Calmness Modulus of Linear Semi-infinite Programs

Abstract

Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand-side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

Marco A. López, a Pioneer of Continuous Optimization in Spain

“…Cánovas, R. Henrion, A. Kruger, J. Parra and M. Théra) are as follows: an expression for the calmness modulus of the mapping in linear programming under canonical perturbations (objective function and right-hand side of the constraints), involving limits of subdifferentials [10]; left-hand-side perturbations of the constraints system added into the analysis in [18]; and the outer limits of subdifferentials of max-functions and calmness moduli for feasible and optimal set mappings dealt with in [8]. -In [13,14,16], the distance to ill-posedness (in terms of the distance to infeasibility and to unsolvability) is studied.…”

Recent contributions to linear semi-infinite optimization: an update

“…In [38] a lower bound on clmS a ((c; b); x) for an LSIO with a unique optimal solution, under SCQ and in Scenario 4, is given in Theorem 6. It turns out that this lower bound equals the exact modulus when T is …nite without requiring either SCQ or the uniqueness of x as optimal solution of P: Also when T is …nite and the optimal set is a singleton, a new upper bound for clmS a ((c; b); x) is proposed in Theorem 13.…”

Recent contributions to linear semi-infinite optimization

“…In [32] a lower bound on clmS a ((c; b); x) for an LSIO with a unique optimal solution, under SCQ and in Scenario 4, is given in Theorem 6. It turns out that this lower bound equals the exact modulus when T is …nite without requiring either SCQ or the uniqueness of x as optimal solution of P: Also when T is …nite and the optimal set is a singleton, a new upper bound for clmS a ((c; b); x) is proposed in Theorem 13.…”