Sharp estimates for approximate and exact solutions to quasi-optimization problems. (4th May 2022)

• Record Type: Journal Article
• Authors: Ait Mansour, Mohamed; Bahraoui, Mohamed-Amin; El Bekkali, Adham
• Is Part Of: Optimization. Volume 71:Number 5(2022)
• Page Start: 1331
• Page End: 1355
• Publication Date: 2022-05-04
• Subjects: Approximate fixed points -- approximate Lim's Lemma -- quasi-minimization -- parametric perturbation -- quantitative stability; Primary: 49J40 -- Secondary: 49J52 -- 49J35 -- 49K40 -- 65K10; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2021.1873986
• Languages: English
• ISSNs: 0233-1934

Abstract

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.