# Proximal and Dynamical Approaches to Equilibrium Problems

Conference paper

## Abstract

The theory of equilibrium problems has emerged as an interesting branch of applied mathematics, permitting the general and unified study of a large number of problems arising in mathematical economics, optimization and operations research. Inspired by numerical methods developed for variational inequalities and motivated by recent advances in this field, we propose several ways (including an auxiliary problem principle, a selection method, as well as a dynamical procedure) to solve the following equilibrium problem:
where

$$(GEP)Find\overline x \in CsuchthatF(\overline x ,x) + \left\langle {G(\overline x ),x - \overline x } \right\rangle \geqslant 0\forall x \in C,$$

*C*is a nonempty convex closed subset of a real Hilbert space*X*,*F*:*C*×*C*→ ℝ is a given bivariate function with*F*(*x*,*x*) = 0 for all*x*∈*C*and*G*:*C*→ ℝ is a continuous mapping. This problem has useful applications in nonlinear analysis, including as special cases optimization problems, variation al inequalities, fixed-point problems and problems of Nash equilibria. Throughtout the paper,*X*is a real Hilbert space, <· , ·> denotes the associated inner product and | · | stands for the corresponding norm. From now on, we assume that the solution set,*S*, of problem (*GEP*) is nonempty. This corresponds to some important situations such as linear programming and semi-coercive minimization problems.## Keywords

