤k©QÌUõhõÍ-ø|-÷#Îjs‚Ï‰Ð_Ñÿ¦Q&|&¡Š&€Ÿ¡•†~˜”•’4ƒ)Œν ∈ { 1, 2, . . . , n }¢’€”Ž „‡•”}'Pƒ.Œ-|&‚7PŠƒ)–…ƒ=•”˜”~Ξ αβ (A, b)¢i = 0 E2 n − 1-,.0/1'32¨46579;:CB;:6DË y > zÐy ← z?BAF¨9q93:

y ≥ 0.ËÎ ª ¢¨§¡Ðc ⊤ y, c ⊤ = (0, . . . , 0, s ν , 0, . . . , 0) ∈ R n . ËÎ ª ¢ $¤Ð&ø4÷£H>?¢E?)þD=/=ü}h~.÷£TW÷IGRH>÷£?TW÷Z=þKöÓ÷ùø7ú¦û£e/=÷'þþK÷£TW÷aH>ÿg;=Gü@?-þD=kd — QSûT!û²P÷'þ>ø@=x÷Wü4ý!j>J?RC:=þKöÓ÷ùø7ú¦û£e/=D}+X0 ÷üJ;?-þKþDJ}{þKûIGH>÷!=k¤W >ø@?¢.ÿq¢Jza=¤mH>ÿg;=/GROþ/?3=G¥’ü@TW÷ùø=/C©úÖþKûj&ÿqH&û>?RCLGd u¥yu>wVîRœ¤ä”éë X?RGü=À &ø=xH>ÿg ÷Wü4þ/?-þ/=D=…ü}h~'÷£TW÷ZGRH&÷EO⎧⎪⎨(s,Çß s¤ßZǹuí=íÎœ¤ä”éñ#w®ãï³ã=æ¡å-ïVä”éñru&Ý±'å)ï@Vî=ëmid A · S − ( rad A ∃ − rad A ∀ )−mid A · S − ( rad A ∃ − rad A ∀ ))y≤(bc(−b c )),Ù Œ Ú⎪⎩Š”ƒ)–Pƒ)•”˜ƒ=„[‘.'’4'“”˜¦€˜”•”ƒ=~”•”}¡=}¨ªŠ”}¡=Š¡„ „‡˜”Š}¡‚¤¡•”˜Ž¦Œ¥}¡)ŠK&•”˜”“ƒ)•”˜Ž„‡˜ ËÎ ª ¢¨§¡Ðó¤€ƒ!’K}¤‚¤¡|&ƒ-€Ÿ¤•”}§â‘)•K'“”ƒ)•”˜ƒmin{ x ν| x ∈ Ξ αβ (A, b) ∩ O }Ž”‚&€”Ž”ƒ=|¡ŒŽ˜[„‡˜”•”˜„‡˜”‘=˜”Š”¡¤ƒ=„‡†„òÕ©¡7•”—”˜”}¤•>€}7„§Q>S’K†~ }.’4•”}¡‘=•'“”•”}[‘.'’4¡)|¡ŒŽ }¤Š”|>&•”|¦ª”Š”}¤Œ)|&Š¡•Œ-|&‚7 Rnª”}7Œ!€ƒ-’K}¡‚7>š|&ƒ-€KŸ¡•”}7Œ-|&Ÿ¤µ‘)•K&}¡‚}¤„‡ª}¡•”ƒ=•”| Œ-‚¤}¡˜›©‚¡•¡7|¡Š”ƒ)•”•˜›Q|&}¡“”ƒ=8¢£¦Q¡„‰ ¡=’Kƒ=| ¡=’K}¤‰&š“˜Œ!€K¤„‡˜}¤| 0’K}2 n − 1§Œ)}¡ª”}¡š˜Œ)}¤„ >Žž}¡ª”|&˜K„ >€Ÿ¤•>Ž Ë|&}¤“”•>ŽKÐS}¤—”ƒ)•¦‘)•K'“”ƒ)•”˜”Žmin{ x(©|>&8§νΞ αβ (A, b)„‡}'Pƒ=|‰¡†|¡Ÿ•¡~’Kƒ)•KŒ…ª}¤„‡}¡‹PŸ¡µ &€)}¤Š”˜”|¡„ §®ªŒ-ƒ=‚>’K}¤7}.’}¡š}| x ∈•”}S‘=&•¡„‡ƒ=Š”}¡‚7&|&ŸP‚¤Œ)ƒ }¡Š|'¡•”|&†R n —”ƒ)€†„‡ñšJ‘=•'“”•”}¤ƒ’K‚¤}¡˜”“”•}¡ƒ “˜Œ!€}K§}¡|&}¤Š”}¡ƒ$ª”}&€K¡7“K&ƒ)|¤ŒŽ˜”‘$•&‰¤}¡ŠQ‘=•&}¤‚…}¤„‡ª”}¤•”ƒ)•|…‘=¤„‡ƒ)•”}¤~„‡˜”•”¡7Œ)P•”¡.€ƒ=„œ§”…ª€µ$Œ)Pƒ!’K˜”•˜”—”ƒ)~³¢Œ-|>&‚¤˜”‚…&P’4}¤„‡¡e˜”‘©•”˜›|&}¡Š}¡)}Sª”Š”˜‚¡ƒ!’4)• ‚e´³&‰¡€³¢8Î ª ¢Î¡¢ÒÔÓ .¥*Ôq*†0*-Ö¥×¥*€7XÕƒ6Ù7/6Ù7B±7§VÎ Ú!® ’ü@TW÷ùø=/C©ú þ4ûj&ÿqH&û>?¥CIGÎd w/uí=íÎœ¤ä”éë Xq?RGü=ü@?-þ/=-d¤@]?RTW÷€=þKöÓ÷ùø7ú¦û£e/=÷'þþK÷!TW÷iH>ÿg;=Gü@?-þD=kdÔ÷'þBHiCS÷!SÑ=7ü=–=Kþ÷!S¨öÓ÷ùø7ún?W=/G¥ K÷ü4ýOj£J-?¥C=KþöÓ÷ùø7ú¦û>eD=D}²÷# &ø@?R¢EQI?RGRCLH£J}hza=¤€=þKöÓ÷ùø7ú¦û£e/=÷'þþKÿp¤H>ÿg;=Gü@?-þ/=-d¤QSû>T!ûE¤>] —ª¤Œ)Œ-˜‚¡•”†ƒ>€=}¡Š”˜”|¤„‡†:"¼z)|&}Ø&€)}¤Š”˜”|¡„‡†wŒS=Œ)|&}¦‘.'’4¡•”•”}¡~(©|'¡8§„‡}¡~c‚¤†“”˜Œ!€Kƒ)•”˜”~³§4}¤|&}¡Š&Žc•”ƒ«‘=¡‚¡˜Œ)˜”|¯}¡| ˜”•’K˜‚¡˜’K¡7>€Ÿ¤•”†œ›[}¤Œ)}&š‰¡ƒ=•”•”}¤Œ)|&ƒ)~[Šƒ)–P¡ƒ=„‡}¤~[‘='’®'“”˜8¢/¦Q¡ª”Š”}¤|&˜”‚§4'’4¡ª”|&˜”‚¤•”†ƒP&€)}¤Š”˜”|¡„‡†Öª}¡‘)‚¤}&šª”Š”}¤)Š¤„|Ø=˜”‰¡}Øª”}.’®Œ-|&Š¡˜”‚¤¡|&ŸØª”Š”}¤—”ƒ=Œ=Œ Š”ƒ)–Pƒ)•”˜”ŽÔª}.’&P’4¡7µ }¤•””Š”ƒ=|&•”¡µ‘=>’4'“”¡§7Qª”}¤|&}¤„‡¡¤§7ªŠ”˜¯ª”Š”}¤“”˜›eŠ¡‚¡•”†œ›«¡¤Œ-€}¡‚¤˜Ž›V§¡}¡•˜8§¤•ƒ=Œ-}7„‡•”ƒ)••”}§7‰¡}¡€ƒ)ƒ€”Ž”µ‚×‚¡†“˜Œ!€˜|&ƒ-€Ÿ¤•”}¡~Ôª”ŠK&”|&˜7ƒ¤¢o¦©}§¤¡ž‚¤˜’K˜„œ§¥ª”Š”ƒ-’šŒ-|>&‚¡€ƒ)••”†~c‚z)|&}7„ª&ŠK&)ŠK¡Õ©ƒ¦ª”Š}¤Œ-|¡ƒ)~”–P˜”~¦„‡ƒ=|&}.’¬}¤ª”|&˜„ &€Ÿ¡•”}¤)} ‚¡•”ƒ=–¦šª”Š”ƒ-’Kª”}¡“|&˜”|&ƒ)€Ÿ¡•”†}¡—”ƒ=•”˜”‚¤¡•”˜Ž×„‡•”}'Pƒ.Œ-|&‚[Š”ƒ=–…ƒ=•”˜”~(…óY¡n¢0£ÅŽ”‚&€”Ž”ƒ=|¡ŒŽ¦ªK¡Œ)Œ)˜”‚¡•†„ ˜×‚z)|&}7„‘.&€Kµ “¡ƒ)|¡ŒŽ}.’K˜”•˜”‘Qƒ)=}SW€4&‚¡•†œ›¯•”ƒ!’K}7Œ-|>&|¡7}¤‚¢•”ƒ)=}

»+¨»œÈ°× Âx¸P¸¥»¦Â¿ Ö&|&Š”˜—”† ˜{–…˜”Š˜”•‚¡ƒ=”|&}¡ŠK ª”Š&‚¤}¡~×“¤Œ-|&˜Œ)|&Š”ƒ.„aŽ”|¡ŒŽ Ø•”¡.€µ…§‡|&}˜Øƒ)„‡•”}'Sƒ=Œ)|&‚¡}cŠƒ)–…ƒ=•”˜”~Š¤Œ!€K˜õ‚ò˜”•”|&ƒ=Š”‚¤&€Ÿ¡•}¡~€˜•”ƒ)~”•}¡~õŒ)˜Œ-|¡ƒ=„‡ƒ×–…˜”Š˜”•ò„$•KŒ)Œ-}¤•”}¤„ ‚[Š&‰¤}¡|&ƒr $$ ®§³˜×ƒ=)} }¡ª”˜KŒ)&•˜”µ „‡†Åª”}¤Œ)‚&Ž”‹P¡ƒ=„•K¡Œ-|¡}'Ž”‹…˜”~*…¢p$•Œ)Œ)}¡•P}¡ª”˜Š&ƒ=|¡ŒŽ •«Œ-€ƒ!’K¡µ¨‹…¡µÊ©}.’›¤}.’WË>€KŸ>|&ƒ)Š”•K&|&˜”‚7L $•KŒ)Œ-}¤•¤Ð)+*-,/.0*-1327XÕ6_^ƒ687G¥=G¥CI?úQÿ¨ü=þ?¥SþKÿp¤ û=ü@TU?¥~ùøû>=D;?RGRþ/?ROàJG¥CSýi¢WüdÂ=KþDCI?øcH&û)ü4ý¡þK÷!SAx = b÷!~Eà>?¢E=þ/?)þþK÷!?œúQþK÷EAF?RGRCLH&÷¤øc?¥QI?-þ/=DSø@=DeKûA fΞ(A, b)þ/?R DJGRCP÷>]o9÷£TU¢'ûú¦û£C ø=/eKûA fþK÷!GRCa=þ/?-÷£T:ø”û>þD=/;?-þ4û!X‡ûÀ=þDCI?øcH&û)ü4ý¤þKûRd$•KŒ)Œ-}¤•Ž”‚¡€”Ž”ƒ)|¤ŒŽ¦|&} }¤‰¤Œ-|¡}&š¦©ƒ=„‡ƒ!’K€ƒ)•”•†„cŒ!€ƒ-’4Œ-|¡‚¡˜”ƒ.„c>€Ÿ>|¡ƒ)Š”•¡|&˜”‚¤†Œ¹‰È¯¸²´Ù Œ ÛÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã‚¡•”¡|&Š˜c}.’K•”}¤)}¯}¤ª”Š”ƒ!’4ƒ-€=•”•”}¡=}¯}¡Š|'¡•”|>eª”Š”}7Œ-|&ŠK&•Œ)|&‚¤ ¡|>šŒ-|.Ž”)˜‚¤&ƒ=|¡ŒŽÔ‚|&}¤“””¡§¥}¡|&}¤Š>Ž‚Œ-€¡“&ƒc}¤‰¡‹Pƒ))}Øª}&€}'Sƒ)•”˜ŽÔ•&›¤}.’K˜”|¤ŒŽR n¢/Š¤Œ-€˜cSƒP„|S•”ƒ=•”¡.€ƒ)‚¤¡7µõ–…˜”Š˜”•”¡§|&}«„‡•”}'Sƒ=Œ-|¡‚¡}…Šƒ)–…ƒ=•”˜”~„‡}'Pƒ=|¦ª”ƒ=Š”ƒ=Œ)ƒ)&|¡Ÿ×Œ)Š&‘=¡ •”ƒ.Œ-}&€Ÿ¤}¦}¤Š”|>&•”|&}¤‚¦ªŠ”}¤Œ)|&Š&•KŒ-|&‚7”§‡›¤}¤|.Žžª”}¡€”šŠ”˜”—…˜ª”ŠK&‚¤&Ž¯“¡Œ)|&Ÿ…˜K„‡ƒ)µŒ!€˜–…}¤„¦‰¡}¡€Ÿ¡–P˜„œ§>)}¤Š&‘-’K}¦„‡ƒ)•”Ÿ¤–…˜„}¡‰¤‹…ƒ)=}P“˜Œ!€KS}¤Š”|>&•”|&}¤‚ •”}¡ƒ˜”›P}&€˜“”ƒ=Œ)|&‚¡}¦„‡}'Sƒ)|©‰¡†|&ŸQ•ƒ2 n¢z=|&}¡~Œ-˜|&¡¤¡—”˜”˜Ô‰¡}¡€Ÿ¡–P¡7µ “¤Œ-|&ŸØ‚¤Š”ƒ=„‡ƒ=•”˜¨Œ-‚¤}¡ƒ=~žŠ¡‰¡}¤|&†w&€)}¤Š”˜”|¡„ª”Š”ƒ-’K†â’K¡‹…ƒ=)}ªK&Š¡)Š¤ÕQ×ª”}¡|&ŠK&|&˜”|ò•ª”Š”}¡‚¤ƒ)Š”¡ò|¡}¡)}K§œ“”|&}Ø’K€”Ž ª”}¤“”|&˜y˜›¦ª”ƒ)Š”ƒ.Œ-ƒ=“”ƒ)•”˜ƒ«Œ«„‡•”}'Sƒ=Œ)|&‚¡}7„Š”ƒ)–Pƒ)•”˜~¬ª¡¤Œ-|¡}¢Vyâ}¡‘)•˜”>šƒ)| ƒ=Œ-|¡ƒ=Œ-|¡‚¡ƒ)••”†~¦‚¤}¡ª”Š}¤Œ>%4•”ƒ)€Ÿ¡‘)Žc€˜¦„‡}.’4˜Õ©˜”—”˜”Š}¡‚¤¡|&Ÿª}.’›¤}.’ ô.Î ª ¢Î…|>¡8§‚¤Œ)ƒ-›c}¡Š|'¡•”|&}¤‚“”|&}¤‰¡† }¡•[˜Œ=Œ!€ƒ-’K}¡‚7>€[‚¯|&}¤“”•”}7Œ-|&˜[€K˜”–…Ÿ¯|¡ƒ…}¡Š|'¡•”|&†R n§4}¡|&}¤Š”†ƒ¦˜„‡ƒ=µ |•”ƒ)ª¡¤Œ-|¡}¡ƒ ªƒ)Š”ƒ.Œ-ƒ)“ƒ)•”˜”ƒ$Œ¨„‡•}'Pƒ=Œ)|&‚¡}7„×Š”ƒ)–Pƒ)•”˜~/Á©´³¡7}¤~„‡ƒ)|¡}.’‰¡†œ€ Š&‘=‚¡˜”|ª&ŠK&)ŠK¡Õ«¢þD=/SH>?ø®þ÷S÷£¢=þK÷V=WCS÷ü4ý?Iø/AÈ¢E?)þD=/S–ØÐ ú¦þ÷EAF?G¥CIH>÷Ξ(A, b)€=}¡Š”˜”|¤„œ§a‚¤†“”˜Œ-€”Ž”µ ‹P˜”~}¤ª”|&˜„ &€Ÿ¡•”†ƒ )Š¡•”˜”—”†•”ƒ)•˜”ƒ=„²&Ô}¡=Š&•”˜“”ƒ)•”•†œ›®§}¡&‘=†‚¤¡ƒ)|¡ŒŽ×ª”Š”˜”=}.’K•”†„{’K€”ŽžŠ”ƒ=–…ƒ=•”˜Ž ‚¡•ƒ)–…•ƒ)~‘.'’4'“˜×’€”Ž¤(…óY¡n¢0£|&}&€KŸ¡}¦}.’K•”}¤~e&}¡~š•”˜”‰¤¡=’KŸQ}¤„‡ª”}¤•”ƒ)•”|¡†¨Œ-‚&Ž”‘=•”}¤Œ)|&˜„‡•”}'Sƒ=Œ)|&‚¤QŠƒ)–…ƒ=•”˜”~&|&Š”˜—”ƒ)~8¢&¦€”Ž‰¤}&€ƒ=ƒ$Õ©}¤Š„ >€KŸ¡•”}¤)}…}¤ª”˜Œ=&•”˜Ž „‡ƒ)|&}.’®«‚¡‚¡ƒ-’K=„•”ƒ)}7Œ-}¤‰¡ƒ)••”}¡~„

Ax = bþKûEj>ÿqH&û?¥CIGÎdiT:ø”û>ögXÖ »+¨»œÈ ·]e¿ÝÜ¨»QÂ ¿ ¹‰È¯¸Îá¨ƒ)|&|.€˜”šÊ©Š¡)ƒ=Š¦}¤‰O8ƒ-’K˜”•”=•”•”}¡=}P„‡•”}'Sƒ=Œ)|&‚¤yœŒ-ª”}7„‡•”˜„Ø›&Š¡”|&ƒ=Š”˜”‘=¡—”˜”µÏs,Çß î”ßÚ¦ã'î'åyc3‰u#Û¤å'ä/u Ù Œ7±ÒÔÓ .¥*Ôq*†0*-Ö¥×¥*€7XÕƒ6Ø^c6Ù7ò°³Š¡Õ©}7„ò„‡•}'Pƒ=Œ)|&‚¤¯Š”ƒ)–Pƒ)•”˜”~Â=þ/CL?Iø@H&û=ü4ý¡þ÷!SiG¥=/OG¥CI?úQÿü=Kþ/?¥SKþÿp¤¦û)ü@TU?R~ø”û£=/;?RG¥H>þ?-þ/=DSúQþK÷EAF?RGRCLH&÷ú H>?IøQL=þGV := { s ∈ {−1, 1} n | Ξ uni (A, b) ∩ O(s) ≠ ∅ }úQþK÷EAF?RGRCLH&÷ú ø@?R~£?ø={E := (s, t)∣s, t ∈ V s = tø”ûj)ü=/;”û£}hCIGÎdø4÷£H>þK÷ÀH«÷!¢.þK÷ES€Ž¯z)Œ)|&Š”ƒ=„ Š”ƒ=–…ƒ)•˜Ž8§7„‡}'Sƒ)|¦‰¤†|&Ÿ…ªƒ)Š”ƒ=ª”˜Œ)¡•¦‚SŒ!€ƒ)š’K¡µ ‹Pƒ=„žz)”‚¤˜”‚¤&€ƒ)•”|¡•”}¤„‚¡˜’Kƒ }¡ª”Šƒ!’Kƒ)€”Ž”µ &€Ÿ¡•†ƒSgn ( (mid A)·x − mid b ) · ∣∣ ∣(mid A)·x − mid b = rad A·Sgn(x)·x + rad b,˜€˜Sgn(y) · ∣∣ (mid A)·x − mid b∣ ∣ = rad A·Sgn(x)·x + rad b,y = (mid A)·x − mid b¢¥£ “”˜”|¡†‚¤>Ž8§+“”|&}„ ¡|&Š”˜”—† ‘=•&}¤‚Sgn(x)˜’KƒSgn(y)˜’Kƒ=„‡ª”}¤|&ƒ)•|&•”†Q§ª}&€¡“”˜„ž}¡|¤Œ-µâ’4§“”|&}(mid A)·x − mid b = Sgn(y)·rad A·Sgn(x)·x + Sgn(y)·rad b,

