On error bounds and Newton-type methods for generalized Nash equilibrium problems

Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.     

Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems

Generalized Nash equilibrium problems (GNEPs) allow, in contrast to standard Nash equilibrium problems, a dependence of the strategy space of one player from the decisions of the other players. In this paper, we consider jointly convex GNEPs which form an important subclass of the general GNEPs. Based on a regularized Nikaido-Isoda function, we present two (nonsmooth) reformulations of this class of GNEPs, one reformulation being a constrained optimization problem and the other one being an unconstrained optimization problem. While most approaches in the literature compute only a so-called normalized Nash equilibrium, which is a subset of all solutions, our two approaches have the property that their minima characterize the set of all solutions of a GNEP. We also investigate the smoothness properties of our two optimization problems and show that both problems are continuous under a Slater-type condition and, in fact, piecewise continuously differentiable under the constant rank constraint qualification. Finally, we present some numerical results based on our unconstrained optimization reformulation.     

Restricted generalized Nash equilibria and controlled penalty algorithm

The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem, in which each player’s strategy set may depend on the rival players’ strategies. The GNEP has recently drawn much attention because of its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. However, a GNEP usually has multiple or even infinitely many solutions, and it is not a trivial matter to choose a meaningful solution from those equilibria. The purpose of this paper is two-fold. First we present an incremental penalty method for the broad class of GNEPs and show that it can find a GNE under suitable conditions. Next, we formally define the restricted GNE for the GNEPs with shared constraints and propose a controlled penalty method, which includes the incremental penalty method as a subprocedure, to compute a restricted GNE. Numerical examples are provided to illustrate the proposed approach.     

Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality

We define the concept of reproducible map and show that, whenever the constraint map defining the quasivariational inequality (QVI) is reproducible then one can characterize the whole solution set of the QVI as a union of solution sets of some variational inequalities (VI). By exploiting this property, we give sufficient conditions to compute any solution of a generalized Nash equilibrium problem (GNEP) by solving a suitable VI. Finally, we define the class of pseudo-Nash equilibrium problems, which are (not necessarily convex) GNEPs whose solutions can be computed by solving suitable Nash equilibrium problems.     

Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search

The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash game where, in addition to the cost functions, also the strategy spaces of each player depend on the strategies chosen by all other players. This problem is rather difficult to solve and there are only a few methods available in the literature. One of the most popular ones is the so-called relaxation method, which is known to be globally convergent under a set of assumptions. Some of these assumptions, however, are rather strong or somewhat difficult to understand. Here, we present a modified relaxation method for the solution of a certain class of GNEPs. The convergence analysis uses completely different arguments based on a certain descent property and avoids some of the technical conditions for the original relaxation method. Moreover, numerical experiments indicate that the modified relaxation method performs quite well on a number of different examples taken from the literature.     

Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems

Using a regularized Nikaido-Isoda function, we present a (nonsmooth) constrained optimization reformulation of the player convex generalized Nash equilibrium problem (GNEP). Further we give an unconstrained reformulation of a large subclass of player convex GNEPs which, in particular, includes the jointly convex GNEPs. Both approaches characterize all solutions of a GNEP as minima of optimization problems. The smoothness properties of these optimization problems are discussed in detail, and it is shown that the corresponding objective functions are continuous and piecewise continuously differentiable under mild assumptions. Some numerical results based on the unconstrained optimization reformulation being applied to player convex GNEPs are also included.     

Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players?? strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.     

Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation

We consider the generalized Nash equilibrium problem (GNEP), where not only the players’ cost functions but also their strategy spaces depend on the rivals’ decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable second-order condition. Some numerical results are presented to illustrate the performance of the method.     

On Intrinsic Complexity of Nash Equilibrium Problems and Bilevel Optimization

In this article we study generalized Nash equilibrium problems (GNEP) and bilevel optimization side by side. This perspective comes from the crucial fact that both problems heavily depend on parametric issues. Observing the intrinsic complexity of GNEP and bilevel optimization, we emphasize that it originates from unavoidable degeneracies occurring in parametric optimization. Under intrinsic complexity, we understand the involved geometrical complexity of Nash equilibria and bilevel feasible sets, such as the appearance of kinks and boundary points, non-closedness, discontinuity and bifurcation effects. The main goal is to illustrate the complexity of those problems originating from parametric optimization and singularity theory. By taking the study of singularities in parametric optimization into account, the structural analysis of Nash equilibria and bilevel feasible sets is performed. For GNEPs, the number of players’ common constraints becomes crucial. In fact, for GNEPs without common constraints and for classical NEPs we show that—generically—all Nash equilibria are jointly nondegenerate Karush–Kuhn–Tucker points. Consequently, they are isolated. However, in presence of common constraints Nash equilibria will constitute a higher dimensional set. In bilevel optimization, we describe the global structure of the bilevel feasible set in case of a one-dimensional leader’s variable. We point out that the typical discontinuities of the leader’s objective function will be caused by follower’s singularities. The latter phenomenon occurs independently of the viewpoint of the optimistic or pessimistic approach. In case of higher dimensions, optimistic and pessimistic approaches are discussed with respect to possible bifurcation of the follower’s solutions.     

A best-response approach for equilibrium selection in two-player generalized Nash equilibrium problems

ABSTRACT

In this paper, we propose a best-response approach to select an equilibrium in a two-player generalized Nash equilibrium problem. In our model we solve, at each of a finite number of time steps, two independent optimization problems. We prove that convergence of our Jacobi-type method, for the number of time steps going to infinity, implies the selection of the same equilibrium as in a recently introduced continuous equilibrium selection theory. Thus the presented approach is a different motivation for the existing equilibrium selection theory, and it can also be seen as a numerical method. We show convergence of our numerical scheme for some special cases of generalized Nash equilibrium problems with linear constraints and linear or quadratic cost functions.     

