Existence and solution methods for equilibria (original) (raw)

European Journal of Operational Research

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algo- rithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them.

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On The Search for Solutions for Equilibrium Problems and Fixed Point Problems

Journal of Insurance and Financial Management, 2023

Fixed point theory and fixed point algorithms are significant in mathematics, and so are equilibrium theory and algorithms. Both fixed point and equilibrium theories have also important applications in economics. Equilibrium problems and the search for equilibrium points are significant as such, but interestingly an equilibrium algorithm can also be used to solving neighbouring problems like e.g., fixed point problems. Therefore, we present in this literature survey equilibrium problem types and for each type we find from the literature algorithms which solve the problems of that type and at the same time the corresponding fixed point problems. In the beginning we present some famous fixed point theorems.

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Generalized Equilibrium Problems

Mathematics

If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f, another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition (f(x,x)=0, for all x∈X) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems.

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Existence results for quasi-equilibrium problems

2014

Recently in Castellani-Guili (J. Optim. Th. Appl., 147 (2010), 157-168), it has been showed that the proof of the existence result for quasimonotone Stampacchia variational inequalities developed in Aussel-Hadjisavvas (J. Optim. Th. Appl., 121 (2004), 445-450) can be adapted to the case of equilibrium problem. This proof was based on KKM techniques. In this paper we define and study the so-called quasi-equilibrium problem, that is an equilibrium problem with a constraint set depending of the current point. Our main contribution consists of an existence result combining fixed point techniques with stability analysis of perturbed equilibrium problems.

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