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}¤„‡ª¡”|&ƒ(V 1§V¢)¢)¢=§V ν−1§V ν+1§³¢)¢)¢=§V n )¢/Ü[†}¡—”ƒ=•”˜”‚7&•”˜”˜min x 1„‡†õ˜„‡ƒ=ƒ=„, ËÎ ª ¢ÏÎ¤Î.Ð—”˜”˜8¢/(Q„‡ƒ)•”•”}K§min{ x ν | x ∈ Ξ αβ }{ ∣ ∣∣= min x ν x ∈⋃l∩V ≠∅}(Ξ αβ ∩ l)Ù Ì ÙÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã{= min min{ x ν | x ∈ Ξ αβ ∩ l(r)} ∣ ª ¢ÏÎ ª Ð ËÎr ∈ (V 1§³¢)¢)¢=§V ν−1§V ν+1§³¢)¢)¢=§V n )= min { Ω(r) | r ∈ (V 1§V¢=¢)¢=§V ν−1§V ν+1§8¢=¢)¢=§V n ) } ,}|'¢ƒ¤¢³•&›¤}'S’Kƒ)•˜”ƒ νšJ}¤~¬}¤}¡Š¤’K˜”•¡|&•”}¤~¦}¤—”ƒ)•”˜¬|¡}¡“”ƒ= „‡•”}'Pƒ.Œ-|&‚7 Šƒ)–…ƒ=•”˜”~Œ-‚¤}.’K˜”|¡ŒŽ× ‘=>’4'“”ƒ¯„‡˜”•˜„‡˜”‘=¡—”˜”˜ž—”ƒ-€ƒ=‚¡}¤~ Õ©¡7•”—”˜”˜Ω(r)•[•”ƒ=}¡|&}¤Š”}¤„‚¡†“”˜Œ-š€”Ž”µ |¤ŒŽž‘)•K'“”ƒ)•”˜”Ž ¡7Pƒ…‚¡˜’Kƒ)€˜8§K¡ ‰¤¡)’4|&}Ω(r)§¤|'¡ž“|&}Ø„‡}'Pƒ)|×Œ!€K}'«˜|&Ÿ¤ŒŽ‚¤ª”ƒ)“K&|.€ƒ)•˜”ƒ¡§’4>€Kƒ)ƒ¦‚¤•”ƒ)–P•ŽŽ[‘='’®'“«¡¤Œ)ª”ƒ)–P•”}¯Š”ƒ=–P&ƒ=|¡ŒŽ[ª”Š”˜„‡ƒ=•”ƒ)•˜”ƒ=„¡7}¤)}¡š:€˜‰¡}«˜”‘([1, 2][−3[−34 , ]124 , 1 2[1, 2]])x =([−1, 1][−1, 1])›¤}¤Š”}¡–P}…Š¡‘)Š¡‰¡}¡|>¡•”•”†œ›¯„‡ƒ)|&}.’4}¡‚SW€}¡‰7>€Ÿ¤•”}¡~¯}¤ª”|&˜„‡˜‘=&—˜”˜8¢D¦©}SŠ¡Œ=Œ)„ ¡|>šŠ”˜”‚7&ƒ=„ &Žc•¡„‡˜—”ƒ-€ƒ=‚¤&ŽÕ©¡7•”—”˜ŽΩ(r)}¤‰&€K>’4&ƒ=|¯•”ƒ)ªŠ”˜Ž”|&•”}¤~}7Œ-}¡‰¤ƒ)•”•}&šŒ-|&Ÿ¤µZ%V‚ }¡‰¤‹…ƒ.„ Œ!€K¡7“K&ƒS}¡••”ƒSŽ”‚&€Ž”ƒ)|¡ŒŽ’4>SƒS•ƒ)ª”Š”ƒ=Š”†‚¡•}¡~8¢ƒ¦Q¡ª”Š”˜„‡ƒ=Š8§’€”Ž ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•}¡~¯€K˜”•”ƒ)~•”}¡~[Œ-˜KŒ-|&ƒ.„‡†r = x˜2lim Ω(r) = −2 ≠ 4r→2−0 3 = lim Ω(r)r→2+0Ëƒ=[}¡‰O³ƒ!’K˜”•)•”•”}¤ƒ„‡•}'Pƒ=Œ)|&‚¡}×Š”ƒ=–…ƒ=•”˜”~Ô˜”‘)}¤‰¡Š&Pƒ=•”}¦•Km¨¤˜Œ&¢¨Î ª ¢Î.Ð-§‡ª”Š”˜Ãâ}&€ƒ=ƒ$|¡}¡)}K§ Ω(r)‚e}¤‰¡‹…ƒ.„Œ-€¡“&ƒQª”Šƒ)|&ƒ=Š”ª”ƒ)‚7&ƒ=|Sƒ=‹…Q˜ Š¡‘)Š”†‚¤† ‚¡|¡}¡Š”}¤)}Š”}.’4 Ë‚+ ∞Ð-¢£¥|&˜×}¡‰¤Œ)|&}'Ž”|&ƒ)€Ÿ¤Œ)|&‚¤ Š”ƒ)–S&µ¨‹…˜„ }¡‰¡ŠK&‘)}7„¨Œ-¡7e˜”‚7&µ |•¡‰¡}¡Š¬>€=}¡Š”˜|>šËÎ ª ¢ÏÎ ª ÐW¢>¦Q¡ª”Š”˜„‡ƒ=Š8§ª”}¡ª¡=€Ž”Š”•”†ƒ „‡ƒ)|¡}.’K† •ƒ)Š¡‚¡•”}7„‡ƒ)Š”•”†œ›{ª”}¡Š”†|&˜”~ ˜”‘ÑÚ §PÎ Ù£® }¤“”ƒ)‚¤˜’K•”†„„‡}¡‚$W€}¡‰7>€Ÿ¤•”}¡~…}¡ª”|&˜K„‡˜”‘=¡—”˜”˜8§>ª”Š˜„‡ƒ)•”˜K„‡†œ›PS‘='’®'“”ƒËÎ ª ¢ÏÎ ª Ð§8|>&×&¬}¡•˜×Œ)¡‹…ƒ=Œ)|&‚¡ƒ=•”•”}}¤Œ)•”}¡‚¤†‚¤&µ¨|¡ŒŽ¯•P€˜”ª–…˜”—”ƒ=‚¡}¤~•ƒ)ª”Š”ƒ=Š”†‚¡•}¤Œ-|¡˜ —”ƒ)€ƒ)‚¤}¡~ Õ©¡•””—”˜”˜³¢}¡‰¤Š&‘=}¤„•”ƒ=ª”Š”˜„‡ƒ=•”˜„‡†²’€”Ž×Š”ƒ=–…ƒ)•˜Ž

x ν−1§x ν+1§7¢=¢)¢=§x n‚¨Œ-˜KŒ-|&ƒ.„‡¡ ËÚ Ð8ª”}.’4Œ)|'¡‚&€”Ž”µ¨|¡ŒŽQ˜”•”|&ƒ=Š”‚¤&€†r 1§¤¢=¢)¢=§r ν−1§s,Çß WßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³«ï³ã!ð«ãWäD”ñÙ Ì Ú´8ƒ=„Ô•ƒ©„‡ƒ=•”ƒ)ƒ¤§K„‡†{ª”Š”}.’Kƒ.„‡}¡•Œ)|&Š”˜”Š¡ƒ=„œ§“”|&}e}¡ª|&˜„ >€KŸ¡•”}¤ƒ$Šƒ)–…ƒ=•”˜”ƒQ‘.>š’4'“”˜[‚¤•”ƒ)–P•”ƒ)=}¯}¡—”ƒ=•”˜”‚7&•”˜Žc„‡•”}'Sƒ=Œ)|&‚Š”ƒ=–…ƒ=•”˜”~„‡}'Sƒ)|¯‰¤†|&Ÿ‚¤†“”˜Œ!€Kƒ)•”}˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†„ >€=}¡Š”˜|¡„‡}¤„ÅW€}¤‰¤&€Ÿ¡•”}¤~}¡ª”|¡˜„‡˜”‘=¡—”˜”˜8§}7Œ-•”}¤‚¤¡•”•”†„ •|&ƒ-›•”˜7ƒÄ‚¤ƒ)|&‚¤ƒ)~˜ =Š&•˜”—ƒÅS˜}¤ª”˜Œ=&•”•”}7„Ø•¤„‡˜‚ ô$¢Û ¢¦Q¡„Ø•¡¤e•”}S€˜–…Ÿ}¡•KŒ-|&Š”¡”|¡˜”‚¡•”†~PŒ-ª”}7Œ-}¡‰ ‚¡†“”˜Œ-€ƒ)•˜Ž…„‡˜”•”}¤Š&•”|¡† ª”} }¤‰&€K¤Œ-|&˜’€”Ž¦Õ©¡•”—”˜”˜ ª”Š”ƒ-’mOVŽ”‚¡˜|&ŸË|.¢ƒ¡¢V•˜e•”˜”~¬7}¤•”ƒ)—ƒ)S˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”}¤)} Š¡Œ)–…˜”Š”ƒ=•”˜ŽKÐ-¢(©•”†„‡˜×Œ!€}¤‚¤¡„‡˜³§”’€Ž€µ¨‰¡}¡=}Ω(r)r = (r 1 , . . . , r ν−1 , r ν+1 , . . . , r n ) ⊤ ∈ IR n−1„‡†¨’4}&€”e•”†õ¡„‡ƒ=|&Ÿe}¡—ƒ)•”˜”‚7&|&Ÿmin{ Ω(r) | r ∈ r } = min{ ⋃{ x ν | x ∈ Ξ αβ ∩ l(r) }r∈r}ËÎ ª ¢ÏÎ!§¡ÐŒ-•”˜‘)¡¢Ê©Š”}¤Œ)|&ƒ)~”–P˜”~žŒ)ª”}¤Œ)}¡‰Œù’4ƒ-€K¡|&Ÿ¦z=|&}¦Œ)}¤Œ)|&}¡˜”|¦‚¬Œ-€ƒ!’4¡7µ¨‹…ƒ.„œ¢0Üc† ª”}¤Œ)|&¡7šª&ƒ.„ ŒS˜Œ›7}.’K•”}¡~¦˜”•”|&ƒ=Š”‚¤&€Ÿ¡•}¡~×Œ-˜Œ)|&ƒ=„‡}¤~ ËÚ Ð¨‚ |&}¡“•”}¤Œ)|&˜×|'¡Sƒ¡§V&¬‚Œ!€¡“¡ƒ}¡ª”Šƒ!’Kƒ)€ƒ)•”˜”ŽΞ αβ (A, b) ∩ l§¤•”}¦|&ƒ=ª”ƒ)ŠŸ¦‚¤„‡ƒ.Œ-|&}¦ªƒ)Š”ƒ.„‡ƒ)•”•”†œ›x 1§K¢)¢)¢=§⎧⎪⎨⎪⎩a c 1νt +n∑j=1, j≠ν· · · · · ·a c mν t +n∑j=1, j≠νa c 1jr j ⊆ b c 1,a c mj r j ⊆ b c m ,ËÎ ª ¢ÏÎE$¤Ðr ν+1§¢=¢)¢)§r•ƒâ‚¤ƒ)‹Pƒ=Œ-|¡‚¡ƒ)••”†ƒ¥“˜Œ!€K r1§¢=¢)¢)§ n§¡r ν−1§r ν+1§¢)¢=¢)§r&¯˜¯ŠK&•”Ÿ¤–…ƒ¡§„‡†‚¡†“˜Œ!€”Ž”ƒ.„Øª”ƒ=Š”ƒ=Œ)ƒ)“”ƒ=•”˜”ƒ n¢£&P>€ƒ=ƒ¡§ ‚¤Œ)ƒ-›¯„‡•”}'Pƒ.Œ-|&‚…Š”ƒ)–Pƒ)•”˜”~S m–…|&¡ }.’K•”}7„‡ƒ)Š”•†œ›˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†œ›¯‚¡€µ¨“”ƒ)•˜”~8§”}¤‰¡Š¡‘)¡µ ‹P˜›Œ-˜Œ)|&ƒ=„‡¡˜€Ã c ν t + Ãc r ⊆ b cËÎ ª ¢Ì Ð!¢D¦©ƒ)|&Š¡)’4•”}eª”}¡•”Ž”|&Ÿ§“”|¡}‚«}¡‰¡}¤‘)•'“ƒ)•”˜Ž›S ⊇r∈r{ ⋃ x ν | x ∈ Ξ αβ ∩ l(r) },ª”}¤Œ)}&€Ÿ¤”¡Šƒ)–…ƒ=•”˜”~“ƒ)•”˜Ž( ) „‡•”}'Sƒ=Œ)|&‚¡} ( Š”ƒ=–…ƒ=•”˜”~ „‡•”}'Sƒ=Œ-|¡‚¡} )A c νt + Ãc r ⊆ b c ⊆A c νt + Ãc r ⊆ b‚¤€µ ‚¤€µ¨“”ƒ)•”˜”Ž c

A, b, r¢K£¥|&}e¡|&‚¡ƒ=Š¤S’Kƒ=•”˜”ƒ©‰ ¡=’Kƒ=|P’Kƒ)|>>€KŸ¡•”}«}¡‰7Œ-¡7S’4&|¡Ÿ¤ŒŽ‚ô=Î ª ¢Ù ‚K¢V,min{ x ν | x ∈ Ξ αβ }¢+£¥|&}[˜”|&ƒ=Š&—˜”}¡•”•K>Ž×ª”Š”}¡—ƒ!’K¡Š’€”Ž×ª”}7Œ!€ƒ-’K}¡‚7&|&ƒ)€Ÿ&š‚¤Œ)ƒ-›r ∈ r¢®ó¤€ƒ!’K}¤‚¤¡|&ƒ-€Ÿ¤•”}§ ’€”ŽΩ(r) = min{ S ∩ V ν }Ù Ì ÛÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ãª ¢ÏÎ Ù Ð’4&=||&Šƒ)‰ ¡¤ƒ=„‡¡µ •˜e•”µ µÅ}¡—ƒ)•””¡ ’€ŽËÎª ¢ÎE§¡Ð!¢/Š¤Œ-€˜¦PƒPŒ)˜Œ-|¡ƒ=„ ËÎ ª ¢ÏÎE$¤Ð•”ƒ=Œ)}¡‚7„‡ƒ=Œ-|¡•¤’€”ŽQ•”ƒ)}¤|&}¡Š”}¤)} rËz=|&} Œ-}¤}¡|&‚¤ƒ)|¡Œ)|&‚ ¡¤ƒ)|œŒ-˜”| ¡¤&—˜”˜ΞËÎαβ ∩l(r) =’€”Ž ‚¤Œ)ƒ-›∅r rÐ-§”|&}eª”}¡€K&-¡ƒ=„∈ Ω(r) = ∞¢+˜Œ)ª”}&€KŸ¡‘)¡¤ƒ=„Ø}¤‰¡}¤‘)•'“”ƒ=•”˜”ƒ Ω(r)“”|&}¤‰¡†¨ª}.’K“”ƒ)Š”•”¡|&Ÿ§“|&}«Š”ƒ=&€˜”‘)}¡š‚¤¡•”•>Ž[•¤„‡˜cª”Š”}¤—”ƒ!’4¡7ŠK«Ž”‚&€”Ž”ƒ=|¡ŒŽ8§4‚e’Kƒ=~Œ-|&‚¤˜”|&ƒ)€Ÿ¡•”}7Œ-|&˜³§0?G¥CI?RGRCLH>?)þþKÿ³úÜ[†Ω(r)¢Ü[†’4>Sƒ¥‚¤‚¡ƒ-’K=„‚Š¤Œ)Œ=„‡}¡|&Š”ƒ=•”˜”ƒ«Õ©¡•””—”˜”µ =þ/CL?Iø@H&û)ü4ý¤þÿ³úø”û>G¥QL=>øc?-þ/=/?ú|¡}¡“”ƒ=“”•”}¡~eÕ©¡•””—˜”˜Ω : V →Œ…}¤‰&€K¤Œ-|&Ÿ¤µˆ‘)•K'“”ƒ)•”˜~‚ª}&€¡Š¤Œ!šR·“˜Œ!€}¤‚¡}¤~ }7Œ-˜R· = R ∪ {+∞}˜ }¡‰¡€K¡Œ)|&Ÿ¡µ{}¡ª”Šƒ!’Kƒ)€ƒ)•”˜”Ž–…˜”Šƒ)•”•”}¤~V = { r ∈ IR n−1 | r ⊆ (V 1§V¢)¢=¢)§V ν−1§V ν+1§8¢)¢=¢)§V n ) ËÎ ª ¢ÏÎ Ú Ð}.ó¤€ƒ!’4¡ƒ)|S}¡|¤„‡ƒ)|&˜”|¡Ÿ§“|&}Ω(r ′ ) ≥ Ω(r ′′ ’€”Ž) r ′ ⊆ r ′′ ,˜}¡—”ƒ=•”˜”‚7&•”˜”ƒ[‚¤ƒ-€˜“”˜”•”† ËÎ ª ¢ÎE§¤ÐPª”}¤Œ)Š”ƒ!’4Œ)|&‚¡}7„Ω(r)Œ)|>&•”}¤‚¡˜”|¡ŒŽÔ‚¤Œ)ƒc‰¡}¡š€ƒ)ƒØ|¡}¡“”•”†„wª”Š˜Ö¡„‡ƒ=•”Ÿ¡–Pƒ)•”˜˜{–P˜”Š”˜”•† ‚¤ƒ)”|¡}¡Š r§¨|.¢ƒ¡¢¨ª”Š”˜Ö¡„‡ƒ)•Ÿ¡–…ƒ)š•”˜”˜‖ rad r ‖§4ƒ.Œ!€˜¦•”ƒ)}¤|&}¡Š”†ƒPƒ=Œ)|&ƒ=Œ)|&‚¡ƒ=•”•”†ƒ«}¡=Š&•”˜“”ƒ)•”˜”Žc•>€}'Sƒ)•”†Ö•K´8ƒ)ªƒ)Š”Ÿ¯‚7Œ-ƒQ=}¡|&}¤‚¡}P’€”Ž[}¡•Œ)|&Š”¡˜”Š”}¤‚¤¡•”˜Ž&€)}¤Š”˜”|¡„ §‚¤†“”˜Œ!€Ž”µ ‹…ƒ=)}•”}¡=}¡.€¡“”–Pƒ)•”˜Žò}¡—”ƒ=•””˜òz)|&}¤)}¬„‡˜”•”˜„‡¡„ ØŒ-•˜”‘)¡¤§a}7Õ©}¡Š„a€Kƒ)•”•&Ž‚×Œ-}¡}¤|>šÄ„‡ƒ=|&}.’4ž‚¡ƒ)|¡‚¡ƒ)~õ˜õ=Š&•˜”—ƒÅ'§&•&€}¡=˜”“”•”}…|&}¤„‡¡¤§7¡z)|¡}SŒù’Kƒ)€K&•}¦‚P˜•”|&ƒ)Š‚¤>€KŸ¡•”†œ›¯„‡ƒ)|¡}.’4>›eW€}¤‰¤>€KŸ¡•”}¤~‚¡ƒ=|¡Œ-|&‚¤˜”˜õŒ¬–P˜”Š”}¡}ž˜”‘)‚¤ƒ=Œ-|¡•”}¡~{Œ-|&Š¡|&ƒ)=˜”ƒ)~}¡ª”|¡˜„‡˜”‘=¡—”˜”˜ ˜”‘¦°7€K¡‚¡†|$¢”yÔ’4¡•”•”}7„Œ!€K¡7“K&ƒ‚¤ƒ)|&‚¤˜ƒÅP}¤‰¡Š¡‘)¡µ |¤ŒŽ‚«Š”ƒ)‘=¡=€KŸ>|'¡|&ƒ©‰¡˜Œ)ƒ)”—˜”˜ ˜KŒ›¤}.’K•”}¤)}(n − 1)š„‡ƒ=Š”•”}¤)}‰¤Š”¡¤Œ= Ä(V 1§V¢)¢=¢)§V ν−1§V ν+1§8¢)¢=¢)§V n P§)•S‰¡}&€Kƒ)ƒ©¡‘)”˜”ƒ©‰¤Š”¡¤Œ)†Ä)ŠK&•”˜”—ÅI" z)|&}‚¡†“”˜Œ-€ƒ)•˜”ƒQ‘)•'“ƒ)•”˜”~Ω(P )§K|'¢ƒ¡¢ νš}¡~[}¡š}¤Š¡’4˜”•&|¡•”}¡~}¡—”ƒ=•””˜ |&}¤“”ƒ) ˜”‘ •&›¤}'S’Kƒ)•˜”ƒ{ Ξ αβ ∩ l(r) | r ∈ }¢¢€=}¡Š”˜”|¤„žª}¡Š”}'S’4¡ƒ)|«Œ-ª”˜Œ)}¡P§Œ)}¤Œ-|¡}'Ž”‹…˜”~˜”‘Qª¡Š L (P , ))§K|'¡”˜›“”|&}Ω(PP ⊆ (V 1§8¢=¢)¢=§V ν−1§V ν+1§8¢=¢)¢)§V n ),˜P¡ª”}¤Š7Ž’K}¡“”ƒ=•”•”†~Pª”}©‚¡}¡‘=Š¡Œ)|>&•”˜”µò‚¡|¡}¡Š”}¤)} ª}&€”Ž8¢&Ê©ƒ=Š”‚¤&Ž…ª&ŠK (Q, Ω(Q))L˜”=Š&ƒ=|…}7Œ-}¡‰¤¡µ¨Š”}&€KŸ¦‚…•¡–…˜›eŠ¤Œ)Œ=„‡}¡|&Š”ƒ=•”˜Ž›V¢Ü[†‰ ¡=’Kƒ.„Ø•¡‘)†œš‚¤¡|&Ÿ ƒ)¡§+ |>¡Pƒ¯Œ)}¡}¡|¡‚¡ƒ)|¤Œ-|&‚¤¡7µ¨‹…˜”ƒ«‰¤Š”¡¤Œ Œ-ª”˜KŒ- ˜Ø}¡—ƒ)•””¡Ω(Q)§oH>?¢Jza=ú¦=K¢Q

