v2006.03.09 - Convex Optimization

480 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSGiven f(x) : X →R and g(x) : X →R defined on arbitrary set X[123,0.1.2]inf (f(x) + g(x)) ≥ inf f(x) + inf g(x) (1263)x∈X x∈X x∈XGiven f(x) : X ∪ Y →R and arbitrary sets X and Y [123,0.1.2]X ⊂ Y ⇒ inf f(x) ≥ inf f(x) (1264)x∈X x∈Yinf f(x) = min{inf f(x), inf f(x)} (1265)x∈X ∪Y x∈X x∈Yinf f(x) ≥ max{inf f(x), inf f(x)} (1266)x∈X ∩Y x∈X x∈YOver some convex set C given vector constant y or matrix constant Yarg infx∈C ‖x − y‖ 2 = arg infx∈C ‖x − y‖2 2 (1267)arg infX∈ C ‖X − Y ‖ F = arg infX∈ C ‖X − Y ‖2 F (1268)C.2 involving absolute valueOptimal solution is norm dependent. [38, p.297]minimize ‖x‖ 1xsubject to x ∈ C≡minimize 1 T tt,xsubject to −t ≼ x ≼ tx ∈ C(1269)minimize ‖x‖ 2xsubject to x ∈ C≡minimizet,xsubject tot[ tI xx T tx ∈ C]≽ 0(1270)minimize ‖x‖ ∞xsubject to x ∈ C≡minimize tt,xsubject to −t1 ≼ x ≼ t1x ∈ C(1271)In R n the norms respectively represent: length measured along a grid,Euclidean length, maximum |coordinate|.