A sequential linear complementarity problem method for generalized Nash equilibrium problems

TL;DR

This paper introduces a sequential linear complementarity problem (SLCP) method for solving generalized Nash equilibrium problems, with proven convergence and promising numerical results.

Contribution

It proposes a new SLCP method with a merit function for GNEPs, including comprehensive convergence analysis and local quadratic convergence.

Findings

The merit function decreases along search directions, ensuring global convergence.

The method demonstrates local quadratic convergence.

Preliminary results show the method's effectiveness and competitiveness.

Abstract

Generalized Nash equilibrium problems (GNEPs) arise in various applications where multiple players minimize individual cost functions subject to coupled constraints. A relatively unexplored approach to solving such problems is via a sequence of (mixed) linear complementarity problems (LCPs). Compared with the nonlinear equilibrium subproblems arising in recently popular penalty-based methods such as augmented Lagrangian methods, these LCPs are often substantially easier to solve. However, the existing literature on this approach is very limited, largely because of the difficulty of assessing the search directions generated by the subproblems and establishing a principled step-length acceptance criterion. This paper proposes a sequential linear complementarity problem (SLCP) method with a comprehensive convergence analysis. To assess the search directions, we introduce a novel merit function analogous to the $\ell_1$ penalty function in sequential quadratic programming. The merit function is shown to decrease along the search directions generated by the subproblems under suitable assumptions, thereby guaranteeing the global convergence of the SLCP method. We further establish local quadratic convergence and analyze the solvability of the subproblems. Preliminary numerical results demonstrate the effectiveness and competitiveness of the proposed method relative to existing approaches.