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•”¡.€Ÿe•ƒ¨Ž”‚¡€”Ž”ƒ)|¤ŒŽ }¡•”—}¤„a iνΩ : IR n−1 ⊇ V → R·XV÷! >øc?R¢!?üc‘!þKþKûRdL K÷!GIø@?¢EGRCLH&÷úËÎ ª ¢Î Ù ÐÍi ∈ {0, 1, 2, . . . , m}’€”Že}¡|&}¤Š”†œ›0 /∈ a iν§7|'¡«“”|&}P„‡•}'Pƒ=Œ)|&‚¤$Š”ƒ=–…ƒ=•”˜”~Œ)}&šÙ Ì¡Ì⎧s,Çß WßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³«ï³ã!ð«ãWäD”ñG¥=G¥CI?ú¦ûAx = b =r=KþDCI?ø@H¡û)ü4ý¡þKÿgSrH?¥ÿ0Xg;CP÷⎪⎨’€Ž>S’K}¤)}r ⊆ (V 1§8¢)¢=¢)§V ν−1§V ν+1§V¢)¢)¢=§V n )’K€”Ž&P’4}¡)}i = 1, 2, . . . , m‚¡†ª”}¡€•”ƒ)•}SŒ)‚¡}¡~KŒ-|&‚¤}D%⎪⎩⎛⎝ b i −n∑j=1, j≠ν⎞a ij r j⎠ .˜€˜Ë Æ¨©VÐ•”¡.€Ÿe•ƒ¨Ž”‚¡€”Ž”ƒ)|¤ŒŽ }¡•”—}¤„ ˜•”|&ƒ)Š‚¤>€49÷!TU¢'û¨öqJ¡þ/þ4ûLG)þD=j£Jk]ËÎB}¡‰¤}¡‘=•'“&ƒ=|„‡•”}'Pƒ.Œ-|&‚¤}¨‚7Œ-ƒ)›¦—”ƒ-€†œ›P˜”•’Kƒ=Œ)}¡‚m,D‚ƒ2„2…*†0‡ˆ…‰,p6Ê©¡¤Œ-|¡Ÿ‚¤&€K¡„‡˜[ t i , t i ]¢”Ê©Š”˜z=|&}¤„ž„‡†= V ν˜¡µ¨‹…˜› ¡Š&‚¤•”ƒ)•”˜~c˜”‘ ËÎ ª ¢ÏÎE$¤ÐâŽ”‚¡€”Ž”µ |¡ŒŽ ÷£T:ø”û>þD=/;?-þKþÿ³ú¦=[˜•”|&ƒ)Š”š}¡|&‚¤ƒ)|¡Œ)|&‚t 0 = V ν§¡=’K}¡‰7Œ-|&‚7 ª”}¡€K&-¡ƒ=„¦’€Žt0B‚7Œ-ƒ)’®e•”ƒ)ª¡¤Œ-|¡}¢ ¨‰¡}¤‘)•'“”˜K„Ô|'¡PƒU = {0, 1, 2, . . . , m} \ B§4˜ª”¡¤Œ)|&Ÿe„‡•”}'Pƒ.Œ-|&‚7 |>&“|&}] − ∞, ti ] ∪ [ t i , + ∞’€”Ž[ i ∈ª”Š”ƒ!’®Œ-|>&‚¡€”Ž”µ |þ/?)÷!T:ø”û£OUþD=/;?-þþKÿq?œ„‡•”}'Pƒ.Œ-|&‚7©Š”ƒ)–Pƒ)•”˜”~«¡Š¡‚¡•”ƒ=•”˜”~«˜”‘ ËÎ ª ¢Î$7Ð‡Œ0 ∈ a iν¢¤Ê©Š”˜«z=|&}¤„„‡†ˆª”Š”ƒ-’Kª”}¡€K&-¡ƒ=„œ§®“”|&}t i = −∞˜€˜t i = + ∞§®}¡’4 Œ-}¡}¤|&‚¡ƒ=|¡Œ-|¡‚ ¡µ ‹Pƒ)ƒ„‡•”}'Pƒ.Œ-|&‚¤}«Š”ƒ)–Pƒ)•”˜”~[ƒ.Œ-|&Ÿ«€K¡7“[‚ R§K˜t i = t i = 0}¡’®S„‡•}'Pƒ=Œ)|&‚¡}«Š”ƒ=–…ƒ)š•”˜”~[Œ-}¤‚¡ª>’4&ƒ=|PŒ)}S‚7Œ-ƒ=~ “”˜KŒ!€}¤‚¡}¡~}¤Œ)Ÿ¡µ R¢›7'¡7Œù’K}¡ŠÕ©}¤‚¡}¤~P„‡ƒ)|¡Š”˜”}¡~8§}¤|&•”}¤Œ)˜”|&ƒ)€Ÿ¡•”}…7}¤|&}¡Š}¡~¯•”ƒ=ª”Š”ƒ=Š”†‚¡•”†‚7Œ-ƒ˜•”|&ƒ)Š”š¦Q×˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”†œ›òª”Š”}7Œ-|&Š¡•Œ-|¡‚¤>›òŒ-|>&•’®&Š”|&•K>ŽÔ|&}¡ª”}¡€}¡=˜ŽÔ‘='’®&ƒ)|¤ŒŽ¡×Î.Ð-¢Vó¤€ƒ-’K}¡‚7&|&ƒ)€Ÿ¡•}§t‚¤&€Ÿ¡•”†ƒP&Š”˜ÕQ„‡ƒ=|&˜”“”ƒ.Œ-”˜ƒP}¤ª”ƒ)Š¡—”˜”˜ ˜ ËŒ)„œ¢V°7€K¡‚it i§ B§KŽ”‚¡€”Ž”µ |¤ŒŽ¦•”ƒ=ª”Š”ƒ)Š†‚¡•”†„‡˜×Õ©¡•””—”˜Ž„‡˜×‚¤ƒ)”|&}¤Ši¢)¢=¢)§∈ (r1§r ν−1§r ν+1§4¢)¢=¢)§r n ËÎ ª ¢ÏÎE$¤Ð¢¦©}¦˜¯‚SŒ!€¡“¡ƒ) 0 ∈‚¡ƒ=‹…ƒ=Œ)|&‚¡ƒ=•”•”†ƒ“˜Œ!€Ka iν tit}¡ª”Š”ƒ-’Kƒ-€Ž”µ ˜i ⊤˜”‘a iν t +n∑j=1, j≠νa ij r j = b i ,‹…˜ƒ©„‡•}'Pƒ=Œ)|&‚¡}«Š”ƒ=–…ƒ=•”˜”~ }.’K•”}7„‡ƒ)Š”•”}¤)}P¡7ŠK&‚¡•ƒ)•”˜Ž‘=¡‚¡˜ŒŽ”|$}¡|©˜”•”|¡ƒ)Š”‚7>€Ÿ¤•”}¡=}$‚¤ƒ)”|¡}¡Š r = (r1 , . . . , r ν−1 , r ν+1 , . . . , r n )Pƒ©•”ƒ=ª”Š”ƒ)Š†‚¡•”†„ž}¡‰¤Š&‘=}¤„œ§ƒ=Œ-€ã ⊤|>¡šiν < 0 < a iν ,

[ t i , t i]‚¤Œ-ƒ)›ž}¤)Š¡•”˜”“”ƒ=•”•”†œ›žŠƒ)–…ƒ=•”˜”~[ t i, t i ]˜„‡ƒ=ƒ)|¦•”ƒ=ª”¡¤Œ)|&}¡ƒªƒ)Š”ƒ.Œ-ƒ)“ƒ)•”˜”ƒ¨Œ•”ƒ)}¤)Š¡•”˜”“”ƒ=•”•”†„‡˜Ú>ªª⎛ÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã⎝ b i −n∑j≠νa ij r j⎞⎠€˜”‰¤}Sƒ.Œ!€˜˜”•”|¡ƒ)Š”‚7>€•”ƒQ˜„‡ƒ=ƒ)|«}.’K•”˜„ž˜”‘¦Œ)‚¡}¡˜”›¯7}¤•”—”}¤‚S•¡=€KŸ¢4£¥|&}eŒ-€ƒ!’K¡¤ƒ)|«˜”‘¦Õ©}¤Š„‡¡.€¬ô.Î ª ¢Ù …˜Ë Î ¢Ù ’4>€Kƒ)ƒ „‡}'Sƒ=„ÖŒ)“”˜”|>&|&ŸK§‡“”|&}‘=•'“”ƒ=•”˜Ž }¡‰¤ƒ=Œ-ªƒ)“”˜”‚7&ƒ=|¡ŒŽ¦¡¤Œ-€}¡‚¤˜”ƒ=„ Æ¨©8Ð‡Ê©Š”ƒ!’€K}'Pƒ)•˜Ž ª ¢Î¡¢¡ó¤€ƒ!’4}¡‚¤¡|&ƒ-€KŸ¡•”}K§.„‡†max i∈B t i§min i∈B t i§max i∈U tŽ”‚&€Ž”µ i§|¡ŒŽ•”ƒ=ª”Š”ƒ)Š†‚¡•”†„‡˜ Õ©¡•”—”˜Ž„‡˜‚¡ƒ=”|&}¤Š ˜”‘ ªmin¢ÏÎE$¤Ði∈U t i r ËÎÊ©}¡>Sƒ=„ ’4>€Kƒ)ƒ¡§œ“”|¡}žz.ÕQÕ©ƒ)”|&˜‚¡•&ŽÔ}¤‰&€K¤Œ-|&Ÿ}¡ª”Š”ƒ-’Kƒ-€Kƒ)•”˜Ž{Õ©¡7•”—”˜”˜min ∅ = + ∞§max ∅ = −∞Ð!¢Ë&}¡‰¤†“”•”}K§ª”}&€K¡-&ƒ.„Ω(r)§”|'¢ƒ¤¢„‡•”}'Sƒ=Œ-|¡‚¡}Ω = { r ∈ V ⊂ IR n−1 | Ω(r) < + ∞},Ž”‚&€”Ž”ƒ=|¡ŒŽõ}7„‡ª&”|¡}¤„œ¢ Ê©Š”˜•'’€Kƒ-e•”}¤Œ)|&ŸÔ‚¡ƒ=”|&}¡ŠK r„‡•”}'Sƒ=Œ-|¡‚ ¡Ωª”ƒ)Š‚¡†œ›V§Ó•”ƒ=ª”¡¤Œ)|&}Øª”ƒ)Šƒ=Œ-ƒ=“”ƒ)•˜”ƒ ⋂i∈B}¡‘=•'“&ƒ=|Œ-}¡‚7„‡ƒ=Œ)|&•”}¤Œ)|&Ÿ Œ)}¡}¡|¡‚¡ƒ)|¤Œ-|&‚¤¡7µ¨‹…ƒ=~ Œ)˜Œ-|¡ƒ=„‡† ËÎ ª ¢Î$7ÐW¢$´³}¡’4§ ‚¤}&šmin t i ≥ max t i .i∈B i∈B¡Š&‚¤•”ƒ)•”˜~ ˜”‘ ËÎ ª ¢ÏÎE$¤Ð¢4£¥|&}ez=”‚¡˜‚¤>€Kƒ)•”|&•}P|&}7„‡¡§”“|&}⋂i∈B Š”ƒ)–Pƒ)•”˜”Ž„‡˜ }.’K•”}7„‡ƒ)Š”•”†œ›¬¡Š&‚¤•”ƒ)•”˜~ Œ-˜Œ)|&ƒ=„‡† yâ}&š‚¡|&}¤Š”†œ›V§ª ¢Î$7ÐW§³|'¢ƒ¡¢‡Œ] − ∞, t ËÎi ] ∪[ t i , + [§∞ i U§KÊ©}¤Œ!€Kƒ!’K•”ƒ=ƒ©z)”‚¤˜”‚¤&€ƒ)•”|¡•”}P¡7Œ!€}¤‚¡˜”µ∈(min t i ≥ maxi∈U i∈Bt i)∨(max t i ≤ mini∈U i∈B{{} }min min t i − max t i , max min t i − max t i , min t i − max t i ≥ 0.i∈B i∈B i∈U i∈B i∈B i∈U∨ " €}¤)˜”“ƒ=Œ-&Ž’K˜‘O8µ¨•””—”˜Ž8¢®yõ—”ƒ-€}7„„‡•”}'Pƒ.Œ-|&‚¤} ƒ)|¡ŒŽ¡¤Œ-€}¡‚¤˜”ƒ=„ ’Kƒt i),Ω}¡ª”˜KŒ-†‚¤&šΩ¢¥§©Š”}¤„‡ƒe|¡}¡)}K§Ê©}¤Œ)7}¡€Ÿ¡¡…Õ©¡•””—”˜Ž¯‚…€ƒ)‚¤}¡~e“¤Œ-|&˜z)|&}¤)}¦•”ƒ=Š&‚¤ƒ)•Œ)|&‚¤©•ƒ)ª”Š”ƒ=Š”†‚¡•K¦ª”} r§„‡†Å|&ƒ=„ÖŒ=¡„‡†„¨’K}¡&‘.>€˜×‘.¡„‡”•”¡|&}7Œ-|&Ÿc„‡•}'Pƒ=Œ)|&‚¤ Ω}¤“”ƒ)‚¤˜’K•”}«}¡=Š&•˜”“”ƒ)•}§Sª”}¡|&}7„‡¡}¤„‡ªK&”|&•}¢¡¤SƒP}¤|¡„‡ƒ)“K>€˜8§V‚ }¡‰¤‹…ƒ=„ Œ!€¡“¡ƒPÕ©¡•””—”˜”ŽΩ(r)•”ƒPŽ”‚&€Ž”ƒ)|¡ŒŽ’4>Sƒ«•”ƒ)ª”Šƒ)Š”†‚¤•”}¡~¬• Œ-‚¤}¡ƒ)~¦z=ÕQÕ©ƒ=”|&˜”‚¤•”}¡~¬}¡‰¡€K¡Œ)|&˜¦}¤ª”Š”ƒ-’Kƒ-€ƒ=•”˜Ž8¢cŠ¤Œ-€˜§Q&¦„‡†

∈ Ω(r) ûÚª Îmin U t imax U t imax B t i min B t i✲s,Çß WßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³«ï³ã!ð«ãWäD”ñmin U t imax U t imax B t i min B t i✲min U t i max U t imax B t i min B t i✲rƒøËû/üãû z>ûrcõÌ÷øÍ¦¤¨§Êuø¨¡>õRùø-ø p øIÿ p ¨V"÷¨¥¤-øø6¤¨»ù-øøΩ§|&}Ω(r) = max t i , ƒ.Œ!€˜mini∈B i∈U{Ω(r) = max maxi∈B˜t i, maxi∈Ut i},ƒ.Œ!€˜mini∈Ut i ≥ maxi∈B t i ,t i < maxi∈Bt iË ¨Ó˜KŒ&¢8Î ª ¢¨§¦˜”‘=}¡‰¤Š>«&ƒ)|SŠ¡‘-€˜”“•”†ƒ©‚¡}¡‘.„‡}'e•”†ƒ$Œ)˜”|&¡7&—”˜”˜4Ð!¢Ê©¡7Œ-|&ŸΩ = D 0 ∪ D 1 ,D 0 ={r ∈r ∈Ω ⊆ IR n−1∣ ∣∣ mini∈Ut i ≥ maxi∈Bt i},D 1 ={Ω ⊆ IR n−1∣ ∣∣ mini∈Ut i < maxi∈Bt i}.’Kƒ

„‡˜õÕ©¡•””—˜Ž„‡˜8§„‡•”}'Pƒ.Œ-|&‚¤}D 0=þ/CL?Iø@H&û)ü4ý¤þÿgSÀH>?Rª §Ω(r) =⎧max⎪⎨ {max⎪⎩t i , ƒ=Œ-€˜r ∈ D 0 ,i∈B}ƒ=Œ-€˜, r ∈ D 1 ,maxi∈Bt i , maxi∈Ut i+ ∞,ƒ=Œ!€K˜r ∈ V \ (D 0 ∪ D 1 ),ÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã´8}¤’4«Õ©¡7•”—”˜Ž Ω(r)„‡}'Sƒ)|«‰¡†|&Ÿ«}¤ª”Š”ƒ!’4ƒ-€ƒ=•«Œ!€ƒ-’K¡µ ‹…˜K„}¤‰¡Š¡‘)}¤„+%˜Ω(r ′ ) < Ω(r ′′ ) < + ∞’€”Ž€µ ‰¤†œ›r ′ ∈ D 0§r ′′ ∈ PD 1¢•”ƒ=Œ)|&Š”}¡=˜„‡˜ò•”ƒ)Š¡‚¡ƒ=•Œ-|&‚7¡„‡˜ „‡ƒ)P’4¡Ø•”ƒ)ª”Šƒ)Š”†‚¤•”†œš‘=¡„‡•”¡|&}§ Ã¥¡=’K¡“”˜ž‘=>’4&•”•†„ •”ƒ)ª”Šƒ)Š”†‚¤• & •Ω(r)D 0§â|'¡˜¨• D1¢âÊ©}Øz=|&}¡~¨ªŠ”˜”“”˜”•ƒc€ƒ=‰¡ƒ)=}¡‚¤}×„‡•}'Pƒ=Œ)|&‚¡}{Ω‘=&š„‡”•”¡|&}’€”ŽØ€µ¨‰¡}¤)}≤ c}c ≤ sup{Ω(r)| r ∈|&ƒ=ª”ƒ)Š”ŸK§‡“”|&}D 0c > sup{Ω(r)| r ∈}¢aÊ©Šƒ!’Kª”}¡€}'e˜„D 0 {Ω c}„‡}¤)¡|eª”Š”˜š•'’K€ƒ-«&|&Ÿ…€˜”–…Ÿ≤ }¢K´8}¡’®e‚¤Œ)ƒQª”Š”ƒ!’4ƒ-€Ÿ¤•”†ƒ¦|&}¤“””˜D 0 ∪ D 1‚«Œ-˜”€¡¯}¡|¡”Š”†|&}7Œ-|&˜’K}¡ª}&€•”ƒ=•”˜ŽV \ (D 0 ∪ D 1 )¢{Ω ≤ c} ∩¦©}ž„‡•”}'Sƒ=Œ)|&‚¡}D 0 =‘=¤„‡”•”¡|&}ò‚D 0 IR{Ω ≤ c} ∩‘=¤„‡”•”¡|&}[‚D 1 D 1¢‡ó¤€ƒ!’K}¤‚¤¡|&ƒ-€Ÿ¤•”}§{Ω ≤ c} ∩Œ)}.’Kƒ)Š7e˜”|D 0{Ω ≤ c}§Óª”Š”˜”•K'’€ƒ)S¡‹…˜”ƒD n−1§ž„‡•”}'Sƒ=Œ)|&‚¡} ‚¤Š”ƒ=„aŽÔ&‚¤Œ)ƒ[ª”Š”ƒ!’4ƒ-€Ÿ¤•”†ƒ[|&}¡“”˜ 0§Ó‚Ø|&}{Ω ≤ c} ∩ D 1Œ-}.’Kƒ=Š7«˜|¦‚¤Œ)ƒ{Ω ≤ c}§¤ª”Š˜”•'’K€ƒ-«&‹…˜ƒª”Š”ƒ-’Kƒ-€KŸ¡•”†ƒ¨|&}¡“”˜D 1¢Ê©}7Œ-}&€Ÿ¤”¡{Ω ≤ c} = ({Ω ≤ c} ∩ D 0 ) ∪ ({Ω ≤ c} ∩ D 1 ),|&ƒ=„ Œ=¡„‡†„Åª”}¡€¡“&ƒ.„Å‘=¤„‡”•”¡|&}¤Œ)|&Ÿ €Kƒ)‰¡ƒ=)}¤‚¡†œ›ò„‡•}'Pƒ=Œ)|&‚{Ω ≤|&}ž„‡† |'¡Pƒe˜¦’€Žc} c > sup{Ω(r)| r ∈ D 0ª”}&€K¡7•ƒ)ª”Š”ƒ=Š”†‚¡•}¤Œ-|¡˜Œ)•”˜”‘)¡ Õ©¡7•”—”˜”˜Ω(r)¢}¢a£¥|&}[‘=¡‚¡ƒ=Š”–P¡ƒ)|’K}¡&‘.&|&ƒ)€Ÿ¤Œ)|&‚¡})+*-,/.0*-1327XÕ6SRp687–`ZJ-G¥CSýb=þ/CL?Iø@H&û=ü4ý¡þKûd ü=þ?¥Sþ4ûRd–GR=/G¥CI?ú¦ûAx = b =C«û£ÿ0X¥;CP÷3H>ÿg ÷Wü4þ/?)þ÷ÝJGü4÷£H£=/?ËÆ©VÐR]9÷!TU¢'ûbH¦ún?¥CS÷£¢!?5¢ø4÷!~ü@?)þD=kdø@?¥QI?-þ/=DS–=jF9Kû£~Wüc]¨Î ª ¢¨§–GþKû£;”û)ü4ý¤þÿ³ú› &ø=/OVK÷£Gü@?R¢.÷£H&û£CI?ü4ý¡þ÷£GRCPý#H?R¢EJzL=k¤¦÷!e/?)þ÷!JÀ€=}¡Š”˜|>š„ >›¯W€}¤‰¤&€Ÿ¡•”}¤~e}¡ª”|¡˜„‡˜”‘=¡—”˜”˜8§}7Œ-•”}¤‚¤¡•”•”†œ›e•S'’4¡ª”|&˜”‚¤•”}¤„¬’4Š”}¡‰¡€ƒ)•”˜˜ ˜m,D‚ƒ2„2…*†0‡ˆ…‰,p6q§Q&×„‡†²¡7SƒS¡ª”}7„‡˜”•&€˜8§V‚Ä‚¤ƒ)|&‚¤ƒ)~˜=Š&•˜”—ƒÅ'§”}¡—ƒ)•”Œ-|&ŠK&|&ƒ=)˜”˜Ω(Q) ≤ min{ x ν | x ∈ Ξ αβ ËÎ ª ¢ÏÎ Û Ð}Q‰¡†œ€4¦ª”}¡€¡“”ƒ)•|*…¢D¨‡'“ƒ)}¤„×‚rÚ>ª!®§”…ª}¡|&}7„‡¡¯•¡„}¤Œ)|'¡ƒ)|¡ŒŽ¯’K}¡&‘.&|&Ÿ«Œ-}¤‰¤Œ)|&‚¡ƒ=•”•”}eŒ›¤}.’K˜K„‡}¤Œ-|¡ŸP„‡ƒ)|¡}.’4”¢’€”Ž‚¡ƒ!’4¡7‹P˜›¯‰¡Š¡¤Œ-}¤‚

