Convergence of over-relaxed contraction-proximal point algorithm in Hilbert spaces

The proximal point algorithm has many applications for convex optimization with several versions of the proximal point algorithm including generalized proximal point algorithms and accelerated proximal point algorithms that have been studied in the literature. In this paper, we propose accelerated versions of generalized proximal point algorithms to find a zero of a maximal monotone operator in Hilbert spaces. We give both weak and linear convergence results of our proposed algorithms under standard conditions. Numerical applications of our results to image recovery are given and numerical implementations show that our algorithms are effective and superior to other related accelerated proximal point algorithms in the literature.

"An algorithm for approximating zeros of m-accretive operators is constructed in a uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a zero of an m-accretive operator. In the case of a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar for approximating zeros of maximal monotone operators. Furthermore, the convergence theorem proved is applied to approximate a solution of a Hammerstein integral equation. Finally, numerical experiments are presented to illustrate the convergence of our algorithm."

Metric subregularity and the proximal point method

Based on the proximal point algorithm, which is a widely used tool for solving a variety of convex optimization problems, there are many algorithms for finding zeros of maximally monotone operators. The algorithm works by applying successively so-called "resolvent" mappings with errors associated to the original object, and is weakly convergent in Hilbert space. In order to acquiring the strong convergence of the algorithm, in this paper, we construct a hybrid Halpern type proximal point algorithm with errors for approximating the zero of a maximal monotone operator, which is a combination of modified proximal point algorithm raised by Yao and Noor and Halpern inexact proximal point algorithm raised by Zhang, respectively. Then, we prove the strong convergence of our algorithm with weaker assumptions in Hilbert space. Finally, we present a numerical example to show the convergence and the convergence speed, which is not affected but accelerated by the projection in the algorithm. Our work improved and generalized some known results.

This paper is mainly devoted to the study and applications of Hölder metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric $q$-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces.

Abstract. Based on the A maximal (m) relaxed monotonicity frame-works, the approximation solvability of a general class of variational in-clusion problems using the relaxed proximal point algorithm is explored,while generalizing most of the investigations, especially of Xu (2002) onstrong convergence of modi ed version of the relaxed proximal point al-gorithm, Eckstein and Bertsekas (1992) on weak convergence using therelaxed proximal point algorithm to the context of the Douglas-Rachfordsplitting method, and Rockafellar (1976) on weak as well as strong con-vergence results on proximal point algorithms in real Hilbert space set-tings. Furthermore, the main result has been applied to the context of theH maximal monotonicity frameworks for solving a general class of vari-ational inclusion problems. It seems the obtained results can be used togeneralize the Yosida approximation that, in turn, can be applied to rst-order evolution inclusions, and can also be applied to Douglas-Rachfordsplitting methods for nding the zero of the sum of two A maximal (m)-relaxed monotone mappings.

We construct a sequence of proximal iterates that converges strongly (under minimal assumptions) to a common zero of two maximal monotone operators in a Hilbert space. The algorithm introduced in this paper puts together several proximal point algorithms under one frame work. Therefore, the results presented here generalize and improve many results related to the proximal point algorithm which were announced recently in the literature.

This paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a Hilbert space which unifies three previously studied strategies: relaxation, inertial type extrapolation and projection step. The first two strategies are intended to speed up the convergence of the standard proximal point algorithm, while the third permits one to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters.

A strongly convergent hybrid proximal method in Banach spaces

In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator's value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem's solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Brezis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Guler's classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an ∈-approximation solution of the problem above.

It is known, by Rockafellar [ SIAM J. Control Optim. , 14 (1976), 877-898], 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(1996), 239-252] 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 non-summable error sequence is also discussed.

Weak and strong convergence for some generalizedproximal point algorithms areproved. These algorithms include the Eckstein and Bertsekasgeneralized proximal point algorithm, a contraction-proximalpoint algorithm, and inexact proximal point algorithms.Convergence rate is also considered.

In this paper, we prove a weak convergence theorem of Ishikawa's\ntype for $m$-generalized hybrid mappings in a Hilbert space. Further,\nby using a new modification of Ishikawa iteration, we\nprove a strong convergence theorem for $m$-generalized hybrid\nmappings in a Hilbert space.

This note is a reaction to the recent paper by Rouhani and Moradi (J Optim Theory Appl 172:222–235, 2017), where a proximal point algorithm proposed by Boikanyo and Morosanu (Optim Lett 7:415–420, 2013) is discussed. Noticing the inappropriate formulation of that algorithm, we propose a more general algorithm for approximating zeros of a maximal monotone operator on a Hilbert space. Besides the main result on the strong convergence of the sequences generated by this new algorithm, we discuss some particular cases, including the approximation of minimizers of convex functionals and present two examples to illustrate the applicability of the algorithm. The note clarifies and extends both the papers quoted above.

Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces