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A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems. [ABSTRACT FROM AUTHOR]

We study existence and asymptotic behavior of a bounded solution of the following Tikhonov regularized second-order difference equation (): where, A is a multivalued maximal monotone operator defined on a Hilbert space and are positive real parameters. We first prove under condition existence of unique bounded solution of (). For asymptotic behavior, we use a suitable assumption on cn and to prove strong convergence of un to the element of minimal norm of Some applications are thereafter discussed with respect to minimization and saddle-point problems. Specially, we study the rate of convergence of optimal values in convex minimization and convex-concave problems. We end the paper by concluding remarks and noticing some research perspectives. [ABSTRACT FROM AUTHOR]

In the finite dimensional setting, we investigate the existence of viscosity subsolutions of Hamilton–Jacobi–Bellman equations related to control problems subject to evolution inclusions governed by time-dependent maximal monotone operators with Young measures. [ABSTRACT FROM AUTHOR]

In this article, 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 the literature 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 operator 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-Fejér-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. [ABSTRACT FROM AUTHOR]

In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equationwhereAis a maximal monotone operator in a real Hilbert spaceHand {cn} and {θn} are positive real sequences. We do not assumeA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing nonempty. These theorems provide new approximation results for zeroes of monotone operators, as well as significantly unify and extend previously known results by assuming much weaker conditions on the coefficients {cn} and {θn}. [ABSTRACT FROM AUTHOR]

We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation:whereAis a maximal monotone operator in a real Hilbert spaceH, and {cn} is a positive sequence of real numbers. We do not assume thatA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing nonempty. These theorems provide new approximation results for zeros of monotone operators, as well as unify and extend previously known results in [2, 3, 7, 10, 18, 22, 23, 25] by considering much weaker conditions on the coefficients {cn}. In particular, our new strong ergodic theorem extends the results of [7] and [23, Theorem 3.3] for first-order difference equations, to the case of second order difference equations, and implies also a new strong convergence theorem. [ABSTRACT FROM PUBLISHER]

The purpose of this article is to prove a strong convergence result associated with a generalization of the method of alternating resolvents introduced by the authors in convergence of the method of alternating resolvents [4] under minimal assumptions on the control parameters involved. Thus, this article represents a significant improvement of the article mentioned above. [ABSTRACT FROM PUBLISHER]

We investigate asymptotic behavior of solutions to the following second order difference equation where A is a maximal monotone operator in a real Hilbert space H and {c i } is a positive real sequence. We show weak and strong convergence of solutions to an element of A −1(0), under appropriate assumptions on A and the sequence {c i }. Our results extend and improve previous results by Morosanu [21] and Mitidieri-Morosanu [18] (see also [19, pp. 156–168]) under more general assumptions on c i . [ABSTRACT FROM AUTHOR]

We introduce two inexact proximal-like methods for solving equilibrium problems in reflexive Banach spaces and establish their convergence properties, proving that the sequence generated by each one of them converges to a solution of the equilibrium problem under reasonable assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we prove three strong convergence theorems of modified Mann iteration for relatively asymptotically nonexpansive mappings, relatively nonexpansive mappings, and maximal monotone operators in a Banach space, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Matsushita, Kim, Martinez-Yanex, Xu, and some others. [ABSTRACT FROM AUTHOR]

We give a sufficient condition, weaker than the others known so far, that guarantees that the sum of two maximal monotone operators on a reflexive Banach space is maximal monotone. Then we give a weak constraint qualification assuring the Brézis-Haraux-type approximation of the range of the sum of the subdifferentials of two proper convex lower-semicontinuous functions in nonreflexive Banach spaces, extending and correcting an earlier result due to Riahi. [ABSTRACT FROM AUTHOR]

We generalize a proximal-like method for finding zeroes of maximal monotone operators in Hilbert spaces with quadratic regularization due to Solodov and Svaiter, making it possible the use of other kind of regularizations and extending it to Banach spaces. In particular, we introduce an appropriate error criterium to obtain an inexact proximal iteration based on Bregman functions and construct a hyperplane which strictly separates the current iterate from the solution set. A Bregman projection onto this hyperplane is then used to obtain the next iterate. Boundedness of the sequence and optimality of the weak accumulation points are established under suitable assumptions on the regularizing function, which hold for any power greater than one of the norm of any uniformly smooth and uniformly convex Banach space, without any assumption on the operator other than existence of zeroes. These assumptions let us, also, obtain similar results in Banach spaces for the Hybrid Extragradient-Generalized Proximal Point method, proposed by Solodov and Svaiter for finite dimensional spaces. We then transpose such methods to generate augmented Lagrangian methods for LP-constrained convex optimization problems in Banach spaces, obtaining two alternative procedures which allow for sequences, and optimality of primal and dual weak accumulation points, are then established, assuming only existence of Karush-Kuhn-Tucker pairs. *The work of this author was partially supported by CNPq grant no. 301280/86 and by Marcel Dassault chair, Centro de Modelamiento Matemático, Universidad de Chile. [ABSTRACT FROM AUTHOR]