Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems

The paper introduces and analyzes the convergence of a new iterative algorithm for approximating solutions of equilibrium problems involving strongly pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. The algorithm uses a stepsize sequence which is non-increasing, diminishing, and non-summable. This leads to the main advantage of the algorithm, namely that the construction of solution approximations and the proof of its convergence are done without the prior knowledge of the modulus of strong pseudomonotonicity and Lipschitz-type constants of bifunctions. The strongly convergent theorem is established under suitable assumptions. The paper also discusses the assumptions used in the formulation of the convergent theorem. Several numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with others.

[1]  L. D. Muu,et al.  Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model , 2009, J. Optimization Theory and Applications.

[2]  B. Martinet,et al.  R'egularisation d''in'equations variationnelles par approximations successives , 1970 .

[3]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[4]  P. L. Combettes,et al.  Equilibrium programming in Hilbert spaces , 2005 .

[5]  Giandomenico Mastroeni,et al.  Gap Functions for Equilibrium Problems , 2003, J. Glob. Optim..

[6]  Dang Van Hieu,et al.  Parallel Extragradient-Proximal Methods for Split Equilibrium Problems , 2015, 1511.02474.

[7]  S. I. Lyashko,et al.  A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming , 2016 .

[8]  Pham Ky Anh,et al.  Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings , 2015, Numerical Algorithms.

[9]  R. Cooke.  Real and Complex Analysis , 2011 .

[10]  L. Muu,et al.  Convergence of an adaptive penalty scheme for finding constrained equilibria , 1992 .

[11]  Le Dung Muu,et al.  Dual extragradient algorithms extended to equilibrium problems , 2011, Journal of Global Optimization.

[12]  Sjur Didrik Flåm,et al.  Equilibrium programming using proximal-like algorithms , 1997, Math. Program..

[13]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[14]  W. Oettli,et al.  From optimization and variational inequalities to equilibrium problems , 1994 .

[15]  G. Mastroeni.  On Auxiliary Principle for Equilibrium Problems , 2003 .

[16]  D. Hieu.  Hybrid projection methods for equilibrium problems with non-Lipschitz type bifunctions , 2017 .

[17]  I. Konnov.  Application of the Proximal Point Method to Nonmonotone Equilibrium Problems , 2003 .

[18]  D. Hieu.  An extension of hybrid method without extrapolation step to equilibrium problems , 2015, 1510.08201.

[19]  A. Moudafi.  Proximal point algorithm extended to equilibrium problems , 1999 .

[20]  Pham Ky Anh,et al.  Modified hybrid projection methods for finding common solutions to variational inequality problems , 2017, Comput. Optim. Appl..

[21]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[22]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[23]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[24]  J. Krawczyk,et al.  Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets , 2004, IEEE Transactions on Power Systems.

[25]  T. D. Quoc,et al.  Extragradient algorithms extended to equilibrium problems , 2008 .

[26]  Anatoly Antipin.  The convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence , 1995 .

[27]  Le Dung Muu,et al.  On Existence and Solution Methods for Strongly Pseudomonotone Equilibrium Problems , 2015 .

[28]  D. Hieu.  Halpern subgradient extragradient method extended to equilibrium problems , 2017 .

[29]  I. Konnov.  Equilibrium Models and Variational Inequalities , 2013 .

[30]  Jean Jacques Strodiot,et al.  A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems , 2013, J. Glob. Optim..

[31]  Van Hieu Dang.  An extension of hybrid method without extrapolation step to equilibrium problems , 2016 .

[32]  Boris Goldengorin,et al.  Optimization and Its Applications in Control and Data Sciences , 2016 .

[33]  L. Popov.  A modification of the Arrow-Hurwicz method for search of saddle points , 1980 .

[34]  Susana Scheimberg,et al.  An inexact subgradient algorithm for Equilibrium Problems , 2011 .

[35]  Phan Tu Vuong,et al.  A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces , 2016, J. Glob. Optim..