v2007.09.13 - Convex Optimization

536 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSGiven f(x) : X ∪ Y →R and arbitrary sets X and Y [147,0.1.2]X ⊂ Y ⇒ inf f(x) ≥ inf f(x) (1457)x∈X x∈Yinf f(x) = min{inf f(x), inf f(x)} (1458)x∈X ∪Y x∈X x∈Yinf f(x) ≥ max{inf f(x), inf f(x)} (1459)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 (1460)arg infX∈ C ‖X − Y ‖ F = arg infX∈ C ‖X − Y ‖2 F (1461)C.2 diagonal, trace, singular and eigen valuesFor A∈ R m×n and σ(A) denoting its singular values, [46,A.1.6][90,1] (confer (36))∑iσ(A) i = tr √ A T A = sup tr(X T A) = maximize tr(X T A)‖X‖ 2 ≤1X∈R m×n [ I Xsubject toX T I]≽ 0= 1 2 minimizeX∈S m , Y ∈S nsubject totrX + trY[ ] X AA T ≽ 0Y(1462)