A sequential linear complementarity problem method for generalized Nash equilibrium problems

TL;DR

This paper addresses solving generalized Nash equilibrium problems ($GNEPs$) by introducing a sequential linear complementarity problem ($SLCP$) method. A novel merit function is designed to guarantee descent and global convergence under an $(\alpha,\beta)$-monotonicity condition, with subproblems that are affine GNEPs ($AGNEPs$) and solvable under mild regularity. Theoretical results establish global convergence and local superlinear convergence, linking the SLCP method to classical SQP when the $GNEP$ reduces to a nonlinear program; the analysis also covers solvability for $AGNEPs$ and the role of restricted private strategies. Numerical experiments compare SLCP to augmented Lagrangian and Newton-type methods, showing competitive accuracy and efficiency, especially on problems with many $AGNEP$ subproblems and favorable problem structure.

Abstract

We propose a sequential linear complementarity problem (SLCP) method for solving generalized Nash equilibrium problems (GNEPs). By introducing a novel merit function that utilizes the specific structure of GNEPs, we establish global convergence of the method. The conditions guaranteeing global convergence are analogous to those for the classical sequential quadratic programming method with exact Lagrange Hessians, making this a natural and reasonable generalization. Moreover, we provide a detailed analysis of the solvability of the mixed linear complementarity subproblems, which are formulated as affine GNEPs. Sufficient characterizations for the local superlinear convergence are also derived, highlighting the efficiency of the proposed method. Finally, numerical experiments demonstrate the practical performance and effectiveness of the SLCP method in comparison with existing approaches.