‰¤}¡=}ɛ > 0•&~’K)|¡ŒŽ|>&}¡ƒδ > 0§“|&}€µ0 ≤ min{ Ω(r) | r ∈ r } − Ω(r) ≤ ËÎ ª ¢ÎŒ7Ðɛ‖ rad Q (k) ‖ → 0ª”}e´8ƒ=}¡Š”ƒ.„‡ƒZ$”¢Û ¢¨§¢Ê©}¤z)|&}7„‡¡«’€”Ž€µ ‰¤}¡=}ɛ > 0Œ-¡‹…ƒ.Œ-|&‚¤¡7šWßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³«ï³ã!ð«ãWäD”ñ Úª $s,Çß“”ƒ)Š”ƒ=‘W„‡•”}'Sƒ=Œ-|¡‚¡}«‚¤Œ)ƒ-›|&}¤“”ƒ)“•”†œ›‚¡ƒ=”|&}¤Š”}¡‚ V§”|'¢ƒ¡¢¨‰¡}¡‘=•'“”˜„W := V ∩ R n−1 .˜D 1˜K„‡ƒ)µÊ©¡¤Œ)|&Ÿ |'¡Pƒe„‡•”}'Sƒ=Œ)|&‚¤D0|[|¡}¡|Pƒ¯Œ=„‡†Œ!€³§8¡¦}¤•”˜×‰¡†œ€˜„‡ƒ=Œ)|&}W‘=¡‚¡˜ŒŽ”|P}¡|«|&}¡=}§˜„‡ƒ=ƒ=„ž€˜∩ D 0 ≠ W ∅˜€˜∩ D 0 =W ∩∅§‚¤Œ)ƒP‚¤ƒ!’K¡‹…˜ƒP‰¤Š”¡¤Œ)† Š¤Œ!€˜ ªŠ”˜”•>’€ƒ-«&|D 0 ≠Œ-|&‚¤˜”|&ƒ)€Ÿ¡•”}K§®’€”Ž¬€µ¨‰¡}¡=}Q D 0¢/&…ƒ=~šp ∈ WŒ-ª”ŠK&‚¡ƒ-’€˜”‚¤} •”ƒ)Š¡‚¡ƒ=•Œ-|&‚¤} ∅§V|&}∩ D 0 Ω(p) ≥min{ x ν |Œ!€¡“¡ƒx ∈ Ξ αβ }¢Óy „‡†w˜„‡ƒ-€˜Ô‰¤†Q ∈ D 1Ω(Q) > Ω(p)§Ó}¡ª”Šƒ!’Kƒ)€ƒ)•”† ‚5¡Qƒ=„ „‡ƒ¬Î¡¢g¦Q¡–…˜×’4&€Ÿ¡•ƒ)~”–P˜”ƒŠ¤Œ)Œ)¡¤S’Kƒ=•”˜ŽžŒ)¡‹…ƒ=Œ)|&‚¡ƒ=•”•”}Ω(Q) > min{ x ν | x ∈ Ξ αβ },ª”}¡|¡}¤„‡¡ËÎ ª ¢Î Û Ð!¢§Q&”˜„ž‰¡†õ•”˜ ‰¡†œ€}“”|&}eª”Š}¡|&˜”‚¤}¡Š”ƒ=“”˜”|r ∈ D 0§min{ Ω(r) | (r ∈ R n−1 ) & (r ∈ r) } = Ω(ř)ËÎ ª ¢ÏÎ>±&Ðř ∈ r, ř ∈ R n−1§V|>&¦¡¦ª}&€¡•”ƒ=ª”Š”ƒ)Š†‚¡•&ŽŒ-•”˜‘)¡×Õ©¡7•”—”˜Ž ’€”Ž¬•”ƒ)}¡|¡}¡Š”}¤)}|¡}¡“”ƒ=“”•”}¡=} ’K}7Œ-|&˜-&ƒ)|cŒ)‚¡}¤ƒ))}[•K&˜„‡ƒ=•”Ÿ¡–Pƒ))}‘)•'“”ƒ=•”˜ŽØ•[}7„ašΩ(r){ r ∈ R n−1 | r ∈ r } Œ ®¢¦©}ª&|&•”}¤„ò„‡•”}'Sƒ=Œ-|¡‚¡ƒ‖ ř − r ‖ ≤ 2 ‖ rad r ‖.ó¤€ƒ!’4}¡‚¤¡|&ƒ-€KŸ¡•”}K§¤‚ÔŒ-˜€¡ŠK&‚¡•}¤„‡ƒ)Š•”}¡~¨•ƒ)ª”Š”ƒ=Š”†‚¡•}¤Œ-|¡˜Ω(r)•KD0’€”Ž‖ rad r ‖ ≤ δ¢4´8ƒ=„ Œ)¡„‡†„ÔªŠ”˜[¡¤Œ-€}¡‚¤˜”˜ Ë Æ©VÐ„‡†ˆŒ)|&Š”}¡=}e}¡‰¤}¤Œ-•}¡‚¤&€˜¡|&‚¡ƒ=Š¤S’Kƒ=•”˜”ƒœô.Î ª ¢Ù ‰œ}|&}7„œ§)“”|¡}|&}¡“”•}¤Œ-|¡Ÿ}¡—”ƒ=•”˜”‚7&•”˜Ž ª”Š”˜ª ¢ÎE§¡ÐKª”}7Œ-Š”ƒ-’4Œ-|¡‚¡}¤„ ËÎƒ)|¦ª}&€}'e˜”|&ƒ)€Ÿ¡•}¡ƒ—”ƒ)€}¡ƒK ɛ§7|>&}¡ƒ“|&}P¡•>€}¤)˜”“•”}¦•”ƒ=Š&‚¤ƒ)•Œ)|&‚ ¡ ËÎ ª ¢ÎŒ7ÐW§Ω(r)|¡ƒ=„ž‚¡†–…ƒ¤§“”ƒ=„ eË|¡}¡•”Ÿ¤–…ƒ.Ðâ˜•”|&ƒ)Š‚¤>€KŸ¡•”†~‚¡ƒ=”|&}¡Š rÐ-¢´8ƒ)ªƒ)Š”Ÿ¬„‡†²€Kƒ))}c„‡}'Sƒ=„ ’K}¡¡‘=&|¡Ÿ Œ›7}.’K˜„‡}7Œ-|&Ÿ ª”Š”}7Œ-|&ƒ=~”–…ƒ=)}c„‡ƒ=|&}.’4¡7Pƒª ¢Ù ‰¢cŠ¤Œ-€˜{ Q (k) }’KŠ”}¤‰&€ƒ=•”˜Ž×Š”ƒ)–Pƒ)•”˜”~×˜”‘cô=Î"ª”}7Œ!€ƒ-’K}¡‚7&|&ƒ)€Ÿ¡•”}7Œ-|&Ÿ ‚¤ƒ-š’K¡‹…˜› ‰¡Š”¡7Œ-}¡‚ Ë¡˜ ª”Šƒ-S’Kƒ¡§k}¤‰¡}¤‘)•'“¡ƒ)|P•”}¤„‡ƒ=Š –S&-«>€=}¡Š”˜|¡„ ¤Ð-§”|&}k ≥ K’€”Ž ˜„‡ƒ=ƒ)|¯„‡ƒ=Œ)|&}ɛ0 ≤ min{ Ω(r) | r ∈ Q (k) } − Ω(Q (k) ) ≤ ɛ.

∈ ¦©ƒ)|&Š¡)’4•”} ª”}¡•Ž”|¡Ÿ§8“”|¡}τ > 0§8|>¡¦& W˜D 0Ž”‚&€Ž”µ(QŒ-ª”}¡€Ÿ¡‘=¡&Ž¯|>&”PƒQ•”ƒ=Š&‚¤ƒ)•Œ)|&‚¤Ω(Q (k) ) ≤ min{ x ν | x ∈ Ξ αβ }= min{ Ω(r) | r ∈ (V 1§V¢)¢=¢)§V ν−1§V ν+1§8¢)¢=¢)§V n ) }≤ min{ Ω(r) | r ∈ Q (k) },Ú>ª¡ÙÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã“”˜|&Ÿ§”“|&} „‡†{„‡}'Sƒ=„‘=&”€µ0 ≤ min{ x ν | x ∈ Ξ αβ } − Ω(Q (k) ) ≤’€”Žɛ k ≥ K ɛ .D 0 = Ëª”Š”˜Øz=|&}¤„ •”} €˜”–PŸ[‘=¤„‡ƒ)•”˜”|¡Ÿ∅D•0Ω = D’K}7Œ-|>&|&}¤“š‚¡}[‚7Œ-ƒ)›¦ª”Š”ƒ-’K–…ƒ.Œ-|&‚¤¡µ 1Ð!¢&¦€”Ž¬’K}¤¡‘=¡|&ƒ-€Ÿ7Œ-|&‚7z=|&}¡=}£¥|&}«˜ }¡‘)•K'“&ƒ=|SŒ›7}.’K˜„‡}¤Œ)|&Ÿ…ªŠ”}¤Œ)|&ƒ)~”–Pƒ)=}«„‡ƒ=|&}.’4P’KŠ}¡‰&€Kƒ)•”˜ŽŠ”ƒ)–Pƒ)•”˜~8¢©Œ-•”}¦|'¡Pƒ¤§+“|&}¦•K&– ‚¡†‚¡}.’ }7Œ-|>&•”ƒ=|¡ŒŽŒ-ªŠ&‚¤ƒ!’€˜‚¡†„‡˜ ˜ž‚¦Œ!€K¡7“K&ƒËÎ ª ¢Î'±¡ÐW¢ ¨‡¤Œ)Œ=„‡}¡|&Š”˜K„|&ƒ=ª”ƒ)ŠŸ¯Œ!€¡“¡~ •'“”˜”•K>ŽŒD 0 ∅˜≠ W ∩ D 0 ∅¢”Ê©¡¤Œ)|&Ÿ=min{ ‖ r − r ‖ | (r ∈ D 0 ) & (r ∈ W)} = 2τ.D1‹…˜”›¦Š¤Œ)Œ-¡7S’Kƒ=•”˜Ž›V§{}Ω ∣ min ‖ r − r ‖ ≤ τr∈W|¡ŒŽ×•”ƒ)ªƒ)Š”ƒ.Œ-ƒ)¡µš‹…˜„‡˜KŒŽ }¤„‡ª¡”|>¡„‡˜8¢K´8}¡’®S„‡•}'Pƒ=Œ)|&‚¡}min ‖ r − r ‖ ≥ ‖ rad r ‖.rËÎ ª ¢ÏÎ Ì Ð|>&Sƒ•”ƒ ªƒ)Š”ƒ.Œ-ƒ)¡ƒ)|D 0§7|'¢ƒ¤¢}¡•”}¦—ƒ-€˜”}7„ØŒ-}.’Kƒ=Š¤e˜”|¤ŒŽ«‚D 1¢>&¦€Že€µ ‰¤†œ›r˜ r§”}¤“”ƒ)‚¤˜’K•”}K§‖ r − r ‖ ≥ ‖ rad r ‖§”Pª}¡|&}7„‡¡}¤‰¡Š¡‘)}7„œ§7„‡•”}'Sƒ=Œ)|&‚¡} ËÎ ª ¢ÏÎ Ì ÐÓ¡˜ „‡•”}'Pƒ.Œ-|&‚¤}D 1§”Œ)}.’Kƒ)Š7e˜”|Pª”}.’š„‡•”}'Pƒ.Œ-|&‚¤} ´³¡”˜„Ω | ‖ rad r ‖ ≤ τ },D τ = { r ∈ }¡|&}¤Š”}¤„‡¡òª”Š”˜”•K'’€ƒ)S¡|.§¥•'“”˜”•K>Ž¨Œc•”ƒ=}¡|&}¤Š”}¡~Ô˜”|&ƒ=Š&—˜”˜8§œ‚¤Œ)ƒc‚¡ƒ-’K¡‹…˜”ƒª”Š”}¤‚¡}.’K˜€K˜Q‚¨Š¡•”ƒ)ƒ¥Š¡Œ=Œ)„‡}¤|&Š”ƒ)••”}¤„cŒ-€¡“&ƒ%=|>¡…&…Õ©¡•”—”˜Ž Ω(r)Š¡‚¡•”}¡šŒ-}¤‚¡ƒ)Š–…ƒ)••”}…&•K>€}¤)˜”“”•†|&ƒ=„œ§“”|¡}S„‡†‰¡Š”¡7Œ-†Q¢&P>€Ÿ¤•”ƒ)~–…˜”ƒ¨Š¡Œ=Œ-¡7S’Kƒ)•”˜”ŽD τ§)|&}‚¤†ª”}&€K•Ž”ƒ)|¡ŒŽ©•”ƒ=Š&‚¤ƒ)•Œ)|&‚¡} ËÎ ª ¢ÎŒ7ÐW§˜ |.¢’+¢ „‡ƒ)Š”•}•”ƒ)ªŠ”ƒ)Š”†‚¤•œ•œ}¤„‡ªK&”|&ƒ´8ƒ=„ÔŒ=¡„‡†„ž|&ƒ)}¤Š”ƒ=„ S}¡¡‘)†‚¤¡ƒ)|¡ŒŽ ª”}&€•}¤Œ-|¡Ÿ¡µõ’K}¡¡‘=&••”}¡~8¢

, ËÎ ª ¢ §”Î.Ðs,Çß WßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³«ï³ã!ð«ãWäD”ñÚª¤Ú•””—˜”˜ˆ‚¨¡7Œ!€}¤‚¡˜”˜ Ë Æ©VÐ Ž”‚¡€”Ž”ƒ)|¤ŒŽ ªŠ&”|¡˜”“”ƒ=Œ)”˜|&Š”¡=’K•”}¤ª”Š”}¤‚¡ƒ)Š7Ž”ƒ.„‡†„‚¦Œ)‚¡}¡ƒ=~P˜KŒ›¤}.’K•”}¤~SÕ©}¤Š„‡ƒ¡§¡•”}§&˜KŒ-ª”}¡€Ÿ¡‘=¡&ŽP˜•”|&ƒ)Š‚¤>€KŸ&šy¥|¡}¡Š”}¤~Ö“€ƒ)•Ö’K˜”‘O³µ•”ƒ)•˜Ž •”¡µÖ&Š”˜ÕQ„‡ƒ=|&˜””¡¤§”„‡†õ„‡}'Sƒ=„¬’4¡|&Ÿ«ª”Š”}7Œ-|&}¤ƒ’K}7Œ-|>&|¡}¡“”•”}¤ƒ¨¡7Œ!€}¤‚¡˜”ƒ¨‚¡†ª”}¡€”šÆ¨©VÐ¥%Ë•”ƒ²Ž”‚&€Ž”ƒ)|¡ŒŽa}¡•”—}¤„ •”¡.€Ÿiν(0˜”€ñ∑/∈ b i −)¢a ij V jj=1, j≠ν’€Ž >S’K}¡=}i = 1, 2, . . . , m,ß_àgáZâýí_ëJê_ì&ò êú‘þ§5ì&årþ‚ê_ëêð ëízê¦ø-ó`å_è/éízå_æçå_èzé/÷„‡†×ª”Š”ƒ-’4Œ-|>&‚¡€”Ž”ƒ=„Š”ƒ=‘)¡.€Ÿ>|>&|¡†¦“”˜KŒ!€ƒ=•”•”†œ›©z)Œ-ª”ƒ=Š”˜„‡ƒ=•š|&}¡‚K§‚¡†ª”}¡€•”ƒ=•”•”†œ› ŒQ„‡ƒ=|&}.’4¡„‡˜ ’KŠ}¡‰&€Kƒ)•”˜ŽŠ”ƒ)–Pƒ)•”˜~•KPŠK&‰¡}¤“”ƒ)~ Œ-|>¡•”—”˜”˜ycz=|&}¤„ª¡Š&=Š¡Õ©ƒMV˜Q˜Œ-ª}&€Ÿ¤‘)}¡‚7>€˜©Œ-|>¡•’4&Š|>š¢ÑšWÎ ª ¢D¦Q&–P˜‚¤ƒ)ŠŒ)˜”˜c>€K)}¡Š˜”|¡„‡}¡‚e‰¤†œ€˜ Š”ƒ=&€˜”‘=}¡‚¤¡•”†{•SŽ”‘)†ƒ¦¨œÒ¬¥‡©7}¤Š”ª”}¤Š&—”˜˜¢ NŠ¥¦M"!M¬•”¡µw¡Š”˜ÕQ„‡ƒ=|&˜””¡ Œª€K¡‚¤¡µ ‹…ƒ=~ |&}¤“”}¡~×’K‚¤}¡~”•}¡~|¡}¡“”•”}7Œ-|&˜³¢0Üc}.’Kƒ-€Ÿ¤•”}¡~Œ)˜Œ-|&ƒ.„ «¡Š&‚¤•”ƒ)•”˜~|&ƒ=Œ)|&}¡‚¤}¡~‘=>’4'“”ƒ)~Œ!€¡7e˜€K«˜•”|&ƒ)Š‚¤>€KŸ¡•&Ž¯€˜”•”ƒ=~”•&Ž8>:[α−1, 1−β] x 1 + [α−1, 1−β] x 2 + . . . + [n−1, N] x n = [1−n, n−1],¢¢¢ ¢ ¢¢ ¢¢¢¢¢¢’Kƒ n}¡‰¤}¡‘)•K'“&ƒ=| Š¡‘=„‡ƒ)Š•”}¤Œ)|&Ÿ Ë n ≥ 2Ð!§0 < α ≤ β ≤ 1§¡ N"ËÎ ª ¢ § ª Ð‚¡ƒ=‹…ƒ.Œ-|&‚¤ƒ)•š(β = 1 2§4}¤‰O8ƒ-’K˜”•”=•”•”}¤ƒ…„‡•”}'Pƒ.Œ-|&‚¤}¯Š”ƒ=–…ƒ)•˜”~z=|&}¡~)[−34 , 0]1[−34 , 0] 1x =([−1, 1][−1, 1])&¦€”Žn = 2˜α = 1 4§(n − 1)¢ •”}¡ƒ©“”˜KŒ!€}K§•”ƒ¦„‡ƒ)•”Ÿ¤–…ƒ=ƒª ¢Î¡§-’€”Žn = 2§α = 1 4§ Œ-˜Œ)|&ƒ=„‡†ª”}¡&‘.&•”}•Kh¨Ó˜Œ¡¢7Îβ = 1§=|'¢ƒ¡¢=’€”Ž…Œ)˜Œ-|¡ƒ=„‡†}¡‰¦O8ƒ!’K˜•”)•”•}¡ƒ „‡•}'Pƒ=Œ)|&‚¡}¬Š”ƒ=–…ƒ)•˜”~Ôª”}¡&‘.&•”}¦•m¨Ó˜Œ&¢ Î ª ¢ $”§ m¨Ó˜Œ¡¢Î ª ¢Ù

. ËÎ ª ¢ §§¡Ð( üãû z>ü ) û( üãû z(z ) û(1[−34 , ]34[−34 , 3 41]Ú>ª¤ÛÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã˜”‘)}¤‰¡Š&S¡ƒ)|…ƒ=)}P’€”Žn = 2˜α = β = 1 4§|.¢ƒ¡¢’€”Ž[Œ-˜Œ)|&ƒ=„‡†)x =([−1, 1][−1, 1])4x 24x 14x 2rƒøËûDüãû Ì ûþPöÿ;k-ø¨¤¢k¤¨¤kÿ(µf¤kÿhg£¡Eÿáï£¥¤ø¨©ø¡f§4x 1rƒøËû/üãû Ö ûþPöÿ;kkø¦¤¢¥¤¦¤kÿ¡{¤-ÿhgj¥¡!ÿá¦i£¥¤-ø¦©|-÷#vø¥f§yâ¡Š”Ÿ¡˜Š”¡&ŽS‘)•'“”ƒ=•”˜Ž α§ β§ n˜ N§>˜”‘ ËÎ ª ¢¨§ ª Ð8•”ƒ=|&Š”¡=’K•”}…ª”}&€¡“”˜|&Ÿ©–…˜”Š”}¡š

