A NEW SOLUTION METHOD FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

Abstract We propose a new method for solving an equilibrium problem where the bifunction is pseudomonotone with respect to its solution set. This method can be considered as an extension of the one introduced by Solodov and Svaiter in [28] from variational inequality to equilibrium. An application to Nash-Cournot equilibrium models of electricity markets is discussed and its computational results are reported. Bài báo đề xuất một phương pháp mới giải bài toán cân bằng với song hàm cân bằng là giả đơn điệu theo tập nghiệm của nó. Phương pháp này là một sự mở rộng của phương pháp Solodov và Svaiter (xem Index terms Pseudomonotone equilibria, Ky Fan inequality, auxiliary subproblem principle, projection method, Armijo linesearch, Nash-Cournot equilibrium model

Solving non-monotone equilibrium problems via a DIRECT-type approach

A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of the \texttt{DIRECT} algorithm, which exploits local bounds of the Lipschitz constant of the objective function, with local minimizations. Unlike most existing solution methods for EPs, no monotonicity-type condition is assumed in this paper. Preliminary numerical results on several classes of EPs show the effectiveness of the approach.Comment: Technical Report of Department of Computer Science, University of Pisa, Ital

A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.This research work was financially supported by Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19

Existence and solution methods for equilibria

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 algorithmic 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

Lagrangeano aumentado exponencial aplicado ao problema de equilíbrio

Orientador : Luiz Carlos MatioliTese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa: Curitiba, 11/09/2015Inclui referências : f. 65-67Área de concentração : Programação matemáticaResumo: Neste estudo propomos um Método de Lagrangeano Aumentado Exponencial para resolução do Problema de Equilíbrio Geral. Tal método é uma extensão dos algoritmos apresentados em (NASRI, 2010). Em nossa proposta, substituímos a penalidade clássica de Rockafellar presente nestes algoritmos pela penalidade exponencial. Em seguida, refizemos a teoria geral em torno do novo algoritmo. A teoria de equivalência é construída via ponto proximal, ou seja, a partir da relação de dualidade entre lagrangeano aumentado e ponto proximal, porém o termo de regularização aqui utilizado é uma quase distância de Bregman, diferente do termo quadrático empregado em (NASRI, 2010). Para contornar possíveis problemas de mau condicionamento causados pela presença da penalidade exponencial, procedemos a um ajuste quadrático da mesma. Em seguida, testamos a nova metodologia por meio de experimentos numéricos, considerando inicialmente o método com a penalidade exponencial e, em seguida, com a quadrática ajustada. Para realizar os testes, escolhemos problemas de equilíbrio e problemas de equilíbrio de Nash generalizados (GNEPs), os quais compõem uma classe particular de problemas de equilíbrio. O método puro resolveu problemas pequenos e, para problemas com dimensões maiores, a versão com quadrática ajustada mostrou-se melhor. Na sequência, discutimos os resultados numéricos e apresentamos as considerações finais. Para finalizar, deixamos algumas perspectivas de trabalhos futuros e uma lista de referências, as quais serviram de suporte para essa pesquisa. Palavras-chaves: Palavras-chaves: Problema de Equilíbrio, Lagrangeano Aumentado, Penalidade Exponencial.Abstract: In this study we propose an Exponential Augmented Lagrangian Method to the resolution of the General Equilibrium Problem. Such a method is an extension of the algorithms presented in (NASRI, 2010). In our proposal, we replaced the classic penalty Rockafellar present in these algorithms by the exponential penalty. Then, we redid the general theory surrounding the new algorithm. The equivalence theory is built via proximal point, ie, from the dual relationship between Augmented Lagrangian and the proximal point, but the regularization term used here is almost distance Bregman, different from the quadratic term used in (NASRI, 2010). To work around possible bad conditioning problems caused by the presence of the exponential penalty, we proceeded to a quadratic adjustment of it. Next, we tested the new method by numerical experiments, considering initially the method with the exponential penalty and then with the quadratic adjusted. To perform the tests, we chose equilibrium problems and Generalized Nash equilibrium problems (GNEPs), which compose a particular class of equilibrium problems. The pure method solved little problems and for problems with larger dimensions, the version with the quadratic adjusted proved to be better. Following, we discuss the numerical results and present the final considerations. Finally, we leave some perspectives of future works and a list of references, which served as support to this research. Key-words: Equilibrium Problems, Augmented Lagrangian, Exponential Penalty