Convex Semi-Infinite Parametric Programming: Uniform Convergence of the Optimal Value Functions of Discretized Problems

The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.