New Perturbed Proximal Point Algorithms for Set-valued Quasi Variational Inclusions

In this paper, by using some new and innovative techniques, some perturbed iterative algorithms for solving generalized set-valued variational inclusions are suggested and analyzed. Since the generalized set-valued variational inclusions include many variational inclusions , variational inequalities and set-valued operator equation studied by others in recent years, the results obtained in this paper continue to hold for them and represent a significant refinement and improvement of the previously known results in this area

Quasi Variational Inclusions Involving Three Operators

In this paper, we consider some new classes of the quasi-variational inclusions involving three monotone operators. Some interesting problems such as variational inclusions involving sum of two monotone operators, difference of two monotone operators, system of absolute value equations, hemivariational inequalities and variational inequalities are the special cases of quasi variational inequalities. It is shown that quasi-variational inclusions are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inclusions and related optimization problems are suggested by using resolvent methods, resolvent equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results

Quasi Variational Inclusions Involving Three Operators

In this paper, we consider some new classes of the quasi-variational inclusions involving three monotone operators. Some interesting problems such as variational inclusions involving sum of two monotone operators, difference of two monotone operators, system of absolute value equations, hemivariational inequalities and variational inequalities are the special cases of quasi variational inequalities. It is shown that quasi-variational inclusions are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inclusions and related optimization problems are suggested by using resolvent methods, resolvent equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results

A first-order stochastic primal-dual algorithm with correction step

We investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I. Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty, 2011} for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi Fej\'er-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models

Parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings

AbstractA new class of parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings is introduced. By applying resolvent operator technique of h-maximal monotone mapping and the property of fixed point set of set-valued contractive mappings, the behavior and sensitivity analysis of the solution set of the parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings are studied. The continuity and Lipschitz continuity of the solution set with respect to the parameter are proved under suitable assumptions. Our approach and results are new and improve, unify and extend previous many known results in this field