A Proximal-Projection Method for Finding Zeros of Set-Valued Operators

In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximal-projection method is, essentially, a fixed point procedure and our convergence results are based on new generalizations of Lemma Opial. We show how the proximal-projection method can be applied for solving ill-posed variational inequalities and convex optimization problems with data given or computable by approximations only. The convergence properties of the proximal-projection method we establish also allow us to prove that the proximal point method (with Bregman distances), whose convergence was known to happen for maximal monotone operators, still converges when the operator involved in it is monotone with sequentially weakly closed graph.Comment: 38 page

Inertial Douglas-Rachford splitting for monotone inclusion problems

We propose an inertial Douglas-Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigate its convergence properties. To this end we formulate first the inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating the set of fixed points of a nonexpansive operator, for which we also provide an exhaustive convergence analysis. By using a product space approach we employ these results to the solving of monotone inclusion problems involving linearly composed and parallel-sum type operators and provide in this way iterative schemes where each of the maximally monotone mappings is accessed separately via its resolvent. We consider also the special instance of solving a primal-dual pair of nonsmooth convex optimization problems and illustrate the theoretical results via some numerical experiments in clustering and location theory.Comment: arXiv admin note: text overlap with arXiv:1402.529

Nonexpansive mappings and monotone vector fields in Hadamard manifolds

This paper briefly surveys some recent advances in the investigation of nonexpansive mappings and monotone vector fields, focusing in the extension of basic results of the classical nonlinear functional analysis from Banach spaces to the class of nonpositive sectional curvature Riemannian manifolds called Hadamard manifolds. Within this setting, we first analyze the problem of finding fixed points of nonexpansive mappings. Later on, different classes of monotonicity for set-valued vector fields and the relationship between some of them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. We will discuss about variational inequality and minimization problems in Hadamard manifolds, stressing the fact that these problems can be solved by means of the iterative approaches for monotone vector fields

A distributed primal-dual algorithm for computation of generalized Nash equilibria with shared affine coupling constraints via operator splitting methods

In this paper, we propose a distributed primal-dual algorithm for computation of a generalized Nash equilibrium (GNE) in noncooperative games over network systems. In the considered game, not only each player's local objective function depends on other players' decisions, but also the feasible decision sets of all the players are coupled together with a globally shared affine inequality constraint. Adopting the variational GNE, that is the solution of a variational inequality, as a refinement of GNE, we introduce a primal-dual algorithm that players can use to seek it in a distributed manner. Each player only needs to know its local objective function, local feasible set, and a local block of the affine constraint. Meanwhile, each player only needs to observe the decisions on which its local objective function explicitly depends through the interference graph and share information related to multipliers with its neighbors through a multiplier graph. Through a primal-dual analysis and an augmentation of variables, we reformulate the problem as finding the zeros of a sum of monotone operators. Our distributed primal-dual algorithm is based on forward-backward operator splitting methods. We prove its convergence to the variational GNE for fixed step-sizes under some mild assumptions. Then a distributed algorithm with inertia is also introduced and analyzed for variational GNE seeking. Finally, numerical simulations for network Cournot competition are given to illustrate the algorithm efficiency and performance.Comment: 21 pages,8 figures, parts are submitted to IEEE CD

A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces

The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are analyzed by employing suitable conditions on the control sequences of parameters and the structural properties of the under lying space. We aim to establish strong and del-convergence results of the proposed iterative algorithm and compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family of total asymptotically nonexpansive mappings in Hadamard spaces. Our results can be viewed as an extension and generalization of various corresponding results established in the current literature

Generalized Forward-Backward Splitting with Penalization for Monotone Inclusion Problems

We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximal monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided the condition corresponded to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a large-scale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems

Forward-Backward splitting methods for accretive operators in Banach spaces

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem