Consider the following cone-constrained convex programming problem. (P) \(\min ~f(x),\) s.t. \(x\in C,\) \(-g(x)\in K,\) where \(X\) is a Banach space, \(Y\) is a locally convex (Hausdorff) space, \(C\) is a closed convex subset of \(X\), \(K\) is a closed convex cone in \(Y\), \( f:X\rightarrow \mathbb{R}\) is a continuous convex function, and \(g:X\rightarrow Y\) is a continuous \(K\)-convex mapping. The purpose of this paper is to present characterizations of the solution set of problem (P), which in particular covers semidefinite convex programs and programs involving explicit convex inequality constraints.
First, the authors establish that the Lagrangian function of (P) is constant on the solution set of (P). Then, they present various simple Lagrange multiplier-based characterizations of the solution set of (P). It is shown that, for a finite-dimensional convex program with inequality constraints, the characterizations illustrate the property that the active constraints with positive Lagrange multipliers at an optimal solution remain active at all optimal solutions of (P). Finally, they present applications of these results to derive corresponding Lagrange multiplier characterizations of the solution sets of semidefinite programs and fractional programs. In particular, they characterize the solution set of a semidefinite linear program in terms of a complementary slackness condition with a fixed Lagrange multiplier. Specific examples are given to illustrate the significance of the results.