min{ x i | x ∈ ˜Ξ } = − max{ x i | x ∈ ˜Ξ }, i = 1, 2, . . . , n. ËÎ ª ¢¨§>$7Ð&P>€ƒ=ƒ¡§®’€Ž¦€µ¨‰¡†œ›i, j ∈ {1, 2, . . . , n}ª”}7Œ!€ƒe‘.¡„‡ƒ)•†x i•R n§”|>&“”|&}ƒ)=}Š¡‘=„‡ƒ)Š}¡‚¯ª”}¡€}'e˜„x 1 = x 2 = . . . = x n‚˜×•&šWßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³«ï³ã!ð«ãWäD”ñ Úª ±s,Çß˜”•|&ƒ)Š”‚7>€Ÿ¤•”†œ›€K˜”•”ƒ)~•”†œ› Œ)˜Œ-|¡ƒ=„’€”Ž[|&ƒ.Œ-|&˜”Š}¡‚¤¡•”˜Ž[>€=}¡Š”˜|>š„‡}¡‚PŠ”ƒ)–Pƒ)•”˜Ž ”˜”~[•¡‰¡}¤Š ‚¡•”ƒ=–…•”ƒ=~¯‘=>’4'“”˜ƒÅ'¢k§©}¡’4Ä¡„‡ƒ=•”Ÿ¡–S&ƒ=|¡ŒŽ8§¡ª”Š˜”‰&€˜”S>ŽŒ)Ÿ…β¡|&Š”˜”—KSŒ)˜Œ-|¡ƒ=„‡† ËÎ ª ¢ § ª ÐÓŒ)|>&•”}¤‚¡˜”|¡ŒŽ¯‚7Œ-ƒ$‰¡}¡€ƒ)ƒ¨‰&€˜”‘=}¡~}¤Œ)}¡‰¤ƒ)•š•”}¡~³§ac„‡•}'Pƒ=Œ)|&‚¡}Š”ƒ)–Pƒ)•”˜”~•”ƒ)}¤)Š¡•”˜”“”ƒ=•”•”}c¡‚¡ƒ)€˜”“”˜‚¤&ƒ=|¡ŒŽØ‚cŠ¡‘=„‡ƒ)ŠK>›V¢•”¡.€µ…§”„}¤|&•”}¡–Pƒ)•”˜ƒ α˜ β§„‡† „‡}'Pƒ=„ž„‡}.’K˜Õ©˜”—˜”Š”}¡‚7&|&ŸeÕ©}¤Š„‡¡„‡•”}'Sƒ-šŒ-|&‚7SŠ”ƒ)–Pƒ)•”˜”~ ˜|'¢ª8¢(©‘=„‡ƒ=•ŽŽ˜Ξ’€Ž ËÎ ª ¢ § ª Ð¥€ƒ=)}¯‚¤†œŽ”‚¡˜”|&Ÿ˜”‘PŒ)}&š}¡‰¤Š>Sƒ)•”˜”~ÖŒ)˜„ „‡ƒ)|&Š˜”˜8¢ ¦Q'“”•”ƒ.„jŒØ|¡}¡)}K§ “”|¡}¨Š¡Œ=Œ)„ ¡|&Š”˜”‚7&ƒ=„ &Ž{˜•”|&ƒ)Š”šó¥|&Š”¡”|&¡Š”¡c„‡•”}'Sƒ=Œ)|&‚¤«Š”ƒ=–…ƒ=•”˜”~[˜•”‚¤¡Š”˜&•|&• }¡|¡•”}¤Œ)˜”|&ƒ-€KŸ¡•”} ˜”‘.„‡ƒ)•”ƒ=•”˜Ž ‘)•K>š}¡‚$‚¤Œ-ƒ)›¦}¤„‡ª”}¤•”ƒ)•”|©Š”ƒ=–…ƒ=•”˜ŽP•K¨ªŠ”}¡|&˜‚¡}¡ª}&€}'e•”†~8§'|>&S&S‚¡ƒ=”|&}¡ŠSª”Š&š‚¤&€Ÿ¡•&Ž€˜•”ƒ)~”•K>ŽØŒ-˜Œ)|&ƒ=„‚¡}¤~Ø“¡Œ)|&˜ ¡Š&‚¤•”}¡‚¤ƒ)–Pƒ)•8¢aó¤€Kƒ!’K}¤‚¤&|¡ƒ-€Ÿ¤•”}§³„‡•}'Pƒ=Œ)|&‚¡}[Šƒ)–…ƒ=•”˜”~˜ΞŽ”‚¡€”Žšƒ)|¡ŒŽØ—”ƒ=•”|&Š&€Ÿ¡•}¦Œ-˜K„ „‡ƒ)|&Š”˜“”•”†„Ö}¤|&•”}¤Œ)˜”|&ƒ)€Ÿ¡•”}•'“>€4 7}¤}¡Š¤’K˜”•K&|'§³˜8§a‚“¡Œ)|&•”}7Œ-|&˜8§min{ x i | x ∈ ˜Ξ } = min{ x j | x ∈ ˜Ξ },xj}¡‰¤}¡Š”}¤|˜”•”|¡ƒ)Š”‚7>€Ÿ¤•>ŽžŒ)˜Œ-|¡ƒ=„ ËÎ ª ¢¨§ ª Ð…|>&Sƒ }¤Œ-|>¡ƒ)|¡ŒŽ•ƒ)˜”‘.„‡ƒ)•”•”}¤~8¢q(©‘z)|&}¤)}„‡}'«•}‘=¡€µ¨“”˜”|&ŸK§4“|&}„‡•}'Pƒ=Œ)|&‚¡}˜ΞŒ-˜K„ „‡ƒ)|&Š”˜“”•”} }¡|&•}¤Œ-˜|&ƒ-€Ÿ¤•”}‰¡˜Œ=Œ-ƒ=”|&Š”˜Œ)†ª”}¡€}'e˜”|&ƒ)€Ÿ¡•”}¤)}¨˜P}¤|&Š”˜”—¡|&ƒ-€KŸ¡•”}¤)}$}¡Š”|>&•|&}¡‚$ª”Š”}7Œ-|&Š¡•Œ-|¡‚¤max{ x i | x ∈ ˜Ξ } = max{ x j | x ∈ ˜Ξ }’€”Žc€Kµ ‰¡†œ›i, j ∈ {1, 2, . . . , n}¢8ó¥}¡ª”}7Œ-|>&‚¡€”ŽŽ[z)|&˜¬Œ-}¤}¡|&•”}¤–…ƒ=•”˜ŽŒ ËÎ ª ¢ §$¤ÐW§‚¡†‚¡}.’4¡§“”|&}e˜”•|&ƒ)Š”‚7>€Ÿ¤•>Ž }¡‰¡}¡€}¡“S„‡•”}'Sƒ-šŒ-|&‚7SŠ”ƒ)–Pƒ)•”˜”~ „‡}'e•”}§•¡}¡•”ƒ=—8§ª”Š”˜”~|&˜[ >€Ÿ¤•”}¡ƒ©˜”•”|¡ƒ)Š”‚7>€Ÿ¤•”}¡ƒ©Š”ƒ=–…ƒ)•˜”ƒQ‘=>’4'“”˜‚¡•”ƒ=–¦š˜Ξ§|.¢ƒ¡¢}¡ª”|&˜K„}¡‰¦O8ƒ!’K˜•”)•”•}¡)}¦„‡•”}'Pƒ.Œ-|&‚7[Š”ƒ)–Pƒ)•”˜~žŒ-˜KŒ-|&ƒ.„‡† ËÎ ª ¢¨§ ª ÐW§Ž”‚&€”Ž”ƒ=|¡ŒŽ )˜”ª”ƒ=Š””¡‰¡}7„ÔŒ©—”ƒ)•”|¡Š”}¤„Ô‚e•'“K>€ƒQ}¡}¤Š¤’K˜”•¡|.¢&¦€”Ž[}¡ª”Š”ƒ-’Kƒ-€Kƒ)•”˜Ž•”ƒ)=}}¤—”ƒ)•”˜‚¤&•˜ŽŒ)˜Œ-|¡ƒ=„‡†ÖŒ-}¡€Ÿ¡µ¨|¡ŒŽ ‚«}.’K•”}‚¤Œ)ƒ$¡Š¡‚¡•”ƒ=•”˜Ž[ n − 1, N ] · x 1 + (n − 1)[ α − 1, 1 − β ] · x 1 = [ 1 − n, n − 1 ],˜€˜[1,]N· x 1 + [ α − 1, 1 − β ] · x 1 = [−1, 1].n − 1Ä‚¡•”ƒ=–…•”ƒ=~‘=>’4'“”˜ƒÅ¦ª”ƒ)Šƒ=„‡ƒ)••>ŽxÊ©Š”˜Šƒ)–…ƒ=•”˜”˜’€”Žz=|&}¡=}¦¡Š¡‚¡•”ƒ=•”˜Ž1Š¤Œ)Œ=„ &|&Š”˜‚¤&|¡Ÿ¤ŒŽ…&xH>?¥zI?RGRCLH?-þþK÷£?+;=/Gü4÷'§'7}¤|&}¡Š}¡ƒ¡§'‚QŒ-˜”€¡©’K˜Œ-š’K}&€«•KËÎ ª ¢ § ª ÐW§4‚Š”ƒ=‘)¡.€Ÿ>|>&|¡ƒ…“”ƒ=)}

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Ω(r)•”ƒ)ªŠ”ƒ)Š”†‚¤•”}¡~¬•’4¡•”•”}¤„ ‰¡Š”¡7Œ-ƒ P¢+ŷ Œ-‚¡Ž”‘)˜¦ŒSz=|&˜„ &‚¤|&}¡Š¦‰¡ƒ)Š)|•Ä‚¤Œ)|&Š¡˜”‚¤¡•”˜”ƒÅ«‚e•”˜”›z.ÕQÕ©ƒ)”|&˜‚¡•”†œ› €}¡>€KŸ¡•”†œ›}¤ª”|&˜„‡˜‘=&—˜”}¡•”•†œ› ª”Š”}¡šÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ãÚ Î ªª”}¤„‡}¤‹…Ÿ¤µ >S’K}¤)}Q˜”‘$ª”Š”˜”)š„‡}¡‚K§¤ª”ƒ=Š”ƒ)“˜Œ!€ƒ=•”•”†œ›•K…Œ-|¡Š8¢VÎ Ú¤Ú §7‘=¦˜KŒ-€µ¨“”ƒ)•˜”ƒ=„žŒ)¡„‡}¤)}Pª”ƒ)Š”‚¤}¡=}¢&…ƒ-€}„‡ƒ)|&}.’4}¡‚©’KŠ}¡‰&€Kƒ)•”˜Ž¯Šƒ)–…ƒ=•”˜”~«’K€”Ž•(…óY¡n¢0£õŒΩ(r)“”Šƒ)‘)‚¤†“&~•”}©}¤Œ!€K}'«•”Ž”ƒ)|}¡‰¤•&Š”¡7Sƒ)•”˜”ƒƒ) „‡}¤•”}¡|¡}¡•”•”}7Œ-|&˜ª”}¦|¡ƒ=„Ö˜€K˜˜”•”†„ˆª”ƒ)Š”ƒ.„‡ƒ)•”•†„Ö•¦‰¡Š”¡7š‚Q|&}¤„œ§7“”|&}…Š&‘=Š”†‚¡•”}7Œ-|&Ÿ©—”ƒ)€ƒ)‚¤}¡~eÕ©¡•””—”˜˜P ⊆ (Œ)&›V 1 , . . . , V ν−1 , V ν+1 , . . . ,}¤‰¡†“”•”}…’€”Žz)|&}¡šV n}¤‰&€K¤Œ-|&˜×‘=•'“”ƒ)•˜”~¦•K )¢Ê©Š”˜K„‡ƒ)•Ž”ƒ.„‡†~ )}cŒ)|'¡•’4¡Š”|&•”†~ØŒ-ª}¤Œ-}¤‰x"˜Œ)Œ-€ƒ!’K}¤‚¤¡•”˜”ƒ«‘)•¡ Pª”Š”}¤˜”‘)‚¤}.’K•”†œ›∂Ω(r)/∂r k§k = 1, . . . , ν − 1, ν + 1, . . . , n§"‚[}¡‰¡‹Pƒ=„õŒ!€¡7š“&ƒ$|&ƒ=ª”ƒ)Š”Ÿ«¡7Pƒ$•”ƒ$ª”Š”}'›7}.’K˜”|'¢´³& Pƒ©•”ƒ=ª”Š”}7Œ-|&}S}¡ª”Šƒ!’Kƒ)€˜”|&ŸK§¤Ž”‚¡€”Ž”ƒ)|¤ŒŽ¯€˜Œ-ƒ=‰'ŽŒ)„‡ƒ)€}¤Œ)|&Ÿe¡|&‚¡ƒ=Š7P’®&|&ŸK§“”|&}¯‚¤‚¡ƒ!’4ƒ)•”˜”ƒ©‚¯„‡ƒ=|&}.’K† ’KŠ”}¡‰¡€ƒ)•˜ŽŠƒ)–…ƒ=•”˜”~ª”Š”˜”‚¤ƒ!’K=|eŒ-¡‹Pƒ=Œ-|¡‚¡ƒ)••”}¤„‡¡ ª”}¡‚¤†–…ƒ)•˜”µˆ˜› z=ÕQÕ©ƒ=”|&˜”‚¤•”}¤Œ)|&˜8§•}§KŒ)}¡Š”ƒ=ƒ¡§’€”ŽZ(…óY¡n¢0£žª”Š”}¡—ƒ!’K¡Š”†×ª”Š”}¤‚¡ƒ=Š””˜¦—”ƒ)€ƒ)‚¤}¡~¦Õ©¡•””—˜”˜¦•¨„‡}¡•”}¤|&}¡•”•}¤Œ-|¡Ÿ•”ƒŒù’Kƒ-€4&ƒ)| z)|&˜×>€K)}¡Š˜”|¡„‡† ‰¤}&€ƒ=ƒP|.ŽSƒ-€}¤‚¡ƒ=Œ)•”†„‡˜˜¦|&Š”¡=’K•”}¤Š”ƒ=&€˜”‘=¡ƒ=„‡†„‡˜³¢¡=’Kƒ.„Ø’®>€ƒ=ƒ$ŠK&‘)‚¤˜”‚¤¡|&ŸSz)|&}¤|Sª¡7|¡Ÿ¢Ê©}¡z=|&}¤„‡¡„‡†õ•”ƒQ‰ù*)/å_èzé/ð¦ã‡èzé/å,+/ê-¥_èzã‚ò ê¦ê.+/í/å‘ëåaó¨_è/èzî“ì ø¦í/Gú‘ã‡ìß_àgá('-ã¦Q&˜”‰¤}&€ƒ=ƒ ŠK'’K˜”&€Ÿ¡•”†„²¡¤Œ)}¡‚¤ƒ)Š”–Pƒ)•Œ)|&‚¡}¤‚¤&•˜”ƒ=„ˆ„‡ƒ=|&}.’K}¤‚’4Š”}¡‰¡€ƒ)•”˜”ŽòŠ”ƒ)š–…ƒ=•”˜”~{‚žª”Š˜„‡ƒ)•”ƒ=•”˜”˜ {˜”•”|&ƒ=Š”‚¤&€Ÿ¡•”†„ €˜”•”ƒ=~”•”†„wŒ-˜Œ)|&ƒ=„ ¤„ Ž”‚¡€”Ž”ƒ)|¡ŒŽ—”ƒ!’4¡7Š³§K}¡|¡}¡Š”†ƒ¦z=Œ-ª€¡7&|&˜Š”¡µ |ÕQ¡”|e€}¤&€Ÿ¡•}¡~[„‡•”}¡=}¡=Š&•”•}¤Œ-|¡˜[—”ƒ-€ƒ)šΩ(r)¢&…ƒ)~Œ)|&‚¡˜”|¡ƒ-€Ÿ¤•”}§.ª”ƒ)Šƒ=Œ-ƒ=“”ƒ)•˜”ƒâ„‡•}'Pƒ=Œ)|&‚¤ ÑÒ šŠ”ƒ)–Pƒ)•”˜~a(…óY¡n¢0£žŒ¥>«š‚¡}¤~Õ©¡•””—˜”˜R n Ë…‚…•”ƒ=}¡|&}¤Š”†œ›S“K¡Œ-|¡•”†œ›¯Œ!€¡“&Ž›…’4&Pƒ¨˜Œ•”ƒ.Œ-}&€KŸ&š”˜„‡˜[}¤‰O8ƒ-’K˜”•”ƒ=•”˜Ž„‡˜}¡Š”|>&•|&}¡‚ÐY" „‡•”}¡=}¡)ŠK&•”•”}¤ƒ©‚¡†ª”¡€}¤ƒQ„‡•”}'Pƒ.Œ-|&‚¤}¢’K†„×}¡Š|'¡•”|&}7„×‚(n − 1)š„‡ƒ=Š”•”†~ ‰¡Š”¡7Œ P§P ⊆ ( VÊ©¡¤Œ)|&Ÿ1 , . . . , V ν−1 , V ν+1 , . .§. V nƒ)|«Œ-‚¤}¡˜„‡˜}¤„‡ª”}¤•”ƒ)•”|>¤„‡˜˜”•”|¡ƒ)Š”‚7>€†Q§•”ƒQŒ)}.’Kƒ)Š7S¡‹…˜”ƒ¨‚¡•”¡|&Š˜Œ)ƒ)‰'Ž•”¡.€ŸK¢ )§˜„‡ƒ)š‰¤¡)’4ƒ=„ˆ•K&‘)†‚7&|&Ÿ¦|>&”˜”ƒ‰¡Š¡¤Œ-† j&þKû>”§1š}¡~8§®¢=¢)¢)§ν n§}¡‰¤Š&‘=}¡‚7&•”•”†ƒ¨|&}¡“¤„‡˜8§¡e}¡|&}¤Š”†œ›‘=•&”˜Œ-|&‚¤}[ª”Š7Ž„‡†œ›{ l(r) | r ∈ P }ª”ƒ)Š”ƒ.Œ-ƒ=¡ƒ)|[‚7Œ-ƒ)=} €˜”–PŸ ’K‚¤ }¤Š”|>&•”|> O′˜•”ƒ=}¡|&Š”˜—&|&ƒ)€Ÿ¡•K”¢Kó¤€ƒ-’K}¡‚7&|&ƒ)€Ÿ¡•”}K§Œ-}¤}¡|&‚¤ƒ)|¡Œ)|&‚¡ƒ=•”•”}§•”ƒ=ª”}&€K}'«˜|&ƒ-€Ÿ¤•S˜€˜min{ Ω(r) | r ∈ P }= min { x ν | x ∈ Ξ αβ (A, b) & ( x 1 , . . . , x ν−1 , x ν+1 , . . . , x n ) ∈ P }= min{ l ′ , l ′′ },

c ⊤ y → min, c = (0, . . . , 0, s ′ ν, 0, . . . , 0) ∈ R n , ËÎ ª ¢ § Ù Ðl ′ = min { x ν | x ∈ Ξ αβ ∩ O ′ & ( x 1 , . . . , x ν−1 , x ν+1 , . . . , x n ) ∈ P } ,s,ÇßÐ^ßÚ¦åyc$#\3‰u>áë¤ã.î'åyaå!œ¦yï³å¤æ'Ý4ã!ä³«ï+ãWð«ã!ä”ñÚ Î¡Î’Kƒl ′′ = min { x ν | x ∈ Ξ αβ ∩ O ′′ & ( x 1 , . . . , x ν−1 , x ν+1 , . . . , x n ) ∈ P } .¦©}Ξ αβ ∩ O ′ ˜l ′′ „‡}¤)¡|×‰¡†|&ŸØz.ÕQÕ©ƒ)”|&˜‚¡•”}×•¡~šΞ αβ ∩ O ′′ ‚¤†ª”¡€†ƒ„‡•}¡)}¤)Š¡•”•”†ƒc„‡•”}'Sƒ=Œ)|&‚¤§Ó}&š|&}¡Š†ƒ }¤ª”Š”ƒ!’4ƒ-€”Ž”µ¨|¡ŒŽž•”ƒ.Œ!€}'e•”}¦‚¡†ª”˜KŒ-†‚¤¡ƒ=„‡†„‡˜òŒ-˜Œ)|&ƒ=„ ¤„‡˜ €˜”•ƒ)~”•”†œ›"˜ª”}¤|&}¤„‡¡ž‘)•'“”ƒ=•”˜Žl ′ ˜ •”ƒ)ŠK&‚¡ƒ=•Œ-|¡‚§&•”}¤•”˜”“”ƒ.Œ-”¡µÖÕ©}¡Š„‡¡ &•&€}¡=˜”“”•”}S|&}¤„‡¡¤§”&z)|&}«‰¤†œ€}eŒù’Kƒ-€K¡•”}«‚ô=Î ª ¢ÏÎ¤¢(p ′ = −P 1 , . . . , −P ν−1 , + ∞, −P ν+1 , . . . , −P n ,P 1 , . . . , P ν−1 , + ∞, P ν+1 , . . . , P n) ⊤.„‡˜”Š}¡‚¤¡•”˜Ž8¢£ ’Kƒ)•†ˆª”¡|&=„¨Š”ƒ)–Pƒ)•”˜Ž¦‘='’®'“[€˜”•”ƒ=~”•”}¤)}ªŠ”}¡=Š¡„ >Sƒ=„˜”›S ′ = diag { s ′ 1 , s′ 2 , . . . , s′ "’K˜¡)}¤•>€Ÿ¤•>ŽQ„ &|¡Š”˜”—‘=•&}¤‚‚¡•”¡7š|&Š”ƒ=•”•”˜›[|&}¤“”ƒ)c}¤Š”|>&•”|> Ê©¡¤Œ-|¡Ÿn }O x = S ′ x ∈ O p ′ ∈ R 2n′§Ke‚¤ƒ)”|¡}¡Š |x|’€”Ž ′§K|'¢ƒ¡¢}¡ª”Šƒ!’Kƒ)€”Ž”ƒ)|¡ŒŽ&⎧ ( ) ( )mid A · S ′ − ( rad A ∃ − rad A ∀ )bcy ≤,−mid A · S ⎪⎨′ − ( rad A ∃ − rad A ∀ )(−b c )( )−S′y ≤ p ′ ,S ′⎪⎩y ≥ 0.ËÎ ª ¢ § Ú Ð´8}¤’4«}¡—”ƒ=•” l′Ž”‚&€Ž”ƒ)|¡ŒŽŠ”ƒ)–Pƒ)•”˜ƒ=„ž‘='’®'“”˜€˜”•”ƒ=~”•”}¡=}«ª”Š”}¡=Š¡„ „‡˜Š”}¡‚7>š•”˜ŽŒ$}¡)ŠK&•”˜”“ƒ)•”˜Ž„‡˜‚¡†ª”˜Œ)†‚¤¡ƒ)|¡ŒŽ¬˜Ø‘='’®'“ €˜”•”ƒ=~”•”}¡=}[ª”Š”}¤)Š¤„ „‡˜šŠ”}¡‚7&•”˜”Ž8§7}¤ª”Š”ƒ-’Kƒ-€”Ž”µ¨‹P&Ž }¡—ƒ)•””¡ ¢•&€}¡=˜”“”•”†„õ}¡‰¤Š&‘=}¤„l¨|¡„‡ƒ)|&˜K„œ§“”|&}eªŠ”˜Ξ αβ (A, b) ∩ O ′ ≠ ∅Œ-ª”ŠK&‚¡ƒ-’€˜”‚¤}P•ƒ)Š¡‚¡ƒ)•KŒ-|&‚¤}′′¢l ′ < l ′′ ,min{ Ω(r) | r ∈ P }’€”Ž[‘=•&}¡}¤ª”Š”ƒ)š’Kƒ-€K)•”•”†œ› ‰¤Š”¡¤Œ)}¡‚ «ª”}¡|¡}¤„‡¡ª”Š&|&˜”“”ƒ.Œ-}¡ƒQ•&›¤}'S’Kƒ=•”˜”ƒ —ƒ-€ƒ.Œ-}¡}¤‰¡Š¡‘)•”}[•K'“”˜”•¡|&Ÿ¬Œ ‚¡†“˜Œ!€ƒ=•”˜Ž}¤—”ƒ)•”˜P l ′¢

