v2006.03.09 - Convex Optimization

A.4. SCHUR COMPLEMENT 445minimize ‖A − X‖ 2X∈ S MFsubject to S T XS ≽ 0(1125)Variable X is constrained to be positive semidefinite, but only on a subspacedetermined by S . First we write the epigraph form (3.1.1.3):minimize tX∈ S M , t∈Rsubject to ‖A − X‖ 2 F ≤ tS T XS ≽ 0(1126)Next we use the Schur complement [174,6.4.3] [153] and matrixvectorization (2.2):minimizeX∈ S M , t∈Rsubject tot[tI vec(A − X)vec(A − X) T 1]≽ 0(1127)S T XS ≽ 0This semidefinite program is an epigraph form in disguise, equivalentto (1125).Were problem (1125) instead equivalently expressed without the squareminimize ‖A − X‖ FX∈ S Msubject to S T XS ≽ 0(1128)then we get a subtle variation:minimize tX∈ S M , t∈Rsubject to ‖A − X‖ F ≤ tS T XS ≽ 0(1129)that leads to an equivalent for (1128)minimizeX∈ S M , t∈Rsubject tot[tI vec(A − X)vec(A − X) T t]≽ 0(1130)S T XS ≽ 0