Lispchitz modulus of the argmin mapping in convex quadratic optimization
María Josefa Cánovas, Masao Fukushima, Juan Parra
TL;DR
This work addresses the stability of the argmin mapping for canonically perturbed convex quadratic programs by deriving a point-based Lipschitz modulus formula. It extends the classical Nürnberger Condition to the convex quadratic setting and introduces the extended family of KKT index sets $\mathcal{L}_{\bar c,\bar b}(\bar x)$ to overcome limitations of minimal KKT sets. The main result establishes that, under a suitable Aubin-type condition, the Lipschitz modulus is given by $\mathrm{lip}\,\mathcal{S}((\bar c,\bar b),\bar x) = \max_{D\in \mathcal{L}_{\bar c,\bar b}(\bar x)} \left\| ( I_n 0_{n\times |D|}) M_D^{-1} \right\|$, providing a computable, data-only formula. The findings specialize to the metric projection problem and offer a principled approach to sensitivity analysis in quadratic and linear programming, clarifying when Nagging NC-type conditions are necessary or not for stability.
Abstract
This paper was initially motivated by the computation of the Lipschitz modulus of the metric projection on polyhedral convex sets in the Euclidean space when both the reference point and the polyhedron where it is projected are subject to perturbations. The paper tackles the more general problem of computing the Lipschitz modulus of the argmin mapping in the framework of canonically perturbed convex quadratic problems. We point out the fact that a point-based formula (depending only on the nominal data) for such a modulus is provided. In this way, the paper extends to the current quadratic setting some results previously developed in linear programming. As an application, we provide a point-based formula for the Lipschitz modulus of the metric projection on a polyhedral convex set.