v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization 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)
3.5. EPIGRAPH, SUBLEVEL SET 249that leads to an equivalent for (583) (and for (580) by (513))minimizeX∈ S M , t∈Rsubject tot[tI vec(A − X)vec(A − X) T t]≽ 0(585)S T XS ≽ 03.5.2.0.1 Example. Schur anomaly.Consider a problem abstract in the convex constraint, given symmetricmatrix Aminimize ‖X‖ 2X∈ S M F − ‖A − X‖2 Fsubject to X ∈ C(586)the minimization of a difference of two quadratic functions each convex inmatrix X . Observe equality‖X‖ 2 F − ‖A − X‖ 2 F = 2 tr(XA) − ‖A‖ 2 F (587)So problem (586) is equivalent to the convex optimizationminimize tr(XA)X∈ S Msubject to X ∈ C(588)But this problem (586) does not have Schur-formminimize t − αX∈ S M , α , tsubject to X ∈ C‖X‖ 2 F ≤ t(589)‖A − X‖ 2 F ≥ αbecause the constraint in α is nonconvex. (2.9.1.0.1)
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3.5. EPIGRAPH, SUBLEVEL SET 249that leads to an equivalent for (583) (and for (580) by (513))minimizeX∈ S M , t∈Rsubject tot[tI vec(A − X)vec(A − X) T t]≽ 0(585)S T XS ≽ 03.5.2.0.1 Example. Schur anomaly.Consider a problem abstract in the convex constraint, given symmetricmatrix Aminimize ‖X‖ 2X∈ S M F − ‖A − X‖2 Fsubject to X ∈ C(586)the minimization of a difference of two quadratic functions each convex inmatrix X . Observe equality‖X‖ 2 F − ‖A − X‖ 2 F = 2 tr(XA) − ‖A‖ 2 F (587)So problem (586) is equivalent to the convex optimizationminimize tr(XA)X∈ S Msubject to X ∈ C(588)But this problem (586) does not have Schur-formminimize t − αX∈ S M , α , tsubject to X ∈ C‖X‖ 2 F ≤ t(589)‖A − X‖ 2 F ≥ αbecause the constraint in α is nonconvex. (2.9.1.0.1)
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