Approximation and exact penalization in simple bilevel variational problems

TL;DR

This work tackles a simple bilevel variational problem where the lower level is a variational inequality and the upper level optimizes a convex objective over a convex set. By introducing inexactness in the lower level and reformulating via Minty-gap (and Stampacchia-gap) approaches, the authors obtain a constrained single-level problem and exploit extended MFCQ to enable exact penalization. They prove a uniform bound on the penalty parameter across all finite-cut approximations, and develop three algorithms that combine exact penalization with cutting-plane refinements of the Minty gap. Numerical experiments on VI-based quadratic problems and network Cournot equilibria demonstrate that the final inexactness is typically well below theoretical bounds and that the proposed methods compare favorably with existing approaches. The framework provides a robust route for solving bilevel variational problems with non-affine lower-level operators and suggests extensions to broader quasi-variational settings.

Abstract

A simple bilevel variational problem where the lower level is a variational inequality while the upper level is an optimization problem is studied. We consider an inexact version of the lower problem, which guarantees enough regularity to allow the exploitation of techniques of exact penalization. Moreover, cutting planes are used to approximate the Minty gap function of the lower level. Algorithms to solve the resulting inexact bilevel problem are devised relying on these techniques and approximations. Finally, their convergence is studied in details by analysing also the effect of the given inexactness.

Figures (2)

• Figure 1: $\varepsilon$-solutions for different $a$ and $b$ for the operator $G_{a,b}$ on the set $[0,1]^2$
• Figure 2: Flowchart of the method

Theorems & Definitions (20)

• Proposition 1
• Proposition 2
• proof
• definition 1
• Proposition 3
• proof
• definition 2
• Proposition 4
• proof
• Proposition 5