Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems, and fixed-point problems in Banach spaces. (English) Zbl 1394.65042
Summary: In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a \(J\)-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone \(J\)-equilibrium problem for a \(J\)-\(\alpha\)-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a \(\phi_*\)-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the
usability of our results.
MSC:
| 65K10 | Numerical optimization and variational techniques |
| 90C25 | Convex programming |
| 47J05 | Equations involving nonlinear operators (general) |
| 47J25 | Iterative procedures involving nonlinear operators |
Keywords:
extragradient method; \(J\)-\(\alpha\)-inverse-strongly monotone operator; \(J\)-equilibrium problem; \(J\)-variational inequality; \(\phi_*\)-Lipschitz-type; line search algorithm; sunny generalized nonexpansive retractionReferences:
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