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Register a new userThe Gauss-Seidel Method for Generalized Nash Equilibrium Problems of Polynomials
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This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss-Seidel method and Lasserre type Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss-Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinars Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method.
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This paper studies Nash equilibrium problems that are given by polynomial functions. We formulate efficient polynomial optimization problems for computing Nash equilibria. The Lasserre type Moment-SOS relaxations are used to solve them. Under generic assumptions, the method can find a Nash equilibrium if there is one. Moreover, it can find all Nash equilibria if there are finitely many ones of them. The method can also detect nonexistence if there is no Nash equilibrium.
This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) 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.
In this paper we propose a new operator splitting algorithm for distributed Nash equilibrium seeking under stochastic uncertainty, featuring relaxation and inertial effects. Our work is inspired by recent deterministic operator splitting methods, designed for solving structured monotone inclusion problems. The algorithm is derived from a forward-backward-forward scheme for solving structured monotone inclusion problems featuring a Lipschitz continuous and monotone game operator. To the best of our knowledge, this is the first distributed (generalized) Nash equilibrium seeking algorithm featuring acceleration techniques in stochastic Nash games without assuming cocoercivity. Numerical examples illustrate the effect of inertia and relaxation on the performance of our proposed algorithm.
We consider a network of prosumers involved in peer-to-peer energy exchanges, with differentiation price preferences on the trades with their neighbors, and we analyze two market designs: (i) a centralized market, used as a benchmark, where a global market operator optimizes the flows (trades) between the nodes, local demand and flexibility activation to maximize the system overall social welfare; (ii) a distributed peer-to-peer market design where prosumers in local energy communities optimize selfishly their trades, demand, and flexibility activation. We first characterizethe solution of the peer-to-peer market as a Variational Equilibrium and prove that the set of Variational Equilibria coincides with the set of social welfare optimal solutions of market design (i). We give several results that help understanding the structure of the trades at an equilibriumor at the optimum. We characterize the impact of preferences on the network line congestion and renewable energy waste under both designs. We provide a reduced example for which we give the set of all possible generalized equilibria, which enables to give an approximation of the price ofanarchy. We provide a more realistic example which relies on the IEEE 14-bus network, for which we can simulate the trades under different preference prices. Our analysis shows in particular that the preferences have a large impact on the structure of the trades, but that one equilibrium(variational) is optimal.
The randomized Gauss--Seidel method and its extension have attracted much attention recently and their convergence rates have been considered extensively. However, the convergence rates are usually determined by upper bounds, which cannot fully reflect the actual convergence. In this paper, we make a detailed analysis of their convergence behaviors. The analysis shows that the larger the singular value of $A$ is, the faster the error decays in the corresponding singular vector space, and the convergence directions are mainly driven by the large singular values at the beginning, then gradually driven by the small singular values, and finally by the smallest nonzero singular value. These results explain the phenomenon found in the extensive numerical experiments appearing in the literature that these two methods seem to converge faster at the beginning. Numerical examples are provided to confirm the above findings.
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