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In this research paper, we propose a new iterative algorithm for finding a common solution to fixed point problems of demicontractive mapping and variational inequality problems which involves monotone and Lipschitz continuous operators in the framework of real Hilbert spaces. We incorporate a viscosity iterative technique, using subgradient extragradient method, we prove under standard assumptions that the iterative sequence generated from our algorithm strongly converges to the solution set, assuming the solution set is consistent. Furthermore, we adopt a self-adaptive stepsize that is being generated at each iteration, which is independent of the Lipschitz constant of the singled-valued operator. Our result is an improvement and an extension of many results in this direction. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new iteration method based on the hybrid method in mathematical programming, extragradient method, and Mann's method for finding a common element of the solution set of equilibrium problems, the solution set of variational inequality problems for a monotone, Lipschitz continuous mapping and the set of fixed points for a nonexpansive semigroup in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by this process. The results in this paper generalize and extend some well-known strong convergence theorems in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new extragradient method with regularization for finding a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in infinitedimensional Hilbert spaces, combining the regularization method and the technique of averaged operator, we prove the sequences generated by the proposed algorithm converge weakly to an element of Fix(S) ∩ Γ under mild conditions. [ABSTRACT FROM AUTHOR]

Using the concept of Bregman divergence, we propose a new subgradient extragradient method for approximating a common solution of pseudo-monotone and Lipschitz continuous variational inequalities and fixed point problem in real Hilbert spaces. The algorithm uses a new self-adjustment rule for selecting the stepsize in each iteration and also, we prove a strong convergence result for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. Finally, we provide some numerical examples to illustrate the performance and accuracy of our algorithm in finite and infinite dimensional spaces. [ABSTRACT FROM AUTHOR]

We consider a mini-batch stochastic Bregman proximal gradient method and a mini-batch stochastic Bregman proximal extragradient method for stochastic convex composite optimization problems. A simplified and unified convergence analysis framework is proposed to obtain almost sure convergence properties and expected convergence rates of the mini-batch stochastic Bregman proximal gradient method and its variants. This framework can also be used to analyze the convergence of the mini-batch stochastic Bregman proximal extragradient method, which has seldom been discussed in the literature. We point out that the standard uniformly bounded variance assumption and the usual Lipschitz gradient continuity assumption are not required in the analysis. [ABSTRACT FROM AUTHOR]

We consider the extragradient method to minimize the sum of two functions, the first one being smooth and the second being convex. Under the Kurdyka-Łojasiewicz assumption, we prove that the sequence produced by the extragradient method converges to a critical point of the problem and has finite length. The analysis is extended to the case when both functions are convex. We provide, in this case, a sublinear convergence rate, as for gradient-based methods. Furthermore, we show that the recent small-prox complexity result can be applied to this method. Considering the extragradient method is an occasion to describe an exact line search scheme for proximal decomposition methods. We provide details for the implementation of this scheme for the one-norm regularized least squares problem and demonstrate numerical results which suggest that combining nonaccelerated methods with exact line search can be a competitive choice. [ABSTRACT FROM AUTHOR]