v2010.10.26 - Convex Optimization

654 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSGiven scalar κ and f(x) : X →R and g(x) : X →R defined onarbitrary set X [199,0.1.2]inf (κ + f(x)) = κ + infx∈Xarg infx∈Xf(x)x∈X(κ + f(x)) = arg inf f(x) (1680)x∈Xinf κf(x) = κ infx∈Xarg infx∈Xf(x)⎫⎬x∈Xκf(x) = arg inf f(x) ⎭ , κ > 0 (1681)x∈Xinf (f(x) + g(x)) ≥ inf f(x) + inf g(x) (1682)x∈X x∈X x∈XGiven f(x) : X ∪ Y →R and arbitrary sets X and Y [199,0.1.2]X ⊂ Y ⇒ inf f(x) ≥ inf f(x) (1683)x∈X x∈Yinf f(x) = min{inf f(x), inf f(x)} (1684)x∈X ∪Y x∈X x∈Yinf f(x) ≥ max{inf f(x), inf f(x)} (1685)x∈X ∩Y x∈X x∈YC.2 Trace, singular and eigen valuesFor A∈ R m×n and σ(A) 1 connoting spectral norm,σ(A) 1 = √ λ(A T A) 1 = ‖A‖ 2 = sup ‖Ax‖ 2 = minimize t(639)‖x‖=1t∈R [ ] tI Asubject toA T ≽ 0tI∑iFor A∈ R m×n and σ(A) denoting its singular values, the nuclear norm(Ky Fan norm) of matrix A (confer (44), (1566), [203, p.200]) isσ(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= 1 2 minimizeX∈S m , Y ∈S nsubject to]≽ 0(1686)trX + trY[ ] X AA T ≽ 0Y