Mixed equilibrium problems and optimization problems

In this paper, we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings. Furthermore, we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions. We also discuss some special cases. Results obtained in this paper improve the previously known results in this area.     

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.     

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.

     

Mixed quasi complementarity problems in topological vector spaces

In this paper, we introduce and consider a new class of complementarity problems, which is called the mixed quasi complementarity problems in a topological vector space. We show that the mixed quasi complementarity problems are equivalent to the mixed quasi variational inequalities. Using the KKM mapping theorem, we study the existence of a solution of the mixed quasi variational inequalities and mixed quasi complementarity problems. Several special cases are also discussed. Results obtained in this paper can be viewed as extension and generalization of the previously known results.     

Likelihood robust optimization for data-driven problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.     

On equilibrium points in bimatrix games

We discuss sensitivity of equilibrium points in bimatrix games depending on small variances (perturbations) of data. Applying implicit function theorem to a linear complementarity problem which is equivalent to the bimatrix game, we investigate sensitivity of equilibrium points with respect to the perturbation of parameters in the game. Namely, we provide the calculation of equilibrium points derivatives with respect to the parameters.     

Mixed quasi nonconvex equilibrium problems

In this paper, we introduce and study a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for solving nonconvex equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier known results for solving equilibrium problems and variational inequalities involving the convex sets.     

Duality for constrained robust sum optimization problems

Mathematical Programming - Given an infinite family of extended real-valued functions $$f_{i}$$ , $$i\in I,$$ and a family $${\mathcal {H}}$$ of nonempty finite subsets of I,  the...     

Uniqueness of equilibrium points in bimatrix games

Bimatrix games are constructed having a given pair (x, y) as the unique equilibrium point within the class of all mixed strategy pairs whose nonzero components are the same as (resp., among) those of (x, y). In each case, necessary and sufficient conditions on (x, y) for the existence of such a game are obtained. All games having the first property are constructed. The work extends and complements recent (separate) works ofMillham [1972],Raghavan [1970] and the author. The methods and results are valid in the context of any ordered field.     

Quadratic optimization problems in robust beamforming

This paper presents a class of constrained optimization problems whereby a quadratic cost function is to be minimized with respect to a weight vector subject to an inequality quadratic constraint on the weight vector. This class of constrained optimization problems arises as a result of a motivation for designing robust antenna array processors in the field of adaptive array processing. The constrained optimization problem is first solved by using the primal-dual method. Numerical techniques are presented to reduce the computational complexity of determining the optimal Lagrange multiplier and hence the optimal weight vector. Subsequently, a set of linear constraints or at most linear plus norm constraints are developed for approximating the performance achievable with the quadratic constraint. The use of linear constraints is very attractive, since they reduce the computational burden required to determine the optimal weight vector.     

Integral global optimization method for solution of nonlinear complementarity problems

The mapping in a nonlinear complementarity problem may be discontinuous. The integral global optimization algorithm is proposed to solve a nonlinear complementarity problem with a robust piecewise continuous mapping. Numerical examples are given to illustrate the effectiveness of the algorithm.     

Strengthened equilibrium for game problems with side interest of participants

A strengthened equilibrium for conflict problems with partially overlapping game sets of participants is proposed, which is of substantial help in determining fair division in cooperative games and makes it possible to refine the hierarchy of all known equilibria.     

Exact relaxations for parametric robust linear optimization problems

We first show that the closedness of the characteristic cone of the constraint system of a parametric robust linear optimization problem is a necessary and sufficient condition for each robust linear program with the finite optimal value to admit exact semidefinite linear programming relaxations. We then provide the weakest regularity condition that guarantees exact second-order cone programming relaxations for parametric robust linear programs.     

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.     

Two-person game with nash equilibrium in optimal control problems

A two-person game with a Nash equilibrium is formulated for optimal control problems with a free right end and a linear differential system. The game is reduced to the calculation of a fixed point of an extremal mapping, which in turn is reduced to a variational inequality with linear constraints generated by systems of linear differential controllable processes. An extra-gradient iterative method is proposed for calculating the Nash equilibrium of the dynamic game. The convergence of the method is proved.     

Continuity for vector optimization problems with equilibrium constraints

The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of the marginal map and the approximate value in VOPEC (e) for vector-valued mapping.     

Unconstrained optimization reformulations of equilibrium problems

We generalize the D-gap function developed in the literature for variational inequalities to a general equilibrium problem （EP）. Through the D-gap function, the equilibrium problem is cast as an unconstrained minimization problem. We give conditions under which any stationary point of the D-gap function is a solution of EP and conditions under which it provides a global error bound for EP. Finally, these results are applied to box-constrained EP and then weaker conditions are established to obtain the desired results for box-constrained EP.