Optimal solution characterization for infinite positive semi-definite programming (Q1334827)

Let (P) denote the general convex programming problem: \[ \text{minimize } C(x)\text{ subject to } x\in X, \] where \(X\) is a nonempty, closed, convex subset of a real Hilbert space \(H\) and \(C\) is a convex, real-valued function on \(H\). Let \(X^*\) denote the set of optimal solutions of (P). The objective is to find additional assumptions on the convex objective function \(C\) which are sufficient to conclude that \(X^*\) is affine whenever \(X\) is. To this end suppose \(C(x)= 1/2\langle x,Q(x)\rangle+ \langle x,c\rangle\) for \(x\in H\), where \(Q: H\to H\) is a bounded linear operator on \(H\) and \(c\in H\). (P) is thus transformed into a quadratic programming problem (P\('\)). The result presented in this paper is the following: If \(Q\) in (P\('\)) is positive semidefinite, and if \(X\) is affine in \(H\), then the set of optimal solutions of (P\('\)), \(X^*\), is also affine in \(H\).