A Log-domain Interior Point Method for Convex Quadratic Games

TL;DR

The paper addresses computing variational generalized Nash equilibria in convex quadratic games by reformulating the problem as a variational inequality and solving it with a log-domain interior-point method (IPM) that tracks a central path via $\mu$ and a damped Newton iteration. The method yields a general-purpose solver that accommodates non-potential, general-sum CQGs with limited structural assumptions, demonstrated on three benchmark CQGs (EV charging, market, traffic routing) showing two-phase convergence and favorable performance for small-to-medium problems. Key contributions include the log-domain central-path formulation, practical implementation details (scaling, regularization, sparse LU reuse), and comparisons with first-order methods, highlighting robustness to problem monotonicity. The work provides a scalable, open binary for solving v-GNEs in engineering and economic networks, with potential for broader application and further optimization.

Abstract

In this paper, we propose an equilibrium-seeking algorithm for finding generalized Nash equilibria of non-cooperative monotone convex quadratic games. Specifically, we recast the Nash equilibrium-seeking problem as variational inequality problem that we solve using a log-domain interior point method and provide a general purpose solver based on this algorithm. This approach is suitable for non-potential, general sum games and does not require extensive structural assumptions. We demonstrate the efficiency and versatility of our method using three benchmark games and demonstrate our algorithm is especially effective on small to medium scale problems.

Figures (11)

• Figure 1: Non-EV power demand and EV charging power for a sample EV charging game.
• Figure 2: VI residual and barrier parameter $\mu$ at each iteration when solving the sample game. The orange curve denotes the value of $\mu$, and the blue curve denotes the value of residual. It demonstrates that the $\mu$ remains constant during the additional Newton steps while the residual is greatly reduced
• Figure 3: Demonstration of the solution to the sample market game. The production and sales level at each location is represented in stacked bar plot along with the demand level.
• Figure 4: VI residual and barrier parameter $\mu$ at each iteration when solving the sample game. The orange curve denotes the value of $\mu$, and the blue curve denotes the value of residual. It demonstrates that the $\mu$ remains constant during the additional Newton steps while the residual is greatly reduced
• Figure 5: A road network with 14 edges. Each edge represents a road link with traffic direction and each vertex represents the starting/ending point of the road links

Theorems & Definitions (2)

• Definition 1
• Remark 1