Sharp estimates for approximate and exact solutions to quasi-optimization problems

Abstract

In this paper, we consider a special implicit set-valued map representing solutions to a parametric quasi-optimization problem, $\left(\mathrm{Q}\mathrm{O}\mathrm{p}\mathrm{t}\right)$ 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 $\left(\mathrm{Q}\mathrm{O}\mathrm{p}\mathrm{t}\right)$ 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 $\left(\mathrm{Q}\mathrm{O}\mathrm{p}\mathrm{t}\right)$ 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.

Keywords:

• Approximate fixed points
• approximate Lim’s Lemma
• quasi-minimization
• parametric perturbation
• quantitative stability

2010 Mathematics Subject Classifications:

• Primary: 49J40
• Secondary: 49J52
• 49J35
• 49K40
• 65K10

Acknowledgements

The authors wish to thank an anonymous referee for his/her useful remarks and valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).