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In this paper, we study the convergence of a proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. We extent the method proposed by Moudafi [Proximal point algorithm...

This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates.

The purpose of this paper is to use a modified proximal point algorithm for solving variational inclusion problem in real Hilbert spaces. It is proven that the sequence generated by the proposed iterative algorithm converges strongly to the common solution of the convex minimization and variational inclusion problems.

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.

We study the split common null point problem in two Hilbert spaces. In order to solve this problem, we propose a new algorithm and establish a strong convergence theorem for it. Our scheme combines the hybrid projection method with the proximal point algorithm.

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings, and it is well known that the classic Krasnoselskii–Mann iteration is weakly convergent in Hilbert spaces. The weak convergence is also known even in Banach spaces. Recently, Kanzow and Shehu proposed a generalized Krasnoselskii–Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces. In this paper, we show that the generalized Krasnoselskii–Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces. As applications, we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.

We study the recently introduced generalized split common null point problem in Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.

This review proposes a proximal algorithm for difference of two monotone operators in finite dimensional real Hilbert space. Our route begins with reviewing some properties of DC (difference of convex functions) programming and DCA (DC algorithms). Next, we recall some main results about a proximal point algorithm for DC programming.

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.Mathematics Subject Classification (2010): 47J25, 47H05, 47H09.Keywords: Maximal monotone operator, contraction proximal point algorithm, alternating method, nonexpansive map, resolvent operator

We modify a hybrid method and a proximal point algorithm to iteratively find a zero point of the sum of two monotone operators and fixed point of nonspreading multivalued mappings in a Hilbert space by using the technique of forward-backward splitting method. The strong convergence theorem is established and the illustrative numerical example is presented on this work. The results of this paper extend and improve some well-known results in the literature.

We consider the convergence rate of the proximal point algorithm (PPA) for finding a minimizer of proper lower semicontinuous convex functions. In the Hilbert space setting, Guler showed that the big-O rate of the PPA can be improved to little-o when the sequence generated by the algorithm converges strongly to a minimizer. In this paper, we establish little-o rate of the PPA in Banach spaces without requiring this assumption. Then we apply the result to give new results on the convergence rate for sequences of alternating and averaged projections.

We study preconditioned algorithms of alternating direction method of multipliers type for nonsmooth optimization problems. The alternating direction method of multipliers is a popular first-order method for general constrained optimization problems. However, one of its drawbacks is the need to solve implicit subproblems. In various applications, these subproblems are either easily solvable or linear, but nevertheless challenging. We derive a preconditioned version that allows for flexible and efficient preconditioning for these linear subproblems. The original and preconditioned version is written as a new kind of proximal point method for the primal problem, and the weak (strong) convergence in infinite (finite) dimensional Hilbert spaces is proved. Various efficient preconditioners with any number of inner iterations may be used in this preconditioned framework. Furthermore, connections between the preconditioned version and the recently introduced preconditioned Douglas---Rachford method for general nonsmooth problems involving quadratic---linear terms are established. The methods are applied to total variation denoising problems, and their benefits are shown in numerical experiments.

We introduce a new general proximal-point algorithm for an infinite family of monotone operators in a real Hilbert space and establish strong convergence of the iterative process to a common null point of the infinite family of monotone operators. Our result generalizes and improves numerous results in the available literature.

In this paper, we propose two new extragradient-proximal algorithms for solving split equilibrium and fixed point problems (SEFPP) in real Hilbert spaces, in which the first equilibrium bifunction is pseudomonotone, the second one is monotone, and the fixed point mappings are nonexpansive. By using the extragradient method incorporated with the proximal point algorithm and cutting techniques, we obtain algorithms for solving (SEFPP). Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of (SEFPP). Our results improve and extend the previous results given in the literature.

The main purpose of this paper is to revisit the proximal point algorithms with over-relaxed A -maximal m -relaxed monotone mappings for solving variational inclusions in Hilbert spaces without Lipschitz continuity requirement to overcome the drawbacks of the paper (Verma, 2009) [5] . We affirmatively answer the open question mentioned in the paper (Huang and Noor, 2012) [6] .

