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PSfrag replacementsPruning a box from Baumann point in an Interval Global Optimization Algorithm 243Thus, we introduce the following notation:BPROBCPB0OB (Original Box): OB = X − c,BPR (Box containing PR): BP R = ([pr L 1 , prU 1 ], [prL 2 , prU 2 ]),PB (Pruneable Box): P B = ([pb L 1, pb U 1 ], [pb L 2 , pb U 2 ]),(3)Figure 4. The new notationfor the easier formulation.A(CP B) =( prU12 − prL 12where BP R is the smallest box which contains the pruneable region(PR). In our new notation CP B = ([pr L 1 /2, prU 1 /2], [prL 2 /2, prU 2 /2]),and the area of the CPB isi.e. half of the area of PR, what is the half of the area of BPR.2.1 Shifting CPB (Centered Pruneable Box)) ( )prU22 − prL 2= 1 2 4 (prU 1 − prL 1 )(prU 2 − prL 2 ) = 1 A(BP R),4As it can be seen in Figure 5, sometimes it is better to shift the CPB to the edge of OB. Thiscan improve the method by reducing the number of the generated subboxes at the cost ofthe reduction of the rejected area compared to the area of CPB. To differentiate the centeredpruneable box (CPB) from the shifted pruneable box, we notate the later as SPB. In the caseswhen OB ∩ CP B ≠ CP B, the area of SPB can be larger than the area of CPB (see the thirdcase in Figure 5).Figure 5.Shifting cases.The areas of the boxes (supposing that the resulting SPB is inside of the original one (OB))can be computed as:( ( ) ( )) ()A(SP B ob U ) = pr U2 1 1 − obU 2− pr Lpr U 1 1 − obU 2ob Uob U2pr U 2 − pr L 222pr U 2( ) ( )= (pr U 1 − pr L 1) 1 − obU 2 obU2(pr Upr U 2 pr U 2 − pr L 2) = 1 − obU 2 obU2A(BP R) (4)2pr U 2 pr U 2( )A(SP B ob I ) = 1 − obI i obIiA(BP R), i = 1, 2, I = L, U (5)i pr I ipr I iIf an SPB is inside the original box then the shifted area only depends on the value obI i, i =pr I i1, 2, I = L, U, obtaining larger area if it is nearer to 1/2. If it equals to 1/2 we obtain theA(CP B), what also means that SP B ob I = CP B.i(ob UThese equations show, that knowing the vector 1, obL pr U 1, obU1 pr L 2, obL1 pr U 2one can choose the best2 pr L 2Pruneable Box (PB), if these are inside the original box. The other cases do not differ toomuch, but those have to be treated differently. This would lead to a case analysis, that cannotbe extended for multidimensional case. In case we use the Baumann point, it is guaranteedthat the CPB is inside OB, therefore the case analysis can be avoided.)