On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators

In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

A modified extragradient method for inverse-monotone operators in Banach spaces

We introduce an iterative procedure for finding a point in the zero set (a solution to 0 ∈ A(v) and v ∈ C) of an inverse-monotone or inverse strongly-monotone operator A on a nonempty closed convex subset C in a uniformly smooth and uniformly convex Banach space. We establish weak convergence results under suitable assumptions.      

A strongly convergent hybrid proximal method in Banach spaces

This paper is devoted to the study of strong convergence in inexact proximal like methods for finding zeroes of maximal monotone operators in Banach spaces. Convergence properties of proximal point methods in Banach spaces can be summarized as follows: if the operator have zeroes then the sequence of iterates is bounded and all its weak accumulation points are solutions. Whether or not the whole sequence converges weakly to a solution and which is the relation of the weak limit with the initial iterate are key questions. We present a hybrid proximal Bregman projection method, allowing for inexact solutions of the proximal subproblems, that guarantees strong convergence of the sequence to the closest solution, in the sense of the Bregman distance, to the initial iterate.     

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas—Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.This paper is drawn largely from the dissertation research of the first author. The dissertation was performed at M.I.T. under the supervision of the second author, and was supported in part by the Army Research Office under grant number DAAL03-86-K-0171, and by the National Science Foundation under grant number ECS-8519058.     

A new approximate proximal point algorithm for maximal monotone operator

The problem concerned in this paper is the set-valued equation 0 ∈T(z) where T is a maximal monotone operator. For given xk and βk > 0, some existing approximate proximal point algorithms take x~(k+1) = xk such thatwhere {ηk} is a non-negative summable sequence. Instead of xk+1 = xk , the new iterate of the proposing method is given bywhere Ω is the domain of T and PΩ(·) denotes the projection on Ω. The convergence is proved under a significantly relaxed restriction supk>0 ηk<1.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. Received: 10 August 2007     

New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators

The purpose of this article is to prove the strong convergence theorems for hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems for hemi-relatively nonexpansive mappings, a new monotone hybrid iteration algorithm is presented and is used to approximate the fixed point of hemi-relatively nonexpansive mappings. Noting that, the general hybrid iteration algorithm can be used for relatively nonexpansive mappings but it can not be used for hemi-relatively nonexpansive mappings. However, this new monotone hybrid algorithm can be used for hemi-relatively nonexpansive mappings. In addition, a new method of proof has been used in this article. That is, by using this new monotone hybrid algorithm, we firstly claim that, the iterative sequence is a Cauchy sequence. The results of this paper modify and improve the results of Matsushita and Takahashi, and some others.     

Inertial proximal algorithm for difference of two maximal monotone operators

In this note, a new algorithm is presented for finding a zero of difference of two maximal monotone operators T and S, i.e., T — S in finite dimensional real Hilbert space H in which operator S has local boundedness property. This condition is weaker than Moudafi’s condition on operator S in [13]. Moreover, applying some conditions on inertia term in new algorithm, one can improve speed of convergence of sequence.     

The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces

It is known, by Rockafellar (SIAM J Control Optim 14:877–898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi (Optimization 37:239–252, 1996) introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with error sequence is also discussed.     

The hybrid projection algorithm for finding the common fixed points of nonexpansive mappings and the zeroes of maximal monotone operators in Banach spaces

The proposal of this article is to construct a new modified block by using the hybrid projection method and prove the strong convergence theorem for this method, which include the fixed point set of an infinite family of weak relatively nonexpansive mappings and zeroes of a finite family of maximal monotone operators in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property. The results presented in this article improve and generalize some well-known results in the literature.     

A contraction proximal point algorithm with two monotone operators

It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature.     

Accelerated proximal point method for maximally monotone operators

Mathematical Programming - This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach....     

On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Let be a real reflexive Banach space with dual and open and bounded and such that  Let be maximal monotone with and and with and A general and more unified eigenvalue theory is developed for the pair of operators  Further conditions are given for the existence of a pair such that

The ``implicit" eigenvalue problem, with in place of is also considered.  The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators No compactness assumptions have been made in most of the results.  The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators.  Applications to nonlinear partial differential equations are included.

     

Remarks on perturbation theory of maximal monotone and m-accretive operators in Banach spaces

In this paper we get and improve some results in the perturbation theory of maximal monotone and m-accretive operators having compact resolvents in Banach spaces, in which the composition of resolvent and perturbation is assumed compact.     

A generalized contraction proximal point algorithm with two monotone operators

Abstract

In this work, we introduce a generalized contraction proximal point algorithm and use it to approximate common zeros of maximal monotone operators A and B in a real Hilbert space setting. The algorithm is a two step procedure that alternates the resolvents of these operators and uses general assumptions on the parameters involved. For particular cases, these relaxed parameters improve the convergence rate of the algorithm. A strong convergence result associated with the algorithm is proved under mild conditions on the parameters. Our main result improves and extends several results in the literature.