1. Sharp estimates for approximate and exact solutions to quasi-optimization problems.
- Author
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Ait Mansour, Mohamed, Bahraoui, Mohamed-Amin, and El Bekkali, Adham
- Subjects
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SET-valued maps , *NASH equilibrium , *CONVEX programming , *STRAINS & stresses (Mechanics) , *MATHEMATICS - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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