v2007.11.26 - Convex Optimization

3.1. CONVEX FUNCTION 203minimize tX∈ S M , t∈Rsubject to ‖A − X‖ 2 F ≤ tS T XS ≽ 0(485)Next we use Schur complement [206,6.4.3] [182] and matrix vectorization(2.2):minimize tX∈ S M , t∈R[]tI vec(A − X)subject tovec(A − X) T ≽ 0 (486)1S T XS ≽ 0This semidefinite program is an epigraph form in disguise, equivalentto (484); it demonstrates how a quadratic objective or constraint can beconverted to a semidefinite constraint.Were problem (484) instead equivalently expressed without the squarethen we get a subtle variation:that leads to an equivalent for (487)minimizeX∈ S M , t∈Rsubject tominimize ‖A − X‖ FX∈ S Msubject to S T XS ≽ 0minimize tX∈ S M , t∈Rsubject to ‖A − X‖ F ≤ tt[S T XS ≽ 0tI vec(A − X)vec(A − X) T tS T XS ≽ 0]≽ 0(487)(488)(489)3.1.7.2.1 Example. Schur anomaly.Consider a problem abstract in the convex constraint, given symmetricmatrix Aminimize ‖X‖ 2X∈ S M F − ‖A − X‖2 F(490)subject to X ∈ Cthe minimization of a difference of two quadratic functions each convex inmatrix X . Observe equality