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.     

Mixed complementarity problems for robust optimization equilibrium in bimatrix game

In this paper, we investigate the bimatrix game using the robust optimization approach, in which each player may neither exactly estimate his opponent’s strategies nor evaluate his own cost matrix accurately while he may estimate a bounded uncertain set. We obtain computationally tractable robust formulations which turn to be linear programming problems and then solving a robust optimization equilibrium can be converted to solving a mixed complementarity problem under the l ₁ ∩ l _∞-norm. Some numerical results are presented to illustrate the behavior of the robust optimization equilibrium.     

Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets

We consider a class of generalized Nash equilibrium problems with quadratic cost functions and common linear constraints for all players. Further we focus on the case where every player has a single strategy variable within a bounded set. For this problem class we present an algorithm that is able to compute all solutions and that terminates finitely. Our method is based on a representation of the solution set as a finite union of polyhedral sets using sign conditions for the derivatives of the cost and constraint functions. The effectiveness of the algorithm is shown in various examples from literature.     

CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.     

Generalized Nash equilibrium problems

The Generalized Nash equilibrium problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field. The work of Christain Kanzow has been partially supported by the program “Identification, Optimization and Control with Applications in Modern Technologies” of the Elite Network of Bavaria, Germany.     

Cone complementarity problems with finite solution sets

We introduce the notion of a complementary cone and a nondegenerate linear transformation and characterize the finiteness of the solution set of a linear complementarity problem over a closed convex cone in a finite dimensional real inner product space. In addition to the above, other geometrical properties of complementary cones have been explored.     

Mixed uncertainty sets for robust combinatorial optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a “true” model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-of-sample performance, the combination of multiple sets can give better results than each set on its own.

     

On robust maximum flow with polyhedral uncertainty sets

We address the problem of determining a robust maximum flow value in a network with uncertain link capacities taken in a polyhedral uncertainty set. Besides a few polynomial cases, we focus on the case where the uncertainty set is taken to be the solution set of an associated (continuous) knapsack problem. This class of problems is shown to be polynomially solvable for planar graphs, but NP-hard for graphs without special structure. The latter result provides evidence of the fact that the problem investigated here has a structure fundamentally different from the robust network flow models proposed in various other published works.     

Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented. Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

Generalized Nash equilibrium problems for lower semi-continuous strategy maps

We consider a class of generalized Ky Fan inequalities (quasi-variational inequalities) in which the involved multi-valued mapping is lower semi-continuous. We present a relaxed version of the generalized Nash equilibrium problem involving strategy maps, which are only lower semi-continuous. This relaxed version may have no exact Nash equilibrium, but we prove that it has an ε-Nash equilibrium for every ε > 0. Easy examples of such problems show no existence of exact solutions, but existence of ε-solutions for every ε > 0. We give positive answers to two questions (in the compact case) raised in a recent paper of Cubiotti and Yao.     

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.     

Generalized Nash equilibrium problems and Newton methods

The generalized Nash equilibrium problem, where the feasible sets of the players may depend on the other players’ strategies, is emerging as an important modeling tool. However, its use is limited by its great analytical complexity. We consider several Newton methods, analyze their features and compare their range of applicability. We illustrate in detail the results obtained by applying them to a model for internet switching.     

SDP reformulation for robust optimization problems based on nonconvex QP duality

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with “single” (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand’s Lorentz positivity.     

Perturbation approach to generalized Nash equilibrium problems with shared constraints

This paper aims to consider an application of conjugate duality in convex optimization to the generalized Nash equilibrium problems with shared constraints and nonsmooth cost functions. Sufficient optimality conditions for the problems regarding to players are rewritten as inclusion problems and the maximal monotonicity of set-valued mappings generated by the subdifferentials of functions from data of GNEP is proved. Moreover, some assertions dealing with solutions to GNEP are obtained and applications of splitting algorithms to the oligopolistic market equilibrium models are presented.     

Partial penalization for the solution of generalized Nash equilibrium problems

In this paper we reformulate the generalized Nash equilibrium problem (GNEP) as a nonsmooth Nash equilibrium problem by means of a partial penalization of the difficult coupling constraints. We then propose a suitable method for the solution of the penalized problem and we study classes of GNEPs for which the penalty approach is guaranteed to converge to a solution. In particular, we are able to prove convergence for an interesting class of GNEPs for which convergence results were previously unknown.     

An improved two-step method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is a noncooperative game in which the strategy set of each player, as well as his payoff function, depend on the rival players strategies. As a generalization of the standard Nash equilibrium problem (NEP), the GNEP has recently drawn much attention due to its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. In this paper, we propose an improved two-step (a prediction step and a correction step) method for solving the quasi-variational inequality (QVI) formulation of the GNEP. Per iteration, we first do a projection onto the feasible set defined by the current iterate (prediction) to get a trial point; then, we perform another projection step (correction) to obtain the new iterate. Under certain assumptions, we prove the global convergence of the new algorithm. We also present some numerical results to illustrate the ability of our method, which indicate that our method outperforms the most recent projection-like methods of Zhang et al. (2010).     

A half-space projection method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method.     

On solving generalized Nash equilibrium problems via optimization

This paper deals with the generalized Nash equilibrium problem (GNEP), i.e. a noncooperative game in which the strategy set of each player, as well as his payoff function, depends on the strategies of all players. We consider an equivalent optimization reformulation of GNEP using a regularized Nikaido–Isoda function so that solutions of GNEP coincide with global minima of the optimization problem. We then propose a derivative-free descent type method with inexact line search to solve the equivalent optimization problem and we prove that our algorithm is globally convergent. The convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of GNEP. Finally, we present the performance of our algorithm on some examples.