RELAXED PROXIMAL POINT ALGORITHMS BASED ON A-AXIMAL RELAXED MONOTONICITY FRAMEWORKS WITH APPLICATIONS

Existence and convergence theorems for a class of multi-valued variational inclusions in Banach 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.

In this paper we consider a new general iterative method for solving the split common fixed point problem and variational inclusion problem. It entails finding a point which belongs to the set of common fixed points of a finite family of demimetric mappings and the common zero point set of a family of monotone operators in a Hilbert space such that its image under a linear transformation belongs to the set of fixed points of a demimetric mapping in a uniformly convex and smooth Banach space in the image space. Strong convergence theorem is established under some suitable conditions. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models' structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way.

A generalization to the Rockafellar theorem (1976) on the linear convergence in the context of approximating a solution to a general class of inclusion problems involving set-valued A-maximal relaxed monotone mappings using the proximal point algorithm in a real Hilbert space setting is given. There exists a vast literature on this theorem, but most of the investigations are focused on relaxing the proximal point algorithm and applying it to the inclusion problems. The general framework for A-maximal relaxed monotonicity generalizes the theory of set-valued maximal monotone mappings, including H-maximal monotone mappings. The obtained results are general in nature, while application-oriented as well.

The proximal point algorithm (PPA) is a classical method for finding zeros of maximal monotone operators. It is known that the algorithm only has weak convergence in a general Hilbert space. Recently, Wang, Wang and Xu proposed two modifications of the PPA and established strong convergence theorems on these two algorithms. However, these two convergence theorems exclude an important case, namely, the over-relaxed case. In this paper, we extend the above convergence theorems from under-relaxed case to the over-relaxed case, which in turn improve the performance of these two algorithms. Preliminary numerical experiments show that the algorithm with over-relaxed parameter performs better than that with under-relaxed parameter.

In this paper, we introduce an efficient algorithm for solving bilevel split variational monotone inclusion problems. The class of problems studied in this paper contains several other classes of well-known problems which have been studied by many authors. Unlike several existing methods in the literature, our method does not require the underlying operators in the lower-level problem and upper-level to be Lipschitz continuous. The proposed method is a combination of the Tseng method and the projection and contraction method. The convergence of our new algorithm is enhanced with double inertial steps. We obtain the strong convergence results of our algorithm under mild conditions on the control parameters of our method. Furthermore, we carry out some numerical tests to show the advantage of our new method over many well-known methods. Our results improve, generalize, and unify several existing results in the literature.

In this paper, we present two iterative algorithms involving Yosida approximation operators for split monotone variational inclusion problems ( S p MVIP ). We prove the weak and strong convergence of the proposed iterative algorithms to the solution of S p MVIP in real Hilbert spaces. Our algorithms are based on Yosida approximation operators of monotone mappings such that the step size does not require the precalculation of the operator norm. To show the reliability and accuracy of the proposed algorithms, a numerical example is also constructed.

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.

The proximal point algorithm (PPA) is a method for solving inclusions of the form $0in T(z)$, where T is a monotone operator on a Hilbert space. The algorithm is one of the most powerful and versatile solution techniques for solving variational inequalities, convex programs, and convex-concave mini-max problems. It possesses a robust convergence theory for very general problem classes and is the basis for a wide variety of decomposition methods called splitting methods. Yet the classical PPA typically exhibits slow convergence in many applications. For this reason, acceleration methods for the PPA algorithm are of great practical importance. In this paper we propose a variable metric implementation of the proximal point algorithm. In essence, the method is a Newton-like scheme applied to the Moreau--Yosida resolvent of the operator T. In this article, we establish the global and linear convergence of the proposed method. In addition, we characterize the superlinear convergence of the method. In a companion work, we establish the superlinear convergence of the method when implemented with Broyden updating (the nonsymmetric case) and BFGS updating (the symmetric case).

A new projection algorithm for variational inclusion problems and its application to cervical cancer disease prediction

A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems

It is well-known that the stability of a first-order autonomous system can be determined by testing the symmetric positive definite solutions of associated Lyapunov matrix equations. However, the research on the constrained solutions of the Lyapunov matrix equations is quite few. In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations. The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman–Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the literature. The third iterative algorithm is based on the famous alternating direction method of multipliers (ADMM), and its convergence is subsequently discussed in detail. Finally, numerical simulation results illustrate the effectiveness of the proposed iterative algorithms.

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.