A proximal point method for a class of monotone equilibrium problems with linear constraints

It is well known that the equilibrium problems which often arise in engineering, economics and management applications, provide a unified framework for variational inequality, complementarity problem, optimization problem, saddle point problem and fixed point problem. In this paper, a proximal point method is proposed for solving a class of monotone equilibrium problems with linear constraints (MEP). The updates of all variables of the proximal point method are given in closed form. An auxiliary equilibrium problem is introduced for MEP via its saddle point problem. Further, we present some characterizations for solution of the auxiliary equilibrium problem and fixed point of corresponding resolvent operator. Thirdly, a proximal point method for MEP is suggested by fixed point technique. The asymptotic behavior of the proposed algorithm is established under some mild assumptions. Finally, some numerical examples are reported to show the feasibility of the proposed algorithm.

[1]  Xiaoming Yuan,et al.  A proximal point algorithm revisit on the alternating direction method of multipliers , 2013 .

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

[3]  Yaping Fang,et al.  A characterization of nonemptiness and boundedness of the solution sets for equilibrium problems , 2013 .

[4]  Bingsheng He,et al.  The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent , 2014, Mathematical Programming.

[5]  Wataru Takahashi,et al.  STRONG AND WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND RELATIVELY NONEXPANSIVE MAPPINGS IN BANACH SPACES , 2009 .

[6]  Massimo Pappalardo,et al.  Existence and solution methods for equilibria , 2013, Eur. J. Oper. Res..

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

[8]  Y. Cho,et al.  THE EXISTENCE OF SOLUTIONS AND WELL-POSEDNESS FOR BILEVEL MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES , 2013 .

[9]  Xiaoming Yuan,et al.  A Note on the Alternating Direction Method of Multipliers , 2012, J. Optim. Theory Appl..

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

[11]  S. J. Li,et al.  Existence of solutions for generalized vector quasi-equilibrium problems , 2010, Optim. Lett..

[12]  F. Giannessi.  Vector Variational Inequalities and Vector Equilibria , 2000 .

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

[14]  Alfredo N. Iusem,et al.  On certain conditions for the existence of solutions of equilibrium problems , 2008, Math. Program..

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

[16]  Boris S. Mordukhovich,et al.  Hybrid proximal methods for equilibrium problems , 2012, Optim. Lett..

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

[18]  B. Logan,et al.  Signal recovery and the large sieve , 1992 .

[19]  Michael Elad,et al.  A generalized uncertainty principle and sparse representation in pairs of bases , 2002, IEEE Trans. Inf. Theory.

[20]  Bingsheng He,et al.  A customized proximal point algorithm for convex minimization with linear constraints , 2013, Comput. Optim. Appl..

[21]  A. Moudafi.  On Finite and Strong Convergence of a Proximal Method for Equilibrium Problems , 2007 .

[22]  Alfredo N. Iusem,et al.  Inexact Proximal Point Methods for Equilibrium Problems in Banach Spaces , 2007 .