v2010.10.26 - Convex Optimization

248 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.5.2 semidefinite program via SchurSchur complement (1511) can be used to convert a projection problemto an optimization problem in epigraph form. Suppose, for example,we are presented with the constrained projection problem studied byHayden & Wells in [184] (who provide analytical solution): Given A∈ R M×Mand some full-rank matrix S ∈ R M×L with L < Mminimize ‖A − X‖ 2X∈ S MFsubject to S T XS ≽ 0(580)Variable X is constrained to be positive semidefinite, but only on a subspacedetermined by S . First we write the epigraph form:minimize tX∈ S M , t∈Rsubject to ‖A − X‖ 2 F ≤ tS T XS ≽ 0(581)Next we use Schur complement [277,6.4.3] [248] and matrix vectorization(2.2):minimize tX∈ S M , t∈R[]tI vec(A − X)subject tovec(A − X) T ≽ 0 (582)1S T XS ≽ 0This semidefinite program (4) is an epigraph form in disguise, equivalentto (580); it demonstrates how a quadratic objective or constraint can beconverted to a semidefinite constraint.Were problem (580) instead equivalently expressed without the squareminimize ‖A − X‖ FX∈ S Msubject to S T XS ≽ 0(583)then we get a subtle variation:minimize tX∈ S M , t∈Rsubject to ‖A − X‖ F ≤ tS T XS ≽ 0(584)