Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems

Abstract

We consider convex semiinfinite programming (SIP) problems with an arbitrary fixed index set T. The article analyzes the relationship between the upper and lower semicontinuity (lsc) of the optimal value function and the optimal set mapping, and the so-called Hadamard well-posedness property (allowing for more than one optimal solution). We consider the family of all functions involved in some fixed optimization problem as one element of a space of data equipped with some topology, and arbitrary perturbations are premitted as long as the perturbed problem continues to be convex semiinfinite. Since no structure is required for T, our results apply to the ordinary convex programming case. We also provide conditions, not involving any second order optimality one, guaranteeing that the distance between optimal solutions of the discretized subproblems and the optimal set of the original problem decreases by a rate which is linear with respect to the discretization mesh-size.

Keywords:

• Stability
• Convex semiinfinite programming
• Optimal value function
• Optimal set mapping
• Hadamard well-posedness
• Rate of convergence
• Mathematics Subject Classifications 2000: Primary 90C34, 90C31
• Secondary 90C25, 90C05, 05C38, 15A15