A Hybrid Proximal Point Algorithm And A Hundle Method For Solving Nonlinear Variational Inclusions

In this paper, we consider the existence of solutions to a class of variational inclusions and the way to solve this class of problems.In the first part, we present a hybrid proximal point algorithm for solving a class of nonlinear variational inclusions by using the techiniques of resolvent operator. During this process, we generalize the concept of resolvent operator, introduce the notion of ( A,η)-maximal accretive, which is a generalization of the A -maximal monotonicity mapping in Hilbert spaces. The iterative points generated by the proposed algorithm can approximate the solution of the variational inclusions. The convergence analysis for this algorithm is also explored. Furthermore, at the end of this part, we also use the hybrid proximal point algorithm to solve a similar generalized nonlinear implicit quasi-variational inclusion problem.In the second part, we ues the nonsmooth bundle method to solve the generalized variational inclusion which is a case of the problem that consists of finding a zero of the sum of two operators on a real Hilbert space. Bundle method is considered as one of the most important and effective methods for solving nonsmooth optimization problems. We will show how to build, step by step, suitable piecewise linear functionsθi≤φto approximate the nonsmooth function by means of a bundle strategy and how to adapt the stopping criterion. we study the convergence of the suggested algorithm by separating the case where the stepsizes go to zero from the case where they are bounded away from zero. In the first situation, we need the assumptions that the operator is paramonotone and multivalued. In the second case, the convergence needs a stronger assumption: F is single-valued and satisfies a Dunn property.