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.

在本文中，我们考虑一个特殊的隐式集值映射，表示参数准优化问题的解决方案，$\left(\mathrm{问}\mathrm{○}\mathrm{p}\mathrm{吨}\right)$简而言之。这个模型在准凸规划和广义纳什均衡中找到了它的动机，它由所谓的二阶堂-矶田函数的上确界建模。我们利用在度量正则性和近似不动点的背景下考虑的著名 Lim 引理的新变体来建立ε -近似解的定量稳定性$\left(\mathrm{问}\mathrm{○}\mathrm{p}\mathrm{吨}\right)$根据 Attouch 和 Wets [变分系统的定量稳定性：III。ε -近似解。数学程序。1993；61:197–214，定理 4.3]。然后将夏普估计扩展到参数精确解$\left(\mathrm{问}\mathrm{○}\mathrm{p}\mathrm{吨}\right)$通过定性稳定性分析，强调 Painlevé-Kuratowski 和 Pompeiu-Hausdorff 在通常的紧凑性和/或完整性条件下将一组近似最小值收敛到一组精确最小值的作用。最后，我们将我们的主要结果应用于多面体凸映射下的非光滑数学程序，并将我们的贡献置于最近的文献中。