Abstract
In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method and the hybrid (outer approximation) method. In this algorithm, only one optimization program is solved at each iteration without any extra-step dealing with the feasible set like as in the hybrid extragradient method and the hybrid Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in the proposed algorithm. The strongly convergent theorem is established and several numerical experiments are implemented to illustrate the convergence of the algorithm and compare it with others.
Highlights
The equilibrium problem (EP) [2] which was considered as the Ky Fan inequality [15] is very general in the sense that it includes, as special cases, many mathematical models such as: variational inequalities, fixed point problems, optimization problems, Nash equilirium point problems, complementarity problems, see [2, 20] and the references therein
In 2014, in order to avoid the condition of the Lipschitz-type continuity the bifunction f, Dinh et al [9] replaced the second optimization problem in the extragradient method by the Armijo linesearch technique and obtained the following hybrid algorithm
The paper proposes a novel algorithm for solving EPs for a class of pseudomonotone and Lipschitz-type continuous bifunctions
Summary
The equilibrium problem (EP) [2] which was considered as the Ky Fan inequality [15] is very general in the sense that it includes, as special cases, many mathematical models such as: variational inequalities, fixed point problems, optimization problems, Nash equilirium point problems, complementarity problems, see [2, 20] and the references therein. The convergence of the extragradient method was proved under the weaker assumption that operators are only monotone (even, pseudomonotone) and L - Lipschitz continuous. Korpelevich’s extragradient method has been naturally extended to EPs for monotone (more general, pseudomonotone) and Lipschitz-type continuous bifunctions and widely studied both theoretically and algorithmically [9, 13, 26, 27, 28, 33, 36].
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