Variational Inequality Equilibrium Problem Dynamical Approach Real Hilbert Space Tikhonov Regularization
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [1]Alvarez, F. (1998), On the minimizing property of a second order dissipative system in Hilbert space. Prépublications de l’Université Montpelier IIGoogle Scholar
- [2]Antipin, A. S. and Flam, S. D. (1997), Equilibrium programming using proximallike algorithms. Math.-Programming,
**78**, (1), 29–41.Google Scholar - [3]Attouch H. and Commetti R. (1996), A dynamical approach to convex minimization coupling approximation with the steepest method. J. l of Differential Equations,
**128**, 2, 519–540.CrossRefGoogle Scholar - [4]Attouch H., Moudafi A. and Riahi H. (1993) Quantitative stability analysis for maximal monotone operators and semigroups of contractions. Nonlinear Analysis: Theory, Methods and Appl.,
**21**, 9, 697–723.CrossRefGoogle Scholar - [5]Aubin J.-P. (1991) Viability theory, Birkhauser, Basel.Google Scholar
- [6]Auslender A. (1987) Numerical methods for non differentiable convex optimization. Math. Prog. Study
**30**, 102–126.CrossRefGoogle Scholar - [7]Bianchi M. and Schaible S. (1996) Generalized monotone bifunction and equilibrium problems. J. Optim. Theory Appli.,
**90**, 1, 31–43.CrossRefGoogle Scholar - [8]Blum E. and Oettli W. (1994) From optimization and variational inequalities to equilibrium problems. The Math. Students,
**63**, 123–145.Google Scholar - [9]Brézis H. (1971) Monotonicity to nonlinear partial differential equations. Contribution to nonlinear analysis, Academic Press, New York, 101–156.Google Scholar
- [10]Browder F. E. (1965) Existence of periodic solutions for nonlinear equations of evolution, Proc. N. A. S.,
**53**, 1100–1103.CrossRefGoogle Scholar - [11]Brack R. E. (1975) Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal.,
**18**, 15–26.CrossRefGoogle Scholar - [12]Burachik R. S., Iusem A. N. and Svaiter B. F. (1997) Enlargement of monotone operators with applications to variational inequalities. Set-Valued Analysis,
**5**, 159–180.CrossRefGoogle Scholar - [13]Cohen G. (1978) Optimization by decomposition and coordination: a unified approach. IEEE Transactions on Automatic Control AC-23, 222–232.CrossRefGoogle Scholar
- [14]Cohen G. (1980) Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl.,
**32**, 277–305.CrossRefGoogle Scholar - [15]Cohen G. (1988) Auxiliary problem principle extended to variational inequalities. J. of Optimization Theory and Applications,
**59**, 325–333.CrossRefGoogle Scholar - [16]Commetti R. and San Martin J. (1994) Asymptotical analysis of the exponential penalty trajectory in linear programming. Mathematical Programming,
**67**, 169–187.CrossRefGoogle Scholar - [17]Commetti R. (1995) Asymptotic convergence of the steepest descent method for exponential penalty in linear programming. J. Convex Anal. 2,
**112**, 145–152.Google Scholar - [18]Flam S. D. and Greco G. (1991) Noncooperative games; methods for subgradient projection and proximal point. In: W. Oettli and D. Pallaschke, (Eds) Advances in Optimization, Lambrecht, Lecture Notes in Econom. and Math. Systems,
**382**, 406–419, Springer Verlag, Berlin.Google Scholar - [19]Flam S. D. (1997) Gradient approches to equilibrium. Lecture Notes In Economics and Mathematical Systems,
**452**, Springer-Verlag, Berlin, 49–60.Google Scholar - [20]Halpern B. (1967) Fixed points of nonexpansive maps. Bull. Amer. Math. Soc,
**73**, 957–961.CrossRefGoogle Scholar - [21]Kaplan A. and Tichatschke R. (1994) Stable methods for ill-posed problems. Akademie Verlag, Berlin.Google Scholar
- [22]Lassonde M. (1983) On the use of KKM multifunctions in fixed point theory and related topics. J. Math. Anal. and Appli.,
**97**, 1, 151–201.CrossRefGoogle Scholar - [23]Lehdili N. and Lemaire B. (1998) The barycentric proximal method. To appear in Communications on Applied Nonlinear Analysis.Google Scholar
- [24]Lemaire B. (1991) About the convergence of the proximal method. Advances in Optimization, Lecture Notes in Economics and Mathematical Systems
**382**, Springer-Verlag, 39–51.Google Scholar - [25]Lemaire B. (1997) [atOn the convergence of some iterative methods for convex minimization. Recent Developements in Optimization, Lecture Notes in Economics and Mathematical Systems,
**452**, 154–167.CrossRefGoogle Scholar - [26]Mouallif K., Nguyen V. H. and Strodiot J.-J. (1991) A perturbed parallel decomposition method for a class of nonsmooth convex minimization problems. Siam J. Control Opt.,
**29**, 822–847.CrossRefGoogle Scholar - [27]Moudafi A. and Théra M. (1997) Finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl.,
**94**, 425–448.CrossRefGoogle Scholar - [28]Opial G. B. (1967) Weak convergence of sequence of successive approximations for nonexpansive mapping. Bull. Amer. Math. Soc,
**77**, 591–597.CrossRefGoogle Scholar - [29]Ould Ahmed Salem C. (1998) Approximation de points fixes d’une contraction. Thèse de Doctorat, Université Montpellier IIGoogle Scholar
- [30]Passty G. B. (1979) [atErgodic convergence to a zero of the sum of monotone operators. J. of Math. Anal. and App.,
**72**, 383–390.CrossRefGoogle Scholar - [31]Rockafellar R. T. (1976) Monotone operators and the proximal algorithm. Siam J. Control. Opt.,
**14**, (5), 877–898.CrossRefGoogle Scholar - [32]Tikhonov A.N. and Arsenine V. Ya (1974) Methods for solving ill-posed problems. Nauka, Moscow.Google Scholar
- [33]Tossings P. (1994) The perturbed Tikhonov’s algorithm and some of its applications.
*M*^{2}*AN*,**28**, 2, 189–221.Google Scholar - [34]Zhu D. L. and Marcotte P. (1996) Co-coercivity and the role in the convergence of iterative schemes fo solving variational inequalities. Siam Journal on Optimization,
**6**, 3, 714–726.CrossRefGoogle Scholar

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