An inertial iterative scheme for solving variational inclusion with application to Nash-Cournot equilibrium and image restoration problems

Abstract

Variational inclusion is an important general problem consisting of many useful problems like variational inequality, minimization problem and nonlinear monotone equations. In this article, a new scheme for solving variational inclusion problem is proposed and the scheme uses inertial and relaxation techniques. Moreover, the scheme is self adaptive, that is, the stepsize does not depend on the factorial constants of the underlying operator, instead it can be computed using a simple updating rule. Weak convergence analysis of the iterates generated by the new scheme is presented under mild conditions. In addition, schemes for solving variational inequality problem and split feasibility problem are derived from the proposed scheme and applied in solving Nash-Cournot equilibrium problem and image restoration. Experiments to illustrate the implementation and potential applicability of the proposed schemes in comparison with some existing schemes in the literature are presented.

Convergence Analysis of Two Parallel Methods for Common Variational Inclusion Problems Involving Demicontractive Mappings

“…Tis method also solved the problems of image deblurring and image recovery. Some recent results for the VIP and related problems are stated in the studies of [14][15][16][17][18][19][20][21][22][23][24][25]. In order to solve the VIP when both the operators are multivalued maximal monotone mappings, two of the most often used splitting algorithms include the Peaceman-Rachford splitting algorithm [26] and the Douglas-Rachford splitting algorithm [27].…”

Accelerated derivative‐free method for nonlinear monotone equations with an application

“…Many authors have shown numerically, that an algorithm with inertial extrapolation step (accelerated version) converges faster than the unaccelerated version (see, References 37‐40). In the literature, several basic algorithms for solving different types of problems have been incorporated with the inertial extrapolation step, resulting to the fast convergence of the sequence generated by the various algorithms (see, References 41‐46). Motivated by the efficiency of the inertial extrapolation in various algorithms, the following question naturally arises:…”