Nondominated Equilibrium Solutions of a Multiobjective Two-Person Nonzero-Sum Game and Corresponding Mathematical Programming Problem

With reference to a multiobjective two-person nonzero-sum game, we define nondominated equilibrium solutions and provide a necessary and sufficient condition for a pair of mixed strategies to be a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. We give a numerical example and demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

Range sets for weak efficiency in multiobjective linear programming and a parametric polytopes intersection problem

ABSTRACT

The aim of this paper is to obtain the range set for a given multiobjective linear programming problem and a weakly efficient solution. The range set is the set of all values of a parameter such that a given weakly efficient solution remains efficient when the objective coefficients vary in a given direction. The problem was originally formulated by Benson in 1985 and left to be solved. We formulate an algorithm for determining the range set, based on some hard optimization problems. Due to toughness of these optimization problems, we propose also lower and upper bound approximation techniques. In the second part, we focus on topological properties of the range set. In particular, we prove that a range set is formed by a finite union of intervals and we propose upper bounds on the number of intervals. Our approach to tackle the range set problem is via the intersection problem of parametric polytopes. Thus, our results have much wider area of applicability since the intersection (and separability) problem of convex polyhedra is important in many fields of optimization.     

Existence of equilibrium for generalized games in choice form and applications

This paper has two central aims: first, to provide simple conditions under which the generalized games in choice form and, consequently, the abstract economies admit equilibrium; second, to study the solvability of several types of systems of vector quasi-equilibrium problems as an application. Our work outlines that there still is much to be gained from using the results concerning the existence of equilibrium of games as tools of research for other optimization problems.     

Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions

In this paper, new classes of generalized (F,α,ρ,d)-type I functions are introduced for differentiable multiobjective programming. Based upon these generalized functions, first, we obtain several sufficient optimality conditions for feasible solution to be an efficient or weak efficient solution. Second, we prove weak and strong duality theorems for mixed type duality.     

A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem

In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7], and some known corresponding results in the literature.     

Global stability of solutions to a free boundary problem of ductal carcinoma in situ

In the paper we present some remarks on the global stability of steady state solutions to a free boundary model studied by Xu (2004) and also prove some new results of global stability of steady state solutions to the model.     

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+auq−bur, subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.     

Goal programming in the context of the assignment problem and a computationally effective solution method

This paper develops the goal programming technique to solve the multiple objective assignment problem. The required model is formulated and an appropriate solution method is presented. The proposed method, which is a decomposition method, exploits the total unimodularity feature of the assignment problem and effectively reduces the computational efforts. Some issues related to the efficiency of a GP solution are stated and some specialized techniques for detecting and restoring efficiency are proposed.     

Strong convergence of projection algorithms for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces

In this article, we introduce two kinds of new hybrid projection algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinitely countable family of relatively quasi-nonexpansive mappings in a Banach space. Our main results improve and extend the result obtained by Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), pp. 2400–2411] and the corresponding results.     

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.