The purpose of this paper is to present a regularization method for finding a common fixed point of a nonexpansive semigroup in a real Hilbert space. A combination of the considered regularization scheme with the proximal point algorithm and another one of the regularization method with an iterative scheme is studied in this research. We also discuss an application of the proposed methods for an initial value problem in Hilbert spaces.

In this paper, we propose a modified proximal point algorithm for finding a common element of the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings and the set of minimizers of convex and lower semi-continuous functions. We obtain weak and strong convergence of the proposed algorithm to a common element of the two sets in real Hilbert spaces. A numerical example to support our main results is also given.

We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.

In a Hilbert space, we study the convergence of a dynamic string-averaging proximal point method to a common zero of a finite family of maximal monotone operators under the presence of perturbations. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact dynamic string-averaging proximal point algorithm generates an approximate solution if perturbations are summable. We also show that if the perturbations are sufficiently small, then the inexact produces approximate solutions.

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.

We prove the solution existence and propose an iterative algorithm for solving equilibrium problems over the common solutions of a monotone equilibrium problem and the fixed points of a strictly pseudocontractive mapping in Hilbert spaces. The algorithm is a combination of the proximal point and Halpern methods. Strong convergence is established.

We propose new iteration methods for finding a common point of the solution set of a pseudomonotone equilibrium problem and the solution set of a monotone equilibrium problem. The methods are based on both the extragradient-type method and the viscosity approximation method. We obtain weak convergence theorems for the sequences generated by these methods in a real Hilbert space.

In this paper, a modified proximal point algorithm with errors, which consists of a resolvent operator technique step with errors followed by a modified orthogonal projection onto a moving half-space, is constructed for approximating the solution of the general variational inclusion in Hilbert space. The convergence of the iterative sequence is shown under weak assumptions. The results improve and extend some known results.

In this paper, we investigate and analyze a proximal point algorithm via viscosity approximation method with error. This algorithm is introduced for finding a common zero point for a countable family of inverse strongly accretive operators and a countable family of nonexpansive mappings in Banach spaces. Our result can be extended to some well known results from a Hilbert space to a uniformly convex and 2−uniformly smooth Banach space. Finally, we establish the strong convergence theorems for the proximal point algorithm. Also, some illustrative numerical examples are presented.

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.

The purpose of this paper is to show that the iterative scheme recently studied by Xu (J Glob Optim 36(1):115---125, 2006) is the same as the one studied by Kamimura and Takahashi (J Approx Theory 106(2):226---240, 2000) and to give a supplement to these results. With the new technique proposed by Mainge (Comput Math Appl 59(1):74---79, 2010), we show that the convergence of the iterative scheme is established under another assumption. It is noted that if the computation error is zero or the approximate computation is exact, our new result is a genuine generalization of Xu's result and Kamimura---Takahashi's result.

This paper illustrates that the conditions and the main proof of two main theorems of Verma [R.U. Verma, The over-relaxed proximal point algorithm based on H-maximal monotonicity design and applications, Computers and Mathematics with Applications 55 (2008) 2673–2679] concerning the strong convergence of the over-relaxed proximal point algorithm for H-maximal monotone mappings in Hilbert spaces are incorrect.

The aim of the paper is to propose an iterative regularization method of proximal point type for finding a common solution for a finite family of inverse-strongly monotone equations in Hilbert spaces.

On the basis of the general framework of H -maximal monotonicity (also referred to as H -monotonicity in the literature), a generalization to Rockafellar’s theorem in the context of solving a general inclusion problem involving a set-valued maximal monotone operator using the proximal point algorithm in a Hilbert space setting is explored. As a matter of fact, this class of inclusion problems reduces to a class of variational inequalities as well as to a class of complementarity problems. This proximal point algorithm turns out to be of interest in the sense that it plays a significant role in certain computational methods of multipliers in nonlinear programming. The notion of H -maximal monotonicity generalizes the general theory of set-valued maximal monotone mappings to a new level. Furthermore, some results on general firm nonexpansiveness and resolvent mapping corresponding to H -monotonicity are also given.

Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] proved strong convergence theorems for nonexpansive mappings, nonexpansive semigroups and the proximal point algorithm for zero-point of monotone operators in Hilbert spaces by the CQ iteration method. The purpose of this paper is to modify the CQ iteration method of K. Nakajo and W. Takahashi using the monotone CQ method, and to prove strong convergence theorems. In the proof process of this article, the Cauchy sequence method is used, so we proceed without use of the demiclosedness principle and Opial’s condition, and other weak topological techniques.