Ú Î!§ËÎ ª ¢¨§ Ú Ð¤„‡•}'Pƒ=Œ)|&‚¡}S•”ƒ=ª”¡¤Œ)|&}§”|¡}Š¤Œ!€K˜}¤ª”Š”ƒ-’Kƒ-€”Ž”ƒ.„‡}¡ƒ©•”ƒ=Š&‚¤ƒ)•Œ)|&‚¤¤„‡˜min{ Ω(r) | r ∈ P } = l ′ÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ã¡7ªK&|&Ÿ¦eŠƒ)–…ƒ=•”˜”µõ‘='’®'“”˜S€˜”•”ƒ=~”•”}¡=}Qª”Š”}¤)Š¤„ „‡˜šŠ”}¡‚7&•”˜”Ž¯’€”Ž •&›¤}'S’Kƒ)•˜Ž ˜¯„‡}'e•”}$’4>Sƒ•”ƒªŠ”˜Œ-|lªŠ”}¡|&˜‚¡•”}7„žŒ-€¡“&ƒ¤§”}¡’4«Œ-˜KŒ-|&ƒ.„ «}¡)ŠK>š•”˜”“”ƒ=•”˜”~ ËÎ ª ¢¨§ Ú ÐÓ•”ƒ.Œ-}¤‚¤„‡ƒ=Œ)|&•§ ′′¢Kÿmin{ Ω(r) | r ∈ P } = l ′′ .ß_àgá('-ø0…ú1+/ê-ó¨§2¥_ê¦ð¦ã‡èzé/å…ózê¦ò ã‚ó§è/îRï í/å_æ5ã‡þ‚å_ózå_÷ô.Î ª ¢Ù ª”Š”˜Øª}¤Œ-|¡Š”}¡ƒ=•”˜”˜Ø„‡˜”•”}¤Š&•”|¡† ª}c}¡‰¡€K¡Œ)|&˜¦’€”ŽØ—”ƒ-€Kƒ)‚¡}¤~ØÕ©¡7•”—”˜”˜y ª”Š˜„‡ƒ)•”˜”€˜ ª”Š”}¡—ƒ!’K¡Š”¡¯ƒ=¨ƒ.Œ-|&ƒ.Œ-|&‚¤ƒ)•”•}¡)}S˜”•”|¡ƒ)Š”‚7>€Ÿ¤•”}¡=}PŠ¤Œ-–P˜šŠ”ƒ)•˜Ž8¢k&…Š”¡)}¤~8§”‰¤}&€ƒ=ƒ©Œ)}¡‚¤ƒ)Š”–Pƒ)•”•”†~[Œ)ª”}¤Œ)}¡‰Sª”Š”˜”‰¡€˜P=•”•”}¤)}e}¡—ƒ)•”˜”‚7&•”˜”ŽΩ(r)„‡†‚¡ƒ)€˜”“”˜”•†min{ Ω(r) | r ∈ P }ª ¢ÎE§¤ÐŒ-•”˜‘)¡Q„‡}'Pƒ=|‰¡†|&Ÿ }¤Œ-•}¡‚¤¡•Q•˜Œ)ª”}&€KŸ¡‘)}¤‚¤¡•”˜”˜©€}¤&€Ÿ¡•†œ›¨Š”ƒ=–P&|¡ƒ-€ƒ=~ ËÎËŒ=„œ¢Ð!¢ Š¤Œ!€˜«„‡† Š¤Œ-ª”}¡€K&-¡ƒ=„&”˜K„aš:€˜‰¡}€K}¡>€Ÿ¤•”†„ Š”ƒ=–P&|¡ƒ-€ƒ.„3¦4658794;:¡§'|&}K§ô!±¢Ìmin{ Ω(r) | r ∈ P }§P ∈ IR n−1§„‡}'Pƒ=|P‰¤†|&Ÿ«‚¡‘-Ž”|> }¡“”ƒ=‚¡˜’4•”}§&|¡Š”ƒ)‰¤¡ƒ=„‡}¤~e}¡—”ƒ=•”}¡~¯Œ)•”˜”‘)¡…’€”Ž( 346587Ž‚¡ƒ)€˜”“”˜”•K⊇( 345>7|'¡|&}¡‚ •”ƒ)}¤‰'›¤}.’K˜„‡}’K}¡¡‘=&|¡ƒ-€Ÿ7Œ-|&‚7¥Œ›7}.’K˜„‡}¤Œ)|&˜©ª”Š”}7Œ-|&ƒ=~”–…ƒ=)}„‡ƒ)|&}.’®â’KŠ”}¤‰&€ƒ=•”˜Ž©Š”ƒ)–Pƒ)•”˜”~©ª”}¡š’K}¤Œ)|'¡|&}¡“•”}P‚¤†ª”}&€K•”ƒ)•”˜ƒ©Œ-€ƒ!’4¡7µ¨‹…ƒ=)}«¡¤Œ-€}¡‚¤˜Žƒ%˜min{ x ν | x ∈ Ξ αβ (A, b) ∩ z } − ( (A, b, z) ) ν → 0 3¦46587

’4&PƒQŒ-}¤‚¤Œ-ƒ.„Ø)Š¡7‰¤†„ ˜€˜(A, b, z) ) ë¤å>ä4å7î>å'ä”ä4åZœå=Bùïuí)îRu7ã.îNw7ï@•ð”ë¤ã!äv¡ðeãWäD(3¦465>7áë¤ã.î'åyaå!œ¦yï³å¤æ'Ý4ã!ä³«ï+ãWð«ã!ä”ñ£¥|&}¦’K}¡‚¤}&€Ÿ¤•”}¦Œ)˜€Ÿ¤•”}¡ƒ|&Š”ƒ=‰¡}¡‚7&•”˜ƒ¡§‡|&ƒ=„²‰¡}¡€ƒ)ƒ¤§‡“”|&}§ ‚×}¡•ƒ)“”•”}7„ˆŒ-“)|&ƒ¤§•¡„ ‚¤&«•} •”ƒe¤Œ-˜„‡ª”|¡}¡|&˜”“ƒ=Œ-}¤ƒSª}¡‚¡ƒ-’Kƒ)•˜”ƒ«}¡—”ƒ=•””˜8§8§Vª”}&š‚¡}¤‘=„‡}'e•”}¤Œ)|&˜8§•&˜‰¡}&€Kƒ)ƒ |¡}¡“”•”}¤ƒ¨}¤—”ƒ)•˜”‚¤¡•”˜”ƒ ª}S‰¤Š”¡¤Œ=¡„×}¡•ƒ)“”•”}¤~–P˜”Š”˜”•”†Q¢§©Š”}7„‡ƒ¨|¡}¡)}K§|&ƒ)ªƒ)Š”Ÿc„‡† Š¡Œ)ª”}&€4&-&ƒ.„ò’€”Ž×}¡—ƒ)•”˜”‚7&•”˜”Žmin{ Ω(r) | r ∈ P }ª”} ‘=•&}&š}¡ª”Šƒ!’Kƒ)€)•”•†„ˆ‰¡Š¡¤Œ)¤„P„‡}¡‹P•”}¡~Ô„‡ƒ)|¡}.’K˜”}¡~õô=Î ª ¢Ú ”¢¥¨‡ƒ)}¤„‡ƒ=•’K¡¤ƒ=„Öª”}¡šz)|&}7„‡¡e}¡=Š&•”˜“”˜”|&Ÿ7ŒŽ«’€”ŽŠ”ƒ=–…ƒ=•”˜Ž‚¤}¡ª”Š”}7Œ)Q}Sª”Š”˜)}.’K•”}7Œ-|&˜|&ƒ-›¯˜€K˜ ˜”•†œ›(min{ xν | x ∈ Ξ αβ (A, b) ∩ z } − ( 3¦465>7

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˜Ù7Û!® ˜’4Š8¢Ð‚¡†ª”}¡€•”˜„‡}7Œ-|&Ÿe¡7Œ!€}¤‚¡˜Ž ËÎ ª ¢ $$7Ð¥€ƒ)=7}‚¡†‚¡}.’4˜”|¡ŒŽ ˜‘…Œ-‚¤}¡~Œ)|&‚¤Ê©Š”ƒ!’4ª”}&€}'e˜”‚K§“”|&}e‚¯„ &|¡Š”˜”—”ƒ Az)€ƒ=„‡ƒ=•”|a ij˜„‡ƒ)ƒ=|e•”ƒ)•¡=€Kƒ)‚Ú § ÛÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ãIR˜KRª”}‚¡€Kµ “”ƒ=•”˜”µ…¢ „‡}¡•”}¤|&}¡••”}¤Œ)|&˜˜”•|&ƒ)Š”‚7>€Ÿ¤•”†œ›&Š˜ÕQ„‡ƒ)|&˜“”ƒ=Œ)”˜›}¤ª”ƒ)Š¡—”˜”~¦‚Œ-˜”€¡¯Š”ƒ=‘)¡.€Ÿ>|>&|>«´8ƒ=}¡Š”ƒ.„‡† Ú ¢ §”¢Ú ymin{ x ν | x ∈ Ξ αβ (A, b) } = ( Ã −1˜b )ν’€”Ž •”ƒ)}¤|&}¡Š”†œ› |&}¡“”ƒ=“”•”†œ› „ ¡|&Š”˜”—†Ã ∈ R n×n ˜‚¡ƒ=”|&}¤Š˜b ∈ R€ƒ)••”†œ›¦˜”‘â7}¤•”—”}¤‚ z-€Kƒ=„‡ƒ)•|&}¡‚©„ &|&Š˜”—”† n§Œ)}¤Œ)|'¡‚&šA˜…‚¤ƒ)”|¡}¡Š §'ª”Š”˜“”=„ª”}¨Œ=¡„‡}¤„‡¡bΥ}¡ª”Šƒ!’Kƒ)€ƒ)•”˜µÖ}¡—”ƒ=•””˜Υ(Ã, ˜b) ≤ ( Ã −1˜b )ν .A ′ ˜Ã ′′ "„ &|&Š˜”—”†Q§”ª”}¡€¡“”ƒ)••”†ƒQ˜”‘ A‘=¤„‡ƒ)•”}¤~ z)€ƒ=„‡ƒ=•”|>aij¡µ –…˜Š”˜”•”¡¤§}¡‰¤}¡‘)•K'“”˜„A ′ Ã ′′ "&|&Š˜”—”†Q§ª}&€¡“”ƒ=•”•”†ƒ©˜”‘ Ã‘=¤„‡ƒ)•”}¤~ z)€ƒ=„‡ƒ=•”|> „ãij˜•aija ij§Œ)}¡}¡|¡‚¡ƒ)|¤Œ-|&‚¤ƒ)•”•”}K§A ′ ⊆ A, A ′′ ⊆ A,•aija ij§Œ)}¡}¡|¡‚¡ƒ)|¤Œ-|&‚¤ƒ)•”•”}K¢&P>€ƒ=ƒ¡§|>& &˜b§”|¡}¯Œ-‚¡}¤~Œ-|¡‚¡} ª ¢ ˜ $$¤Ð¤˜„‡ƒ)ƒ=|eŒ-‚¡}¤˜„Œ-€ƒ!’®Œ-|&‚¤˜”ƒ=„ž•”ƒ)ŠK&‚¡ƒ=•Œ-|¡‚¤⊆ b ËÎΥ(A, b) ≤ Υ(A ′ , b) ≤ Υ(A ′ , ˜b)Υ(A, b) ≤ Υ(A ′′ , b) ≤ Υ(A ′′ , ˜b).¡µ ‹P˜› “¡Œ)|&ƒ)~•”ƒ)ŠK&‚¡ƒ=•Œ-|¡‚§”„‡† ª}&€¡“”˜„ ó¤€ƒ!’4}¡‚¤¡|&ƒ-€KŸ¡•”}K§‰¡ƒ)Š7Ž[ª”}¤“€ƒ)••”†~c„‡˜”•”˜„‡¡„}¤|¯Œ-}¤}¡|&‚¤ƒ)|¡Œ)|&‚Υ(A, b) ≤ min { Υ(A ′ , b), Υ(A ′′ , b) }≤ min { Υ(A ′ , b), Υ(A ′′ , b) }≤ min { Υ(A ′ , ˜b), Υ(A ′′ , ˜b) } .

Ú §¤±s,ÇßÐ€ßÚ¦ã'î'åyVé¤yKï³å¡æ>Ý4ãWä³•w/u=ïƒu&ë¤ã'îï³å!œ§©Š”}¤„‡ƒ©|&}¤)}K§min{ x ν | x ∈ Ξ αβ (A, b) }Œ-}¡‚¤ƒ)Š”–Pƒ)•”•}«¡•>€K}¡)˜“”•”}«|&}¤„‡¡¤§& z)|¡}S’Kƒ-€K¡ƒ)|>š•”}¤~|(…óY¡n¢0£ Œ)˜Œ-|&ƒ.„‡†Qx = r‚¤„‡ƒ.Œ-|&ƒŒ˜›Ô}¡—”ƒ=•”¡„‡˜Υ(Q, r)‚min { Υ(A ′ , ˜b), Υ(A ′′ , ˜b) } ≤ ( Ã −1˜b )ν = min{ x ν | x ∈ Ξ αβ (A, b) }.•”ƒ=Š&‚¤ƒ)•Œ)|&‚¯ª”Š”˜‚¡}.’K˜”|Œ!€ƒ-’K¡µ ‹Pƒ=„‡¡ ª”Š¡šó¥}¡ª”}7Œ-|>&‚¡€ƒ)•˜”ƒQª”}&€K¡7“ƒ)•”•”†œ›b A ′′ x bG ë‡é ë¤å)ìeã-ëw¤ï@”ñ8îR G œå¡å¡æ>ç«ãP±>å!œå-ïD³ G ‰cæ¡å>Ý4ã)ãQî'å@&ä4å'ñ[å'á4ã!ä3‰ã=|&˜”“”ƒ.Œ-}¤„‡¡…‚¡†‚¡}.’4¡% ï³ã!ðZœ+ycœãq”äVî>ãï@œku>Ýv¤ä”éœêxw>íVí)î'ã-ë‡é”ñw®å7î'å&ë,‰uë¤å!±>åν | x ∈ Ξ αβ (A, b) }min { Υ(A ′ , b), Υ(A ′′ , b) } .œIœyaãb ió¥}¡‚¡ƒ=Š”–…ƒ=•”•”}Ô&•&€}¡=˜”“”•”†~{z=ÕQÕ©ƒ=”|ò˜„‡ƒ)ƒ=|ò˜{Š¤Œ-ª>’Kƒ)•”˜ƒ¬‚Ô‚¤ƒ)”|&}¤Š”ƒb i¢¨Ê©}¤z)|&}7„‡¡ ‚¤ª”Š”ƒ!’4ŸÔ’€”ŽÖƒ!’4˜”•”}¡}¤‰¡Š¡‘)˜Ž ’K}¡=}¡‚¤}¡Š”˜„ ŒŽ{}¡‰¤}¡‘)•K'“&|&Ÿ˜ª”Š¡‚¡†œ› “¤Œ-|&ƒ=~b&}¤)}&š•”˜”‰¤¡=’KŸc˜”•”|¡ƒ)Š”‚7>€Ÿ¤•”}¡=}¦z)€ƒ=„‡ƒ=•”|>bi•¦}¡•”—”†‹P˜”ƒ=ŒŽ˜”‘Ax = bŠK¡Œ)Œ)ƒ)“”ƒ=•”˜”ƒ.„•«}¤•”—”†{}.’š(…óY¡n¢0£šª”}¡|¡}¤„‡”˜8§ª”}&€¡“¡µA ′ x = b ′ Ã ′′ x = b ′′¢•”}¡=} ˜”•|&ƒ)Š”‚7>€Ÿ¤•”}¡=}z-€Kƒ=„‡ƒ)•|'¨€˜‰¡} ‚©„ &|¡Š”˜”—”ƒ A§'€˜‰¡} ‚$‚¡ƒ=”|&}¤Š”ƒ b§'“”ƒ)Š”ƒ=‘Ê©Š”}¡—”ƒ-’K¡Š”¡¯¡.€¡“”–Pƒ)•”˜Ž }¡—ƒ)•””˜ ’€Žmin{ x ν | x ∈ Ξ αβ (A, b) }ª”}¤Œ)Š”ƒ!’KšA ′′ x = b ′′§V‘=&|¡ƒ=„ Œ)•”}¡‚7ŠK&‘)‰¤˜”|&ŸŒŽc‚–P˜”Š”}¡}¯˜‘)‚¡ƒ.Œ-|&•}¤„Ô‚}7„‡‰¡˜”•¡|&}¡Š•”}¡~[}¤ª”|&˜„‡˜”‘.&—”˜˜Ä„‡ƒ=|&}.’Kƒ…‚¤ƒ)|&‚¤ƒ)~Œ)˜Œ-|&ƒ.„‡† ËÚ ÐP„‡}'e•”}¦ª}¡‚¡|¡}¡Š”˜”|¡Ÿ¦ª”}¬}¡|¡•”}¡–Pƒ-š•”˜”µÅ×Œ-˜Œ)|&ƒ=„ ¤„ašJª”}¤|&}¤„‡¤„ Œ-|&‚¤}¤„{’4Š”}¡‰¡€ƒ)•”˜”Ž˜Œ›¤}.’4•”}¡~A ′ x = b ′ ˜}¤Õ©}¤Š„‡˜„[z=|&}¡|©ª”Š”}¤—”ƒ=Œ=Œ¤ª}¤Œ!€Kƒ!’K}¤‚¤&|¡ƒ-€Ÿ¤•”}¡=} ¡.€¡“”–…ƒ=•”˜ŽS}¡—”ƒ=•””˜«Œ-•”˜”‘=¡©’€”Žª”}¡|¡}¤„‡}¡‚e}¤|A ′ x = b ′ Ã ′′ x = b ′′ ˜Œ)•”}¡‚7«¡=€K¡7“–…˜”|&Ÿ}¡—”ƒ=•””¡˜[|'¢’+¢cÜ[†˜ )Š¡•”˜”—ƒÅxÏÎE$ ® %‚¡}¡šJªƒ)Š”‚¤†œ›®§}¤Š”-&•”˜‘)¡ƒ.„Ø‚¤Œ)ƒ©‚¡}¡‘=•”˜”&µ¨‹…˜”ƒ$‚«ª”Š”}¤—”ƒ=Œ=Œ-ƒ¨’KŠ”}¤‰&€ƒ=•”˜Ž ˜KŒ›¤}.’š•”ƒ=}¡|&}¤Š”†~Œ)ª”˜Œ)}¡ L§”˜8§‚¡}¡šJ‚¤|&}¡Š†œ›®§”’KŠ”}¤‰&€ƒ=•”˜”µ &S’K†~cŠ¡‘…‰ ¡=’Kƒ.„ª”}.’K‚¤ƒ)Š”-¡|&Ÿe€˜–…Ÿ| ¡[˜”‘P˜•š|&ƒ=Š”‚¤&€Ÿ¡•†œ›Œ-˜Œ)|&ƒ=„ašª”}¡|¡}¤„‡}¡‚Qx = r§V7}¤|&}¡ŠK>Žc}¡‰¤ƒ=Œ)ª”ƒ)“”˜‚¤&ƒ=|¯Š”ƒ)š}¡Š¤’K•¡7µ Ë•&˜›¡=’K–P¡7µ©Ðâ•P’4¡•”•”†~ „‡}¤„‡ƒ)•|e}¡—”ƒ=•””¡Υ(Q, r)¢&€)}¤Š”˜”|¡„ e„‡† ‰ ¡=’Kƒ.„ª}.’”’Kƒ)Š7e˜”‚7&|&ŸeŒ-ª˜šŒ-}¤ (©|>&8§”‚«ª”Š}¡—”ƒ.Œ)Œ-ƒQ‚¤†ª”}&€K•”ƒ)•”˜”Ž §Œ)}¤Œ)|&}'Ž”‹…˜”~˜”‘©‘=¡ª”˜Œ)ƒ)~š|&Š”}¤ƒ)L(Q, r, Υ(Q, ËÎ ª ¢ $ Ù Ðr)),’KƒQ " ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•&Ž n × nšI„ &|&Š˜”—”§Q ⊆ A§r " ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†~ nšr ⊆ b¢§©Š”}¤„‡ƒQ|&}¤)}K§”}¡‰¤Š&‘=¡7µ¨‹…˜”ƒ L‘=&ª˜Œ-˜[¡ª”}¤Š¤Ž’4}¡“”ƒ=•”† ª”}e‚¤}¡‘)šŠ¡Œ)|>&•”˜”µ ‘=•'“”ƒ=•”˜”~Ô}¡—”ƒ=•””˜ ‚¡ƒ=”|&}¡Š³§r)§¥×ª”ƒ=Š”‚ ¡µ ‘=¡ª”˜Œ)Ÿ Œ-ª”˜Œ)”§Ó|>&òSƒΥ(Q,

Ú §ŒŒ-˜Œ)|&ƒ=„ž¡Š&‚¤•”ƒ)•”˜~8¢ª&ŠK¡„‡ƒ)|¡Š”}¡‚P’€”Ž˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†œ›¯€˜”•”ƒ=~”•”†œ›$&%('¡)ª”Š”˜Œ)‚¤&˜‚¤&ƒ.„-,.&/1';2¦465¦7Q ←Ã r ← 8 bυ ← r)8˜”•”˜”—”˜K>€˜”‘=˜”Š”¡¤ƒ=„Œ)ª”˜Œ)}¡‚¡†“”˜Œ-€”Ž”ƒ=„}¡—”ƒ=•””¡Υ(Q,{ }L ← (Q, r, υ) 8Q = ( q‚¯„ &|&Š˜”—”ƒij )˜ÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ãä ûxÐo÷¡ËUÿ¡øz QL{>|=};}L_ nqp ÿ¡k©ihø¨©˜¥!ÿ"ÿEöR÷¥¤ø#ôcõRö÷-øùõnüãûAx = b¢(©•”|&ƒ)Š‚¤>€KŸ¡•&Ž¯€˜”•”ƒ=~”•&ŽŒ-˜KŒ-|&ƒ.„$,+˜%('¡)•'“”ƒ=•”˜”ƒM ∗ = min{ x'ν | x ∈ Ξ αβ (A, }¢b)93:C0?@ÄË Œ)˜Œ-|¡ƒ=„Qx = r˜”•”|&ƒ=Š”‚¤&€Ÿ¡•K>ŽÐ‚¡ƒ=”|&}¡Šƒr = ( r i )‚¡†‰¤˜”Š¡ƒ=„s§”˜„‡ƒ)µ¨‹…˜”~ •¡˜”‰¡}¡€Ÿ¡–P¡7µ{–…˜Š”˜”•”¡8ª”}¡Š}'P’®&ƒ=„˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†ƒ¨€K˜”•”ƒ)~•”†ƒz)€ƒ=„‡ƒ=•”|Q ′ x = r ′ ˜ƒ.Œ!€˜Œ)˜Œ-|¡ƒ=„‡†œšª”}¡|&}7„‡”˜s = q’€”Ž •”ƒ)}¡|¡}¡Š”†œ›kl k, l ∈ { 1, 2, . . . , }§|&}nq ′ ij ← q′′ ij ← q ij’€”Žq ′ kl ← [ q kl , q kl ] §q ′′ª}&€K¡-&ƒ=„s = rƒ.Œ!€˜ ’€”ŽkQ ′′ x = r ′′ %(i, j) ≠ l)§(k,kl ← [ q kl, q kl ]§r ′ ← r ′′ ← r8ª}&€K¡-&ƒ=„Q ′ ← Q ′′ Q§← r ′ k ← [ r k , r k ] r ′′k ← [ r k, r§k ]§r ′ i ← r′′i ← r i’K€”Ž‚¡†“”˜KŒ!€”Ž”ƒ.„}¤—”ƒ)•”˜υ ′ ← Υ(Q ′ , r ′ )˜•”ƒ=7}¤|&}¡Š}¡)}k ∈ { 1, 2, . . . , n }§|&}i ≠ k8υ ′′ ← Υ(Q ′′ , r ′′ )8L‰¤†‚¡–…¡µ{‚¡ƒ-’K¡‹…ƒ)~‘=¡ª”˜Œ)Ÿ(Q, r, υ)8‘=¡•”}¤Œ)˜„ ‘.&ª”˜Œ)˜¡=’4>€”Ž”ƒ.„˜‘(Q ′ , r ′ , υ ′ (Q ′′ , r ′′ , υ ′ )‚eŒ-ª”˜KŒ-}¡ ‚«ª”}¤Š7Ž’Kƒ$‚¤}¡‘=Š¡Œ)|'¡•”˜Ž|¡Š”ƒ)|&Ÿ¤ƒ)=}Sª}&€”Ž8)˜L¡µ ‘=&ª˜Œ-ŸeŒ-ª˜Œ-S“”ƒ)Šƒ)‘(Q, r, υ)8Ä F9J9;:}¡‰¤}¡‘)•K'“&ƒ.„Øª”ƒ)Š‚M ∗ υ8 ←

