Convex generalized Nash equilibrium problems and polynomial optimization

This paper studies convex generalized Nash equilibrium problems that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing generalized Nash equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.

Online distributed tracking of generalized Nash equilibrium on physical networks

In generalized Nash equilibrium (GNE) seeking problems over physical networks such as power grids, the enforcement of network constraints and time-varying environment may bring high computational costs. Developing online algorithms is recognized as a promising method to cope with this challenge, where the task of computing system states is replaced by directly using measured values from the physical network. In this paper, we propose an online distributed algorithm via measurement feedback to track the GNE in a time-varying networked resource sharing market. Regarding that some system states are not measurable and measurement noise always exists, a dynamic state estimator is incorporated based on a Kalman filter, rendering a closed-loop dynamics of measurement-feedback driven online algorithm. We prove that, with a fixed step size, this online algorithm converges to a neighborhood of the GNE in expectation. Numerical simulations validate the theoretical results.