Descent methods for equilibriumproblems in a Banach space

Chadli, Ouayl; Konnov, Igor; Yao, Jen‐Chih

doi:10.1016/j.camwa.2003.05.011

Cited by 53 publications

(37 citation statements)

References 3 publications

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“…The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems and the equilibrium problems as special cases; see, e.g., [1,3,4,10,22]. Some methods have been proposed to solve the equilibrium problems and the mixed equilibrium problems, see, e.g., [2,5,6,7,10,12,14,15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION AND A MIXED EQUILIBRIUM PROBLEM

Yao¹,

Zhou²,

Liou³

2009

Journal of the Korean Mathematical Society

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Abstract. We introduce two iterative algorithms for finding a common element of the set of fixed points of an asymptotically k-strict pseudocontraction and the set of solutions of a mixed equilibrium problem in a Hilbert space. We obtain some weak and strong convergence theorems by using the proposed iterative algorithms. Our results extend and improve the corresponding results of Tada and Takahashi [16] and Kim and Xu [8,9].

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Section: Introductionmentioning

confidence: 99%

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION AND A MIXED EQUILIBRIUM PROBLEM

Yao¹,

Zhou²,

Liou³

2009

Journal of the Korean Mathematical Society

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“…Given a mapping T: C H, let Θ (x, y) = 〈Tx, y -x〉 for all x, y C. Then, z EP(Θ) if and only if 〈Tz, y -z〉 ≥ 0 for all y C. Numerous problems in physics, optimization, and economics reduce to finding a solution of problem (1.1). Equilibrium problems have been studied extensively [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP(Θ) is nonempty and derived a strong convergence theorem.…”

Section: Introductionmentioning

confidence: 99%

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Ceng

Guu

Yao

2012

Fixed Point Theory Appl

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Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the setFix (S i ) of fixed points of a finite family of nonexpansivein a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element ofFix (S i ) ∩ GMEP , which is the unique solution of a variational inequality.

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“…Theorem 5 and Corollary 1 provide the basic tools to design a solution method in the footsteps of descent algorithms for EPs (see, for instance, [49,12,9]): if the current iterate x k is not a stationary point of the gap function ϕ, a step along the descent direction y(x k ) − x k is taken exploiting some inexact line search. Algorithm (0) Choose β, γ ∈ (0, 1), x 0 ∈ X and set k = 0.…”

Section: Descent Algorithmmentioning

confidence: 99%

“…Though descent type methods based on gap functions have been extensively developed for EPs (see, for instance, [49,44,12,43,45,6,8,9] and Section 3.2 in the survey paper [7]), the analysis of gap functions for QVIs is focused on smoothness properties [19,30,59,18,35] and error bounds [2,33] while no algorithm is developed. A descent method has been developed in [38] for jointly convex GNEPs; anyway, the choice of restricting to the computation of normalized equilibria makes the problem actually fall within the EP (and not the QEP) framework.…”

Section: Introductionmentioning

confidence: 99%

Gap functions for quasi-equilibria

Bigi

Passacantando

2016

J Glob Optim

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An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimates of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm.

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Descent methods for equilibriumproblems in a Banach space

Cited by 53 publications

References 3 publications

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION AND A MIXED EQUILIBRIUM PROBLEM

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION AND A MIXED EQUILIBRIUM PROBLEM

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Gap functions for quasi-equilibria

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