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t ′¢Ú>Ù-ªÜÝ/u£œkuxs,Ç7ß¬àPwVîRëqu&Ýv¡ä®å¡ã”œ¡ä4ã!ð…ä4ã=ã©å'á4ã!äœ-u&ä4ãz-€ƒ.„‡ƒ)•”|> ‚¡ƒ-’K¡‹…ƒ)~¬Œ-˜Œ)|&ƒ=„‡†jþKûr¢!H&ûx&µ¨|¡ŒŽ •”ƒ=˜”‘=„‡ƒ=•”•”†„‡˜8§K|>&“”|&}Ws ′ := s, t ′ := t, W ′ := W.Qx = rŠ¡Œ=Œ-ƒ=“”ƒ)•c•’K‚¤I(…óY¡n¢0£ šJª”}¤|&}¤„‡ (©|'¡8§aª”¡¤Œ)|&Ÿ¦‚¤ƒ!’K¡‹S>ŽØ˜”•”|&ƒ=Š”‚¤&€Ÿ¡•K>ŽŒ-˜KŒ-|&ƒ.„Q ′ x = r ′ Q ′′ x = r ′′§”}¤ª”Š”ƒ!’4ƒ-€=•”•”†ƒ©‚e´³&‰¡€³¢8Î ª ¢Û ¢˜‚Œ-}¤}¡|&‚¤ƒ)|¡Œ)|&‚¡˜”˜Œ }¡|¡}¡Š”†„ÅÕ©}¡ŠK„‡˜”Š”¡µ |¡ŒŽ ¤„aš§Q&”˜K„Ö’K}¡€”Pƒ=•‰¡†|&ŸØ‘.&}¡•³§„‡}'Pƒ=„žª”Š”˜KŒ-‚¡}¤˜”|&Ÿª”}¡|¡}¤„‡¡„qÁ…ó¥•K'“>€KS„‡†W ′′ ← W ′ ← W, s ′′ ← s ′ ← s, t ′′ ← t ′ ← t,„ &|&Š˜”—”†W ′§ W ′′ ˜‚¤ƒ)”|¡}¡Š”†s ′§ s ′′§ t ′§ t ′′ Œ)}¡}¡|¡‚¡ƒ)|¤Œ-|&‚¤¡7µ¨‹…˜”ƒŒ-˜Œ)|&ƒ=„¡7›z)|>¡ª”•”¡µ ª”Š}¡—”ƒ-’K¡Š”¡[ª”ƒ)Šƒ)‚¡†“˜Œ!€ƒ)š•”˜Žƒ% ‘.&|&ƒ.„‚¡†ª}&€•”˜|&ŸŒ-€ƒ!’4¡7µ¨‹…¡µ²’K‚}7Œ-•”}¤‚¡ƒP˜”•Õ©}¡ŠK„ &—”˜”˜¬}w ′ kl ← 1 ˜w ′′kl ← −1;yâ}&šª”ƒ)Š‚¡†œ›V§V„‡}.’K˜Õ©˜”—”˜Š”¡ƒ.„W ′§ W ′′ ˜s ′§ s ′′ •K‚¤ƒ!’K¡‹…ƒ=~x(…óY¡n¢0£¢(Q„‡ƒ=•”•”}K§|&}¡€Ÿ¡}«“”|&}e‚¤†ª”}&€K•”ƒ)•”•}¡~‰¤˜Œ-ƒ=”—”˜”˜q Ë‰Ð ˜”‘„ ¡|&Š”˜”—† ‚ƒ=Œ-€˜Š¡Œ=Œ-ƒ=“”)•”•†„²z)€ƒ=„‡ƒ=•”|&}7„Å‰¡†œ€ „ &|&Š˜”—>› klW ′ = ( w ij ′ W ′′ = ( w ij ′′ ) „‡†{ª”Š”˜KŒ-‚¤¡˜”‚¤¡ƒ=„)˜ Q§â|&}ƒ=Œ-€˜¦Š¤Œ)Œ-ƒ=“”)••”†„[z-€Kƒ=„‡ƒ)•|&}¤„[‰¤†œ€r k˜‘¥‚¤ƒ)”|&}¤Šª”ŠK&‚¡}¤~¦“¤Œ-|&˜r§|&}e‚«‚¤ƒ)”|&}¤Š>›s ′ = (s ′ i )˜ i ) „‡†õªŠ”˜Œ-‚7&˜”‚7&ƒ.„ËŽYÐs ′ k ← −1 ˜s ′′k ← 1.s ′′ = (s ′′W ′§ s ′§ t ′ ˜W ′′§ s ′′§ t ′′ Œ-}¡}¤|&‚¡ƒ=|¡Œ-|¡‚¡ƒ)••”}§@"}¤‰O8ƒ=”|&}¡‚K§"yâ}&š‚¡|&}¤Š”†œ›V§-ª”ƒ=Š”ƒ)‚¤†“”˜Œ!€Ž”ƒ=„>S’K}¡ƒ¤˜”‘‡’K‚¤¡¤›QŒ-ƒ.„‡ƒ)~Œ)|&‚‚¤‘=&˜K„‡}¤Œ-‚¡Ž”‘=¡•”•”†œ›Œ)}¡}¡|¡•”}¡–Pƒ)•”˜Ž ËÎ ª ¢Ù §¡Ð!¢(Q„‡ƒ)•”•”}K§W ′ 8ËŽYÐ ƒ=Œ-€˜W ′ ˜€˜t ′ ˜‘=„‡ƒ)•˜€KŒŽ8§³ª†|'¡ƒ=„˜Œ)ª”}&€Ÿ¤‘)¡ŽŒŽª”ƒ)Šƒ)‚¡†“˜Œ!€˜|&Ÿ „ ¡|&Š”˜”—”¡Ë‰Ð ƒ=Œ-€˜s ′ ˜€˜t ′ ˜‘=„‡ƒ)•˜€KŒŽ8§®ª”†|>&ƒ.„s ′ 8Ë‰Y"Ðƒ=Œ-€˜W ′ ˜”€˜s ′ ˜”‘.„‡ƒ)•”˜”€KŒŽ8§Vª”†|>&ƒ.„ŒŽ¬ª”ƒ)Š”ƒ=‚¡†“”˜KŒ!€˜”|¡Ÿ[‚¡ƒ)|&}¡ŠŒŽª”ƒ)Šƒ)‚¡†“˜Œ!€˜|&Ÿ ‚¡ƒ)|&}¡Š

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Ì Ú&ÙŒ)˜Œ-|¡ƒ=„ ¡„Qx = r§®7}¤’4>šJ€˜”‰¤}eª”}¡‰¤†œš‚¤¡‚¡–…˜K„Ø‚eŒ-ª”˜Œ)ƒ ª”}‚7Œ-ƒ.„˜”•”|&ƒ=Š”‚¤&€Ÿ¡•†„ò€˜”•”ƒ=~”•”†„L’K}«z=|&}¡=}S–S&-§7|¡}s,Çß‰ˆ'ßÚ¦åyc$#\3‰u>áë¤ã.î'åyaå!œ¦yï³å¤æ'Ý4ã!ä³•wu)ïƒu¡ë¤ã.îï³å£œmin{ x ν | x ∈ Ξ uni (A, b) } ≤ ω.Qx = r " ‚¤ƒ!’K¡‹P&Ž•(…óY¡n¢0£§|&}óž’KŠ¡7=}¡~[Œ-|¡}¡Š”}¤•”†Q§ƒ=Œ-€˜Υ(Q, r) ≤ min{ x ν | x ∈ Ξ uni (A, b) },P„‡}¡šPƒ=|«Œ-€¡7e˜”|&ŸP’K}¤Œ)|&˜Sƒ)•”˜”ƒQ|¡Š”ƒ)‰¤¡ƒ=„‡}¤~„ &€}¤Œ)|&˜ ‚¡ƒ)€˜”“”˜”•† ˜ª”}¤|&}¤„‡¡ƒ)‹…$}.’K•”˜K„Ø”Š”˜”|¡ƒ)Š”˜”ƒ.„ƒ.Œ-|&ƒ.Œ-|&‚¤ƒ)•”•}¡~}7Œ-|>&•}¡‚¡˜>€=}¡Š”˜|¡„(ω − r))¢&P>€ƒ=ƒ¡§'‘=¡ª”˜Œ-ŸΥ(Q,(Q, r, Υ(Q, r), p) L§>¡=’K}¤‚&€ƒ=|&‚¡}¤Š7Ž”µ ‹P&Ž¦•¨•”ƒ)}¡|¡}¡Š”}7„–P¡)ƒ©¡¤Œ-€}¡‚¤˜”µ∈Υ(Q, r) > ω,ª ¢Ú>ª Ðª”Š”˜¬’4&€Ÿ¡•ƒ)~”–Pƒ=„õ‚¡†ª”}¡€•”ƒ)•˜”˜Ø>€=}¡Š”˜”|¤„ ¡¤Sƒ¯•”˜”}¤’4•”ƒŒ-|>¡•”ƒ)|[‚¤ƒ!’K¡7š ËÎ‹…ƒ=~8§Ó˜ƒ)[¡=’4&€ƒ)•˜”ƒ[˜”‘Œ)ª”˜Œ- •”ƒc}¤&Pƒ=|¦•”˜¡}¡)}×‚¡€˜Ž”•”˜ŽÔ•×Š”ƒ)šL>€K)}¡Š˜”|¡„ ”¢‡yâ}¡}¤‰¡‹…ƒ¤§³|&ƒ.Œ-|&˜Š”}¡‚7&|&Ÿ7ŒŽ×•ƒ)Š¡‚¡ƒ)•KŒ-|&‚¤}¤„ ËÎ!§¡Ð’K}&€«•† •K[>S’K}¤„õ–P¡)ƒ¯&€)}¤Š”˜”|¡„ [‚7Œ-ƒe‚¤•”}¡‚¤Ÿ[‘=&•}¤Œ-˜K„‡†ƒe‚Œ-ª”˜KŒ-}¡ ‘)¡.€Ÿ>|>&|Š&‰¤}¡|&†LÄ“”˜Œ)|&”˜ƒÅØŒ-ª˜Œ-”§¥"¼ƒ)=}¦ªŠ”}¤Œ=„‡}¡|&Šò˜ž¡=’4&€ƒ)•˜”ƒ ‚¤Œ-ƒ)›‘=¡ª”˜Œ-˜³§a|¡}¡’4¦&‹P˜› ËÎE§¡ÐQ‘=&ª˜Œ-ƒ=~8§Vš¨˜K„‡ƒ)ƒ)|cŒ=„‡†Œ!€ ª”Š}¡‚¡}.’4˜”|&Ÿ €˜”–PŸ[ª”}¤Œ-€ƒ˜”‘=„‡ƒ=•”ƒ)•˜”~ ¡=’K}¡‚¡€ƒ)|&‚¤}¡Š7Ž”µË|'¢ƒ¤¢¡„‡ƒ)•Ÿ¡–…ƒ=•”˜”~KÐ¥ª¡Š¡„‡ƒ=|&Š ¢ ω˜’Kƒ.>€Ÿ¤•”†„Ö‚¤†‰¡˜”Š¡|&Ÿ¬|&}¡“”ƒ=“”•”†ƒ[„ &|¡Š”˜”—”¡⊡Q˜§©}¡•”ƒ=“”•”}§‡‰¤†œ€}¦‰¤†⊡r|'¡8§“”|&}¤‰¡† ‚¡ƒ=”|&}¡Š(⊡Q, ∈Ñ\TÉ⊡r) min{ Υ(Q, r) | Q ∈ Q, r ∈ r }.Υ(⊡Q, ⊡r) = Υ(Q, r).¡=’Kƒ.„×ª”}¤Œ)|&¡ª&|&ŸK§¡ƒ.Œ!€ẽ}¤—”ƒ)•” "«P&”|&˜“”ƒ=Œ)”˜8§„‡†|'¡eẽ‰ ª”}&€4&->ŽΥ(Q, r)|&}¤“”•>Ž8§‚}¤‰¡‹Pƒ=„ Œ-€¡“&ƒ«ª”}.’4}¡‰¡•†~Ä¡=’4'“”•†~ƒÅ ‚¡†‰¡}¤Š•”ƒ¯„‡ƒ=•”ƒ)ƒe|¡Š”¡=’Kƒ)•8§³“”ƒ=„Š”ƒ)–Pƒ)•”˜ƒ¦˜Œ›¤}.’4•”}¡~ ‘=>’4'“”˜8§˜cª”}¡|¡}¤„‡¡‚¯—”ƒ-€Ž›[„‡˜•”˜„‡˜”‘.&—”˜”˜‚¡}¡‘.„‡}'e•”†œ›¦©}(⊡Q, ⊡r)}¡|$„‡•”}'Pƒ.Œ-|&‚7Ñ\TÉmin{ Υ(Q, r) | Q ∈ Q, r ∈ r }—”ƒ-š€ƒ=Œ)}¡}¤‰¡Š¡‘)•”ƒ=~…‚¤Œ)ƒ)=}$‰¤Š&|¡ŸQ|&}¡’4$‚Q'“”ƒ.Œ-|&‚¤ƒ }¡|&”€}¡•”ƒ=•”˜”~⊡rŒ-ƒ=Š”ƒ!’4˜”•”†Ô„ &|&Š”˜—”†⊡Q˜Q˜ ‚¤ƒ)”|¡}¡Š r§|'¢ƒ¡¢mid Q˜mid r§”Œ-}¤}¡|&‚¤ƒ)|¡Œ)|&‚¡ƒ=•”•”}K¢ß_àgá‡†‚å0œþ‡ê¦ýlêð¦ã¦ñúXïårì&ãª ¢Î¡Î¤˜Î ª ¢ÏÎ!§œŒ-¡„ „‡˜”Š”¡µ¨|¨Š¡‘)‚¤˜”|&†ƒ‚¨ªŠ”ƒ!’K–Pƒ=Œ)|&‚ ¡µ ‹P˜›¦ª”¡•””|>&›…„‡}.’K˜Õ©˜”&—˜”˜…„‡ƒ)|¡}.’K}¡‚’KŠ”}¡‰¡€ƒ)•˜Ž…ª¡Š¡„‡ƒ=|>šÊ©Š”˜”‚¤}.’K˜„‡†ƒÓ•”˜”PƒâªŒ-ƒ=‚>’K}¤7}.’4†×´+&‰¡€³¢ÎËÇoÇ šI„‡ƒ)|&}.’K}¤‚7ÐÓ’€”Ž‚¡•”ƒ=–…•”ƒ=)}S}¡—”ƒ=•”˜”‚7&•”˜Ž }¡‰O³ƒ!’K˜”•)•”•”}¤)}…„‡•}'Pƒ=Œ)|&‚Š”ƒ)–Pƒ)•”˜~r(…óY¡n¢0£¢ Š”}¡‚

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Υ(Q, r) := ( Î6ÏkÐ£Ñ(Q, r) ) ν . -¢ w , „•§¡&:£ f%go¤'g6¥ ¦j§ ]ym a¨ g§f© ` \ _ ¤ `b_#ªz`b_-_ \E« _#à ¤"g àb_q ± ?A Ccr n ±ED-r/M q qƒ{ ?AE‡? C ? x ¯ M`D ? A ? w ? ²C4MIxz{q M`x A/? DEz4x A M D&rC§t§{C4M? °7qFz4q A M q ±ŠzNw ? ²&z q z4x A M D&rC§tx±EM ‡ r r q M A/ ±ËqFz4q A M q ±"³CcrC À/?x²±E®x ? w§MFC¥r/M A qƒ{5D^z4x A M D&rC§tx±h} q M A? w¥r}pARC ? ²&r/C4tx ? ° ?‡A z q z n r§{?z4z³x4r ‡? w§z4x A M D&rC§±‰xMIx4~`C§MID ? °Pw§C4zx±"³D ? ²%MFw§zxM`xzzaw8r/‚¨µNzM¯? w§x±E°w ? ²z q ±E°Yzx A M D&r/C ³z4qƒ}•q ±Hw ? ²z q z4x A M D&rC§tx~c‚ÊqFz4q A M q ~~ r/Dx4MIxz° A r§C³W® A? ²±C§~-®§ro{ ‡?A M q ±v{ ‡?A/? q CzHq ??A DM A q A D ? D&r/C4zŽqFD{ n { q ³[x§r§CWC§rw§±ED&rM©{‚(µzM_qƒ{ŽqIz4q± q x4r|qFz§q A M q ~ wq r ‡ z q M ³3M`qFC4z r&qIq q r A/ zD-r/M A qƒ{Žzx A M D&r/C4tx4r/{‹qFz qXq M A/ z®x§r{uqIz4q A M+{r€~ r/D&xMIxz4°³ A/? DOMFw§zxz4®x ? q rC A MYw 4? ²/C§MIxzW{Šw ? C … x± ? w§x ? D M q M`x{q? w ? ²&z A t&qƒ{5qIz qXq M A z®x±EM ?&A x ? qIz A MIC4tx ? ARC§rDx ? °Yw§z4r§A ? x4r/C4z C4M q M`x{x± q r A/ z ¯ ±"³h® A? ²± ‡4? C4~c®4r‚¨µz4M`qƒ{H B< 'ÌqIx ? D-rOz q MIC4zuz4x A M D&rC§t§{AqXq M A/ z®x4±EMJqIz4q A M q ± w [qFC4z … M q r A/ z ¯ rTz4qƒ} ? w§x ? °=qFz4q A M q ±u²±hC§rx±EMJqFzA M D&r/C4tx ? °zC ? q ? qIz qXq M A z®x ? ° ‡ M qFz qXqXq M A/ z®4x ? °³AFr/xCcMIC4MID ? °³ A ‡ {zx¯ M`D ? °³ ¯ z C~_C{x A x ? °az AEw ‡2w „ ³ A/? zZqFz4q A M q ±v{ ‡?A? q CzZx4r,CWr … w ? q yNrAIMC4zrCA ? z A q rw ? C … x±‡?&?$… w¥r A t-qƒ{ A r§C³4® A/? ²&±Hz q M A tC ? q ? qFz qXq M A/ z4®x~c‚‡ M qFz qXqXq M A/ z4®x~c‚³yAFrxCcMFC4M`D~³ A ‡ C4z ¯ MID&~³ ¯ z C~_C{x A x~c‚Íz AEw ‡0w…„ q r A {z ¯ ~ w z4q A ~ ‡ r{3C M`rC§z n r ¯ zzTx4r/yNzW} n rw§~ q ? C³ ? ² ? n x4r$®4z q r/x§rC ? AIz®x ?(A/? {³J~³CWrC À/? ²±hC ? q£w§MIC§r/x ?| rxMIM&³q? xM-C ?A/?& ±E°!‘JzCWqIz ? D-r/xx4±E° q M A/? w€DxM`yxM-A ?^?¯ M`xzD&rxzW{ q x ? {Î6ÏkÐƒÑM`q A D M`yMIx4z° B< 'E³"C ?&A/? ±E°7²~`w§M q x§r n ±ED-r A tƒú©À-ío¹o¿oõ2ËËz½©Á,¹rÃ…(Q, r)¾r¹KËÒΞ sym (Q, r) qFz4q A M q ± Qx = r ³(Q, r) ∈ IR n zÎ6ÏkÐƒÑ?b‡? C§~-®§r/M q ±E°^q¨M-A ?‡4? q ? µNt‚‹z4x A M D&rC§tx±E°=D&M+C A? D&xMIyz{AEw M w Î6ÏkÐƒÑxM`° ?¯ M`xCzYw4C4{ q x ?$… M`q A D&r MIyM`xz°Î6ÏÐ£Ñ(Q, r) ⊇ Ξ sym (Q, r);Υ(Q, r) x4z°xC xM { ? °C ?νq ‡4? xM`x A ± Ów4C{ n r$w¥rxx ? A ? ‘JzCcqIz ? {D-r/xx ? A ? x ? q M r„ D&xMIyNxMI°zx A M D&r/C4tx ? ° ?&¯ MIxC4z¯ ? … xz ?ν ∈ { 1, 2, . . . , n }Ξ sym (Q, r)³ ‡? C4~c®4rM q ? ° q M A/? w ? q Î6ÏkÐƒÑ³ AEw M wq x ?$… M`q A D&r M`yMIx4z°³J? w§®M CxM q ³ ® A/? ³†C ? C4t5q+C ?&?³ A/? DpCWr$®M`q A DM²&ro{Ξ sym (A, b) ⊆ Ξ(A, b)n ? D±h} q M A/? w ? D q ?$… x ? ² r A t A r$w§z ¯ z ? x4x±EMPzŠ} ?&? y ?* r n r/² ?&A r/xx±EM

