v2006.03.09 - Convex Optimization

C.3. INVOLVING DIAGONAL, TRACE, EIGENVALUES 481(Ye) Assuming existence of a minimum element (p.99):minimize |x|xsubject to x ∈ C≡minimize 1 T (α + β)α,β, xsubject to α,β ≽ 0x = α − βx ∈ C(1272)All these problems are convex when C is.C.3 involving diagonal, trace, eigenvaluesFor A∈ R m×n and σ(A) denoting its singular values, [38,A.1.6][73,1]∑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 ≽ 0I(1273)For X ∈ S m , Y ∈ S n , A∈ C ⊆ R m×n for C convex, and σ(A) denotingthe singular values of A [73,3]minimizeA∑σ(A) isubject to A ∈ Ci≡minimizeA , X , Ysubject totrX + trY[ ] X A≽ 0YA TA ∈ C(1274)For A∈ S N + and β ∈ Rβ trA = maximize tr(XA)X∈ S Nsubject to X ≼ βI(1275)