Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

TL;DR

This paper develops a second-order formula for the graphical derivative of solution maps in parameterized equilibria with nonconvex conic constraints, and applies it to analyze stability properties like isolated calmness.

Contribution

It introduces a precise second-order formula involving Lagrange multipliers for the graphical derivative of solution maps with nonconvex conic constraints, advancing stability analysis.

Findings

Derived a pointwise second-order formula for graphical derivatives.

Characterized Lipschitzian stability (isolated calmness) using the formula.

Applied the formula to stability analysis of solution maps.

Abstract

The paper concerns parameterized equilibria governed by generalized equations whose multivalued parts are modeled via regular normals to nonconvex conic constraints. Our main goal is to derive a precise pointwise second-order formula for calculating the graphical derivative of the solution maps to such generalized equations that involves Lagrange multipliers of the corresponding KKT systems and critical cone directions. Then we apply the obtained formula to characterizing a Lipschitzian stability notion for the solution maps that is known as isolated calmness.