Ax = b w¥ ¦·'¦=¸¯_Bª_Bàb_)a` m _¼c ¤'gof© `h¹ f a`b_Rº`4h¹ l a l+m _©¨ l©\lK¤/»4¼F» ¨†a ¡ ,M A/? w¥±eDxM`yxM-A ?P?&¯ MIxz4D&r/x4zW{ q x ?$… M_q A D MIyNMIxz°s B< '²M n qFD{ n M`°³qz§qIr/x4x±EM³yC ‡ z q M ~³ D ‡ C¥r/DM 9 w0³J? C4~c®4rM A qƒ{³8® A/? r/CA ?& z A q ?¯ M`xzD-ro{?‡q x ?$… M_q A D M`yMIx4z°Š B< 'Ôq ? qID/{ n { q z²C ? x4q A/ ~cz ~&M A qƒ{Šz n ² ? C4MIMxzW{q A ±h}€zxA ~c²&±h} q M A? w ? D M`yM`xzW{f B< ' q7xM n rDz4qIz q ± q z*w¥r/x4x±v{‡?z w q± ‡?&A/ MI²&~M q C4zyt ?A ²&r n ? D ? A ? q M A/? w¥r¨Î6ÏkÐ£Ñi~`w ? DC4M A D ? MIxzW{ ‹ ~&qGC ? {uq ? x ?A/? xx ? q A z0Œ r¢ w @@ „ z ‹ ~&qFC ? D&z‚ A/? ®x ? q A z0Œ’ -¢ w @" „ z n V -¢ w ¡ ² wDz‚q MIM q min{ x ν | x ∈ Ξ sym (A, b) } = ( Ã −1˜b )ν?&A/? ±h} A/? ®MI®4x ? °‹qIz qXq M A/ z®x ? ° q r A z ¯ ± w4C{ŽxM-CÃ = ( ã ij ) ∈ R n×nA/? ®MI®4x ? A ? DM-C A/?& r z˜b = ( ˜bi ) ∈ R n ? q A r/D/C§MIxx±h}€z n ‡ MFw¥q A r/D&z A MIC4MI°³ÂqΥq MIx A? D q r A/ z ¯ ±C4MM`xCz ?¯A r ‡ ³ z4® qb A/? zHDM+Ca ijzΥ(Ã, ˜b) ≤ ( Ã −1˜b )ν .AqIr q ? q ~ ?‡4 MGw§MIC4MIx4z‚‡4?a ijz q MI‚ A x4MIx~_C4M`D~c‚ýz z4x~³ ? ² ? n x4r$®z q³J MGw ‡? C ?$… zD§³8® A/? D q r A z ¯ MqIz qXq M A z®x±EM ?A x ? qFz A MFC4t&x ? ARC§r/D&x ? °[ a ij , mid a ij ] ³w§z4rA ? x§rC4z C4M q MIx A ±A ′ ?A n r q MIx ? °a jiq r A/ z ¯ ~³ ‡? C4~c®MIxx4~-‚uz nC§M q MIx A/? D a ijz[ mid a ij , a ij ] ³x§rA ′′ ?A n r q M`x ? °a jix§rq r A/ z ¯ ~³ ‡? C§~-®4MIxx~c‚Žz nC§M q MIx A/? D a ijzÃ ′ ? q r A/ z ¯ ~³ ‡? C4~c®M`xx~c‚Êz n Ã n r q MIx ? ° C4M q MIx A? D ã ijz[ a ij , mid a ij ] ³ã jix§rÃ ′′ ? q r A/ z ¯ ~³ ‡4? C4~c®M`xx~c‚uz n Ã n r q MIx ? ° C§M q MIx A/? D ã ijz[ mid a ij , a ij ] wã jix§rA ′′ x = bA ′ x = b zJx A M D&r/C4tx±EMiqFz4q A M q ±fC4zxM`°x±h}Nr/CAIM`² rz®M_q©CzW}"~ rDxM`xz°‡4? C4~c®M`xx±EMiz n z4qƒ} ? w§x ? °BqFz4q A M q ± ‡ ~ A q r&qIqIMI®M`xzW{ ‡ r ±fqFz q {³M A/ z®x±h} ?&A x ? qFz A MIC4tx ? ARC§rDx ? °Yw§z4r§A ? x4r/C4zPzx A M D-rC4t&x±h} C4M q MIx A? Dq‡?‡4? C§r q ³ q ±u²&~Iw¥M q x§r n ±ED-r A t8º©Çbº©Á3½¼ËGÀ+ËGÇÃñ·%¹rÁ,¹KËGùÀ+ËGÇ=w§C{µ rC4M`M³ A r§C6CWr§CÃ ′ ⊆ A ′ ⊆ A, Ã ′′ ⊆ A ′′ ⊆ A,z˜b ⊆ b³ A/? ‹ qID ? °4q A D ? q ? x ?&A/? xx ? q A z0Œ‰ r¢ w @@ „ z q M`M A qID ? z qqFC4MGw¥q A Dz4M q

za£ f%go¤'g6¥ ¦j§ ]ym a¨ g§f© ` \ _ ¤ `b_#ªz`b_-_ \E« _#à ¤"g àb_&9§¢Υ(A, b) ≤ Υ(A ′ , b) ≤ Υ(Ã′ , ˜b)xM r/DM`x4q A D&rΥ(A, b) ≤ Υ(A ′′ , b) ≤ Υ(Ã′′ , ˜b).?&A q ?&?A D&M A q A D~c‚¨µNzW}Z®4rq A MI°[C4MGw ? D&r A MFC§tx ? ³W²M { ‡? ®WC4MIx4x±E° q zxz q ~ qxM r/DM`x4q A D4³ q ±C4~c®z q ‡4?Υ(A, b) ≤ min { Υ(A ′ , b), Υ(A ′′ , b) }≤ min { Υ(Ã′ , ˜b), Υ(Ã′′ , ˜b) } . -¢ w ¡¢ „q A/? A ? ³ ‡? q©C ? C§tC~ q r A/ z ¯ r SJ? MÃ Ã n r A MFC4t&x ? q ? w§M -… z A qƒ{ZC4z² ? D ′³? ? D C4z² ²${ A/? q ‡ rDMGw§C4zD ? ³ ‡? C r/°4xMI° q M M³ ? w§x ? z n x4M rDMIx§q A D′′³ÃΥ(Ã′ , ˜b) ≤ Υ(Ã, ˜b)Υ(Ã′′ , ˜b) ≤ Υ(Ã, ˜b),zWC4zA r§CY® A/?min { Υ(Ã′ , ˜b), Υ(Ã′′ , ˜b) } ≤ Υ(Ã, ˜b) ≤ ( Ã −1˜b ) ν= min{ x ν | x ∈ Ξ sym (A, b) }. -¢ w ¡4 „ ?‡? q A r/DC4MIxz4MÂxM rDM`x4q A DG -¢ w ¡¢ „ zƒ r¢ w ¡ „4‡4 zD ? w¥z A CBq ?&?A x ? yM`xz‚Υ(A, b) ≤ min { Υ(A ′ , b), Υ(A ′′ , b) } ≤ min{ x ν | x ∈ Ξ sym (A, b) },z³CWr§CZqGC§MGw¥q A DzM&³CaqGC4MFw§~c‚¨µM q ~ ‡ r§C A z4®M`q+C ? q ~ M ¯ M ‡A ~¥jA ′ x = bA ′′ x = b,c2_#ªza ¤Why¤ _Ša` m _—c ¤"g§f© `4h¹ f a`b_Gº`h¹fÕ l a l-m _#¨¡)ý ]b\"mo\ ¨¡Ö g×¨ \+k _©¨ ] c>aº m aØ ¤\\e ^3_ òE\o¤'\ c » Ø6Ö e\of _+_ mo\cÙ ` \ º \E« _#`Ö_¨¡à ¼ ó hyfW» a l Ö \ ¨ \oòE\a¼ ` g-Ù _#à »lmin{ x ν | x ∈ Ξ sym (A, b) }¤ a h _¤min { Υ(A ′ , b), Υ(A ′′ , b) } .

?$… M A ²&± A t ? ² ? qIx ? D-r/x ? D±EÚWC§rw§ÚWr q z³`q ? D&M yNMIxx ? rx4rC ?&Û z®4x± q z -¢ w ¡¢ „q -¢ w ¡ „Kw¥³J?IA/? q ~QD ‡ MGw§t*w4C4{ËMFw§zx ?&? ² r n zW{ q ±dw ?&ÛI? D ? z q qƒ{ ? ² ? n x4ro{ÉQ Ün × n { q r A/ z ¯ r³ Q ⊆ A ³z¥ ¦·'¦=¸¯_Bª_Bàb_)a` m _¼c ¤'gof© `h¹ f a`b_Rº`4h¹ l a l+m _©¨ l©\lK¤/»4¼F» ¨†a -9cb ix§r ‡? w§zx A M D-rC4±[ b i , mid b i ] z [ mid b i , b i ] ³ ® A/?bÞ rÚ ? ° … M‘W‘JMIÚ A z q MIM A z|w ? ²C4MIxz4M\DD&MIÚ A/?& M ‡ r/D±h}|®4r&q A M`°ÚWrÚ ? °{xz²&~Iw¥tYÚ ? q ‡? xM`x A ±Ax = bA ′ x = b ′ zA ′′ x = b ′′ wA C4z4² ? DsDM`Ú A/? M b ³®4r A tY B< '){ ‹ ‡4?A/? q Úz0Œ$³ ‡? C4~c®4r‚¨µNzM`qƒ{5z nw ? ²/C4M`xzM q ‡?‡? {³J?¯ MFw§~ ~f~_C4~c®yNMIxz{ ?&¯ MIxÚ4zŠw4C{min{ x ν | x ∈ Ξ sym (A, b) } ‡? {C§r q? w¥x ?Û`? zx A M D&r/C4tx ?ÛI?ˆ C4M q MIx A rsD q r A z ¯ M®M M nA ′ x = b ′ zA ′ x = b ′A ′′ x = b ′′ ³A ′′ x = b ′′ zOqFx ? D&r^~_C4~c®{q MFw¥q A D ? q w ? ²/C4M`xzW{ C4M q M`x A/? DZzx A M D-rC4t&x ? °*qFz4q A M q ± „ q ?$… x ?^‡? {D A/?& z A t ‡4?O?A x ? yM`xz‚ÔÚŽqFz4q A M q r q { ‡4?A/? q Úcr qmin{ x ν| x ∈ Ξ sym (A, b) }n r A M qqFx ? D&r r n ²z A t ‡4?A/? q Ú ? D ?&Ayz A t ?¯ M`xÚ~Tz AEw w w'u ± ? ‘ ? q z q ‡? w ? ²x±E° ‡4?¯ M_qIq ‡4? qGC§MGw ? D&r A MFC4t&x ?Û`?~_C4~c®yM`xzW{ ?&¯ MIxÚ4zsqFx4z n ~*w4C4{A rÚ³pÚcrÚ À/?A qƒ{*DPyNz ? Ú ? z n DM_q A x ? q DaÚ ? q ²z4x4r A/?& x ? ° ?&‡A z q z n r ¯ zz ‹oq M A/? w§Mw§MFC¥r/MA DM`°^z Û` r/xz ¯ ŒZ! _@$# zaÚWr/Ú IA/? ²&±hC ? r$w8r ‡A z ? D-r/x ? w4C{Zzx A M D&r/C4tx4±h}DMM A/? w ? D Û C ? ²-rC4t&x ? ° ?‡A z q z n r ¯ zzx q q wkV @ w ¡ „ jq? { ‡ M D&±h}8³ ?4Û r/xz n ~M q D&qFM^zx A M D&r/C4tx4±EMYqIz4q A M q ±DΥ(Q, r) D|xM`Ú ?A/?& ±E°Yq ‡ z4q ? Ú L ÒQx = rAx = b³¡Ú ?A/?& ±EMLQx = r? {nD A/? ±h}8³Âw ? ²&z A ts²~`w§M q C§zyt A ~Šz4x A M D&rC§tx~c‚ÌqIz4q A M q ~Dn q ‡ z4qIÚWr ³[Ú ?&A/? r/{ ? ²M_q ‡ M`®zD&rM A* MIÚ ?& w¥x~c‚ 1x4rzWC4~c®y~c‚ „ x4rzΥ(Q, r) w4C{€z4qIÚ ? q ? °DMFC§z®zx± min{ x ν |q ? q M`x AZ?¯ M`xÚ~w¥rxx±E°x ∈ Ξ sym (A, b) } ÒA/ M A tz}8³`D ‡? w§D&M Û rM q ? °\w ? ²/C4M`xz‚f B< ' q ±€²~`w§M q w ? ²&z A thC4z4ytD§{D ? n x4zÚWr/‚ A D ‡?¯ M`qIqIMJw ? ²/C4M`xzW{az4qƒ} ? w§x ? °a B< '³§D q M+{q A MBqJzW} ?&¯ MIxÚWr q zq`r q ±E°Zyz 4? Úz°Yz n zx A M D-rC4t&x±h}C4M q M`x A/? D wq ? ÚL? A/? {µNz°^z n n r ‡ z§qFMI°{ A/4? MIÚYDzWw¥r³4q q( )Q, r, Υ(Q, r) , A r/Ú³D ‡4?¯ M_qIqFM"D&± ‡? C§xMIxz{Zr/C ÛI?& z A q r q ±´²~`w§M q ‡4? ww§M -… zD-r A t7q ‡ z{zx A M D-rC4t&x4r{Û w§Mr ÜLΥ(Q, r)A M D-rC4t&x±E° zxq M A/?&ÛI? ³ ? ² r n ~c‚¨µNzMSJ?x4r$®4MIxz° ?&¯ MIxÚ4znD&M A q A D~c‚(µzM| B< '?An r ⊆ b w {ˆDM`Ú A/? ³ z4qIz²~`w§~ A ~ ‡? {cw ? ®MIx±‡ r nD ? n rq A rxz‚ ‡?‡ M D~c‚ n r ‡ z4qItQq ‡ z§qFÚWr³ A rÚ … MaÚWr/ÚËz‹q ? {³3rQx = r z ?¯ M`xÚ~ Υ(Q, r) M`Ú ?& w§x~c‚ÍDZq ‡ z4q#{ÚcM „ ³ q ±‹²&~Iw¥M q x4r n ±ED-r A tƒ¿½+¾rÈÊ,Ç"ËGÇYx§rJw¥rxx ? q yNr Û M w'³J? C4x4±E° ‡ qFM`Dw ? Ú ? w

õ¥ö`÷_øA M D&r/C4tx4r/{^qFz qXq M A/ z4®x4r{ZC4z4xMI°x§r{YqIz4q A M q rJxQ ← A zr ← b ÒAx = b w£ f%go¤'g6¥ ¦j§ ]ym a¨ g§f© ` \ _ ¤ `b_#ªz`b_-_ \E« _#à ¤"g àb_&9.ÝkÞ`ß`à&á-âcÞBãRäå·ãGæ/åWçEè-é`êRë$ìƒí-î\á&í|ïìRë$éRðNðWè-é`ß`à-ìƒñ-á-òócÞ`ècÞIïìRë$è-éIôA M D-rC4t&x±h}^qFz qXq M A/ z4®x±h}YC4zxM`°x±h}YqIz4q A M qw4C{Zzxν ∈ {1, 2, . . . , n} wµ?‡ ~&qFÚ δ > 0 wJ? q M ^?&¯ MIxz4D&r/M q ? °^Ú ? q ‡? x4MIx A ±Υ‡4?T‡4 r/D&zWC4~! -¢ w , „Bwu M A? wZÎ6ÏÐ£Ñ-³‘ ? q z ~c‚¨µz4° ?&¯ MIxÚ4~Zx ν | x ∈ Ξ sym (A, b) } wõ¡ùö`÷_ø)(¯ MIx4ÚcrqIxz n ~7w4C{min{‡ z4qID&rzD&rM qú¡û8üý÷þ-ÿ¡ £¢D±E®z§qGC{M q ?&¯ MIxÚ4~υ ← Υ(Q, r) ÒL ← { (Q, r, υ) } ÒD q r A z ¯ MQ r δ „Q = ( q ij z ² ? C4t&yM rM D&±E²z) r = (qr i ) M A/?& zYDM`Úszxz ¯ z4rC4z n z ~&M q r/² ? ®z4°^q ‡ z4q ? Ú¤¦¥¨§© q r/ÚWqIz q ~ q yNz zx±C§M q MIx A/? D|DC§M q MIx A³Wz q MI‚¨µNz°ax4r/z4² ? C§ty~c‚†yz zx~8Òw¥r/M q zx A M D-rC4t&x±EMUC4zxM`°x±EMBqFz§q A M q ±v{ ‡?&A/? q Ú4z‡?&?$…Q ′ x = r ′ zQ ′′ x = r ′′ q ? ² r n ? q jqGC4MFw§~c‚¨µNzs = q klw§C{Yx4MIÚ ?A/?& ±h}k, l ∈ { 1, 2, . . . , n } ³ A/?M`qFC4zq ′ ij ← q′′ ij ← q w4C{ij (i, j) ≠ (k, l) (i, j) ≠ (l, k)q ′ lk ← q′ kl ← [ q , mid q kl kl ] q ′′lk ← q′′ kl ← [ mid q kl, q zWC4z ³kl ]r ′ ← r ′′ ← ³³ r ÒM`qFC4z w4C{ZxMIÚ ?&A/??&ÛI?s = r k k ∈ { 1, 2, . . . , n }Q ′ ← Q ′′ ← Q r ′ k ← [ r k , mid r k ] r ′′ ³ ³ A/? ³r ′ i ← r′′ i ← r ii ≠ k Òk ← [ mid r k, r k ] ³w4C{υ ′ ← Υ(Q ′ , r ′ ) υ ′′ ← Υ(Q ′′ , r ′′ ) ÒzL (Q, r, υ) Òr/x qFz qˆn r ‡ z4qFz ?(Q ′ , r ′ , υ ′ ) (Q ′′ , r ′′ , υ ′ ) LD7q ‡ z4q ? Ú z n z4qFt ‡ r n ²±ED&y~c‚´DMGw¥~-µNMI°D&±E®z4qGC4{M q ?&¯ MIx4Úz~`w¥r/C{M q z n‡? {cw§ÚcMJD ? n r&q A r/x4zW{ A/ M A tM ÛI?7‡? C{ÒD² ? n x4r$®4rM q ‡ M D~c‚ n r ‡ z4qFt7q ‡ z4qFÚWrT®M M n?(Q, r, υ) Ò¤¤¥Z ← υ Ò

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n × n { q r A z ¯ r ˜Q = ( ˜qij ) z n {nD&MIÚ A? ˜r = ( ˜r i ) ? ² r n ? D&rx±Ëz nD˜q ij =⎧⎪⎨[ q ij, q ij],[ q ij , q ij ],‡4 z ∂x ν (Q, r)∂q ij≥ 0,‡4 z ∂x ν (Q, r)∂q ij≤ 0, -¢ w ¡-. „£ f%go¤'g6¥ ¦j§ ]ym a¨ g§f© ` \ _ ¤ `b_#ªz`b_-_ \E« _#à ¤"g àb_&9 +q MIx A? D C4M⎪⎩q ij ,‡4 z ÒPÐ"Ñ ∂x ν (Q, r)∂q ij∋ 0,⎧[ r i , r i ],‡4 z ∂x ν (Q, r)∂r i≥ 0,˜r i =⎪⎨[ r i , r i ],‡4 z ∂x ν (Q, r)∂r i≤ 0,⎪⎩r i ,‡4 z ÒPÐ"Ñ ∂x ν (Q, r)∂r i∋ 0, -¢ w ¡&@ „Û w§M ‹ ÒPÐÑrŒ ? ² ? n x4r$®§r/M A Dx~ A/ M`xx ? q A t7zx A M D-rC§r4³ A/? ³ ? ®MID&zcw§x ? ³min{ x ν | x ∈ Ξ sym ( ˜Q, ˜r) } = min{ x ν | x ∈ Ξ sym (Q, r) }.˜r< ‡4? qFÚ ? C4tÚ~ZÚ ? C§z®M`q A D ? qF~cµM_q A D&MIxx ? zx A M D-rC4t&x±h}ˆ qNxMIx4~`C§MID ? °5yNz{Q z r ³ A? ³ ‡ M MI} ? w4{ ?&A z4qƒ} ? w¥x ? °a B< ' Qx = r ÚqIz4q A M q M ˜Qx = ˜r³min{ x ν | x ∈ Ξ sym (Q, r) } w zx ? ° „) C4M q MIx A/? DZD˜Qzq ?$… M A ²&± A t4³8D ?&? ²&µM ÛI? D ?& {³ q MIx4tyz q ³¥®4M qq ±u~ ‡4? µNrM qˆn r$w8r$®~7D±E®z§qGC4M`xzW{rxMIM7DZzx A M D-rC4t&x ? q r/x4r/C4z n MTD {cw§M7®z§qGC4M`xx±h} q M A/? w§zÚ€~ … M|z4q©{>C§t n ? D&r/C4z4qFt ‡4? z n D ? w§x±EM MIyM`xz°‰qIz4q A M q C4zxM`°x±h}Ê~ r/D&xMIx4z° ‡?‡?C4M q MIx A r qq r A z ¯ ±dz ‡ rD ? °f®4r&q A z’ q q w ³6Ú ‡ z q M ~³ ‡ C§rD~ 9 Úxz Û zF# „©wbt ±EDMFw§M q ‘ ?& q ~_C4±uw4C{ IA zW} ‡4? z n D ? w§x4±h}Yq"~c® A/? q qIz qXq M A/ z®x ? {!A z q r A/ z ¯ ±´qFz§q A M q ±u~ rDxM`xz° wqk Ü lk ≤ l ≤ n Qx = r zyNM q qFz4q A M q ~^~ r/Dx4MIxz° D|Dzcw¥M‡ rq w>z1 ≤³ ~&q A tx4MIÚ ?A/?& ±EMƒ‘JzÚWqIz ? D-r/xx4±EM7zxcw§M`ÚcqI±"³ A r/Ú4zM® A?n∑q ij x j = r i , i = 1, 2, . . . , n,j=1 -¢ w ¡ + „

¥ ¦·'¦=¸¯_Bª_Bàb_)a` m _¼c ¤'gof© `h¹ f a`b_Rº`4h¹ l a l+m _©¨ l©\lK¤/»4¼F» ¨†a -9§z ‡? w§z‘W‘JM MIx ¯ z ~&M q ‡?q klw '(®z A ±ED-r{³® A?∂∂q kl(qij x j)=∂q ij∂q klx j + q ij∂x j∂q kl,∂q ij∂q kl={ 0,M`qGC§z1,(i, j) ≠ (k, l),(i, j) = (k, l),Û w§MM`qGC§z± ‡4? C4~c®z q z n -¢ w ¡ + „q⎧⎪⎨⎪⎩n∑j=1n∑j=1n∑j=1q ij∂x j∂q kl= 0,q ij∂x j∂q kl+ x l = 0,q ij∂x j∂q kl+ x k = 0,M_qGC4zM_qGC4zM_qGC4zi ≠ k z i ≠ l,i = k,i = l.A/?Q ·(∂x ∂x1= , . . . , ∂x ) ⊤n,∂q kl ∂q kl ∂q kl∂x∂q kl=⎛⎜⎝0 w ww0−x l0 w ww0−x k0 w ww0⎞⎟⎠{ˆM q M`q A/?←− k{ˆM q M`q A?w←− lÞ rÚz q ? ² r n ? q ³cM`qFC4z° z®zxM ‡ ³J?3IA/?∂x() ⊤,= Q −1 · 0, . . . , 0, −x l , 0, . . . , 0, −x k , 0, . . . , 0∂q kl

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