13,213 research outputs found
A relaxed-inertial forward-backward-forward algorithm for Stochastic Generalized Nash equilibrium seeking
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
Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming
AbstractSample average approximation (SAA) is one of the most popular methods for solving stochastic optimization and equilibrium problems. Research on SAA has been mostly focused on the case when sampling is independent and identically distributed (iid) with exceptions (Dai et al. (2000) [9], Homem-de-Mello (2008) [16]). In this paper we study SAA with general sampling (including iid sampling and non-iid sampling) for solving nonsmooth stochastic optimization problems, stochastic Nash equilibrium problems and stochastic generalized equations. To this end, we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem. Exponential convergence of estimators of both optimal solutions and M-stationary points (characterized by Mordukhovich limiting subgradients (Mordukhovich (2006) [23], Rockafellar and Wets (1998) [32])) are established under mild conditions. We also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic Nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem
Algorithms for generalized potential games with mixed-integer variables
We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches
Distributed Learning for Stochastic Generalized Nash Equilibrium Problems
This work examines a stochastic formulation of the generalized Nash
equilibrium problem (GNEP) where agents are subject to randomness in the
environment of unknown statistical distribution. We focus on fully-distributed
online learning by agents and employ penalized individual cost functions to
deal with coupled constraints. Three stochastic gradient strategies are
developed with constant step-sizes. We allow the agents to use heterogeneous
step-sizes and show that the penalty solution is able to approach the Nash
equilibrium in a stable manner within , for small step-size
value and sufficiently large penalty parameters. The operation
of the algorithm is illustrated by considering the network Cournot competition
problem
Computing all solutions of Nash equilibrium problems with discrete strategy sets
The Nash equilibrium problem is a widely used tool to model non-cooperative
games. Many solution methods have been proposed in the literature to compute
solutions of Nash equilibrium problems with continuous strategy sets, but,
besides some specific methods for some particular applications, there are no
general algorithms to compute solutions of Nash equilibrium problems in which
the strategy set of each player is assumed to be discrete. We define a
branching method to compute the whole solution set of Nash equilibrium problems
with discrete strategy sets. This method is equipped with a procedure that, by
fixing variables, effectively prunes the branches of the search tree.
Furthermore, we propose a preliminary procedure that by shrinking the feasible
set improves the performances of the branching method when tackling a
particular class of problems. Moreover, we prove existence of equilibria and we
propose an extremely fast Jacobi-type method which leads to one equilibrium for
a new class of Nash equilibrium problems with discrete strategy sets. Our
numerical results show that all proposed algorithms work very well in practice
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