Let A be a maximal monotone operator in a real Hilbert space H and let {un} be the sequence in H given by the proximal point algorithm, defined by un=(I+cnA)−1(un−1−fn), ∀n≥1, with u0=z, where cn>0 and fn∈H. We show, among other things, that under suitable conditions, un converges weakly or strongly to a zero of A if and only if lim inf n→+∞|wn

Let T : K → H be a continuous ( η ) – pseudomonotone and ( η , L ) – relaxed monotone mapping from a nonempty closed invex subset K of a finite-dimensional Hilbert space H into H . Let f : K → R be proper, invex and lower semicontinuous on K and let h : K → R be continuously Frechet-differentiable on K with h ′, the gradient of h , ( η , α ) – strongly monotone and ( η , β ) – Lipschitz continuous on K . Let x ∗ ∈ K be any fixed solution of the variational inequality problem (VIP): find an element x ∗ ∈ K such that 〈 T ( x ∗ ) , η ( x , x ∗ ) 〉 + f ( x ) - f ( x ∗ ) ⩾ 0 for all x ∈ K . Then the sequence { x k } generated by the proximal point algorithm converges to x ∗ .

Strong convergence theorems are obtained for the CQ method for an Ishikawa iteration process, a contractive-type iteration process for nonexpansive mappings, and the proximal point algorithm for maximal monotone operators in Hilbert spaces.

Conditions are given for the viability and the weak convergence of an inexact, relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a Hilbert space. In turn, new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming.

In this article, we incorporate inertial terms in the hybrid proximal-extragradient algorithm and investigate the convergence properties of the resulting iterative scheme designed to find the zeros of a maximally monotone operator in real Hilbert spaces. The convergence analysis relies on extended Fejer monotonicity techniques combined with the celebrated Opial Lemma. We also show that the classical hybrid proximal-extragradient algorithm and the inertial versions of the proximal point, the forward-backward and the forward-backward-forward algorithms can be embedded into the framework of the proposed iterative scheme.

In this paper, we study a class of generalized mixed quasivariational in-clusions. By using the properties of the resolvent operator associated with maximal monotone mapping in real Hilbert space, we have established an existence theorem of solutions for generalized mixed quasivariational inclusions, suggesting a new iter-ative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solutions of the generalized mixed quasivariational inclusions. The results proved in this paper are more general than the previously known results in this area.

In this paper, we study a class of generalized quasivariational inclusions. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert space, we have established an existence theorem of solutions for generalized quasivariational inclusions, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions. As special cases, some known results in this field are also discussed.

In this paper, a modified proximal point algorithm for finding common fixed points of averaged self-mappings in Hilbert spaces is introduced and a strong convergence theorem associated with it is proved. As a consequence, we apply it to study the split feasibility problem, the zero point problem of maximal monotone operators, the minimization problem and the equilibrium problem, and to show that the unique minimum norm solution can be obtained through our algorithm for each of the aforementioned problems. Our results generalize and unify many results that occur in the literature. MSC:47H10, 47J25, 68W25.

The purpose of this paper is to provide some remarks for the main results of the paper Verma (Appl. Math. Lett. 21:142-147, 2008). Further, by using the generalized proximal operator technique associated with the (A, η, m)-monotone operators, we discuss the approximation solvability of general variational inclusion problem forms in Hilbert spaces and the convergence analysis of iterative sequences generated by the over-relaxed (A, η, m)-proximal point algorithm frameworks with errors, which generalize the hybrid proximal point algorithm frameworks due to Verma. MSC: 47H05; 49J40

Under some weaker conditions, we prove the strong convergence of the sequence generated by a modified regularization method of finding a zero for a maximal monotone operator in a Hilbert space. In addition, an example is also given in order to illustrate the effectiveness of our generalizations. The results presented in this paper can be viewed as the improvement, supplement, and extension of the corresponding results.

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.

We consider the relaxed and contraction-proximal point algorithms in Hilbert spaces. Some conditions on the parameters for guaranteeing the convergence of the algorithm are relaxed or removed. As a result, we extend some recent results of Ceng-Wu-Yao and Noor-Yao.