Sharp estimates for approximate and exact solutions to quasi-optimization problems. (4th May 2022) - Ait Mansour, Mohamed; Bahraoui, Mohamed-Amin; El Bekkali, Adham
Abstract : In this paper, we consider a special implicit set-valued map representing solutions to a parametric quasi-optimization problem, ( Q O p t ) for short. This model finds its motivation in quasi-convex programming and generalized Nash equilibria modelled by the supremum of the so-called Nikaido–Isoda functions. We exploit a new recent variant of the celebrated Lim's Lemma considered in the context of metric regularity and approximate fixed points to establish quantitative stability for ε -approximate solutions to ( Q O p t ) under parametric perturbations in the spirit of the result presented for convex programming in the seminal contribution by Attouch and Wets [Quantitative stability of variational systems: III. ε -approximatesolutions. Math Program. 1993;61:197–214, Theorem 4.3]. Sharp estimates are then extended to parametric exact solutions to ( Q O p t ) by means of a qualitative stability analysis stressing the role of Painlevé-Kuratowski and Pompeiu-Hausdorff convergence for sets of approximate minima to a set of exact ones under usual compactness and/or completeness conditions. Finally, we apply our main result to a non-smooth mathematical program under polyhedral convex mappings and situate our contribution in the close recent literature.