262 research outputs found
A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces
We present an iterative method for fixed point
problems, generalized mixed equilibrium problems, and
variational inequality problems. Our method is based
on the so-called viscosity hybrid steepest descent
method. Using this method, we can find the common
element of the set of fixed points of a nonexpansive
mapping, the set of solutions of generalized mixed
equilibrium problems, and the set of solutions of
variational inequality problems for a relaxed
cocoercive mapping in a real Hilbert space. Then, we
prove the strong convergence of the proposed iterative
scheme to the unique solution of variational
inequality. The results presented in this paper
generalize and extend some well-known strong
convergence theorems in the literature
Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator
In a Hilbert framework, we introduce continuous and discrete dynamical
systems which aim at solving inclusions governed by structured monotone
operators , where is the subdifferential of a
convex lower semicontinuous function , and is a monotone cocoercive
operator. We first consider the extension to this setting of the regularized
Newton dynamic with two potentials. Then, we revisit some related dynamical
systems, namely the semigroup of contractions generated by , and the
continuous gradient projection dynamic. By a Lyapunov analysis, we show the
convergence properties of the orbits of these systems.
The time discretization of these dynamics gives various forward-backward
splitting methods (some new) for solving structured monotone inclusions
involving non-potential terms. The convergence of these algorithms is obtained
under classical step size limitation. Perspectives are given in the field of
numerical splitting methods for optimization, and multi-criteria decision
processes.Comment: 25 page
Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems
We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method
Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems
We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others
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