Optimization with Parametric Variational Inequality Constraints on a Moving Set

TL;DR

This paper introduces a novel approach to optimize problems constrained by parametric variational inequalities on moving sets, establishing solution regularity and proposing a convergent smoothing algorithm validated on portfolio management data.

Contribution

It demonstrates Lipschitz continuity of the solution function, proves metric regularity automatically, and develops a new smoothing implicit gradient algorithm with convergence guarantees.

Findings

Solution function is Lipschitz continuous.

The solution set is nonempty and bounded.

The proposed SIGA converges to a stationary point.

Abstract

This paper focuses on optimization problems constrained by Parametric Variational Inequalities (PVI) defined on a moving set. Unlike most existing works on mathematical programs with equilibrium constraints, the equilibrium constraints have parameters not only in the function but also in the related set. We show that the solution function of the PVI is Lipschitz continuous with respect to the upper-level decision variables and the solution set of the optimization problem is nonempty and bounded. Moreover, we prove that the metric regularity of the constraints holds automatically, which allow us to characterize stationary points without any additional assumptions. A Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI. We prove the convergence of SIGA to a stationary point of the optimization problem and numerically validate the efficiency of SIGA by portfolio management problems with real data.