v2010.10.26 - Convex Optimization

564 CHAPTER 7. PROXIMITY PROBLEMSorthant R N−1+ requires: (E.9.2.0.1)δ(Υ ⋆ ) ρ+1:N−1 = 0δ(Υ ⋆ ) ≽ 0δ(Υ ⋆ ) T( δ(Υ ⋆ ) − π(δ(R T ΛR)) ) = 0δ(Υ ⋆ ) − π(δ(R T ΛR)) ≽ 0(1340)which are necessary and sufficient conditions. Any value Υ ⋆ satisfyingconditions (1340) is optimal for (1339a). So{ {δ(Υ ⋆ max 0, π ( δ(R) i =T ΛR) ) }, i=1... ρi(1341)0, i=ρ+1... N −1specifies an optimal solution. The lower bound on the objective with respectto R in (1339b) is tight: by (1307)‖ |Υ ⋆ | − |Λ| ‖ F ≤ ‖Υ ⋆ − R T ΛR‖ F (1342)where | | denotes absolute entry-value. For selection of Υ ⋆ as in (1341), thislower bound is attained when (conferC.4.2.2)R ⋆ = I (1343)which is the known solution.7.1.4.1 SignificanceImportance of this well-known [129] optimal solution (1316) for projectionon a rank ρ subset of a positive semidefinite cone should not be dismissed:Problem 1, as stated, is generally nonconvex. This analytical solutionat once encompasses projection on a rank ρ subset (216) of the positivesemidefinite cone (generally, a nonconvex subset of its boundary)from either the exterior or interior of that cone. 7.10 By problemtransformation to the spectral domain, projection on a rank ρ subsetbecomes a convex optimization problem.7.10 Projection on the boundary from the interior of a convex Euclidean body is generallya nonconvex problem.