Numerical solution of an equilibrium problem of based on the generalized level method

An equilibrium problem is studied whose special case is finding a Nash point in a noncooperative multiperson game. A numerical algorithm for solving this problem is described. Conditions on the problem are stated under which an estimate is obtained for the convergence rate of the algorithm to a unique solution of the problem. The results are used for a numerical analysis of noncooperative games.     

The concert queueing game: strategic arrivals with waiting and tardiness costs

We consider the noncooperative choice of arrival times by individual users, who seek service at a first-come first-served queueing system that opens up at a given time. Each user wishes to obtain service as early as possible, while minimizing the expected wait in the queue. This problem was recently studied within a simplified fluid-scale model. Here, we address the unscaled stochastic system, assuming a finite (possibly random) number of homogeneous users, exponential service times, and linear cost functions. In this setting, we establish that there exists a unique Nash equilibrium, which is symmetric across users, and characterize the equilibrium arrival-time distribution of each user in terms of a corresponding set of differential equations. We further establish convergence of the Nash equilibrium solution to that of the associated fluid model as the number of users is increased. We finally consider the price of anarchy in our system and show that it exceeds 2, but converges to this value for a large population size.     

Collusive game solutions via optimization

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported. It is with great honor that we dedicate this paper to Professor Terry Rockafellar on the occasion of his 70th birthday. Our work provides another example showing how Terry's fundamental contributions to convex and variational analysis have impacted the computational solution of applied game problems. This author's research was partially supported by the National Science Foundation under grant ECS-0080577. This author's research was partially supported by the National Science Foundation under grant CCR-0098013.     

运输问题的合作对策求解方法

产地间或销地间往往存在竞争,在这种情况下,使用运输问题最优化方法是不合理的。因此,从个体理性的视角提出运输问题的合作对策求解方法,方法将运输问题看作是一个博弈问题,各个产地或销地是博弈的局中人,求解其纳什均衡与纳什讨价还价解。在此基础上,说明了运输问题的非合作形式是一个指派问题,并证明指派问题的最优解是一个纳什均衡点。接着,通过实验验证运输问题的最优解是一个纳什讨价还价解,满足产地或销地的自身利益。在此基础上,针对纳什讨价还价解不唯一的问题,从决策者的视角给出最大可能激励成本的计算方法。最后,为弥补纳什讨价还价解不唯一及纳什讨价还价解不允许出现子联盟的缺陷,给出运输收益分配或成本分摊的Shapely值计算方法。     

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.     

System of nonsmooth variational inequalities with applications

In this paper, we consider a system of vector variational inequalities and a system of nonsmooth variational inequalities defined by means of Clarke directional derivative. We also consider the Nash equilibrium problem with vector pay-offs and its scalarized form. We present some relations among these systems and problems. The existence results for a solution of system of nonsmooth variational inequalities are given. As a consequence, we derive an existence result for a solution of Nash equilibrium problem with vector pay-offs.     

Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming

Sample 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.     

An Ergodic Algorithm for the Power-Control Games for CDMA Data Networks

In this paper, we consider power control for the uplink of a direct-sequence code-division multiple-access data network. In the uplink, the purpose of power control is for each user to transmit enough power so that it can achieve the required quality of service without causing unnecessary interference to other users in the system. One method that has been very successful in solving this purpose for power control is the game-theoretic approach. The problem for power control is modified as a Nash equilibrium problem in which each user can choose its transmit power in order to maximize its own utility, and a Nash equilibrium is an ideal solution of the power-control game. We present a noncooperative power-control game in which each user can choose the transmit power in a way that it gets the sufficient signal-to-interference-plus-noise ratio and maximizes its own utility. To ensure the existence of a solution, we also propose the variational inequality problem which is connected with the proposed game. On a linear receiver, we deal with the matched filter receiver. Next we present a new ergodic algorithm for the proposed power control because the existing iterative algorithms can not be applied effectively to the proposed power control. We also present convergence analysis for the proposed algorithm. In addition, applying the proposed algorithm to the proposed power control, we provide numerical examples for the transmit power, the signal-to-interference-plus-noise ratio and so on. Numerical results for the proposed algorithm shall show that as compared with the existing power-control game and its method, all users in the network can enjoy the sufficient signal-to-interference-plus-noise ratio and achieve the required quality of service.      

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

The noncooperative multi-leader-follower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may not exist, due to the nonconvexity in each players problem; the other is that such a nonconvex Nash game is computationally intractable. In order to obtain a viable formulation that is amenable to practical solution, we introduce a class of remedial models for the multi-leader-follower game that can be formulated as generalized Nash games with convexified strategy sets. In turn, a game of the latter kind can be formulated as a quasi-variational inequality for whose solution we develop an iterative penalty method. We establish the convergence of the method, which involves solving a sequence of penalized variational inequalities, under a set of modest assumptions. We also discuss some oligopolistic competition models in electric power markets that lead to multi-leader-follower games.Jong-Shi Pang: The work of this authors research was partially supported by the National Science Foundation under grant CCR-0098013 and ECS-0080577 and by the Office of Naval Research under grant N00014-02-1-0286.Masao Fukushima: The work of this authors research was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports of Japan.     

System of Vector Equilibrium Problems and Its Applications

In this paper, we introduce a system of vector equilibrium problems andprove the existence of a solution. As an application, we derive someexistence results for the system of vector variational inequalities. We alsoestablish some existence results for the system of vector optimizationproblems, which includes the Nash equilibrium problem as a special case.