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.     

Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization

This paper deals with the convergence analysis of a second order proximal method for approaching critical points of a smooth and quasiconvex objective function defined on a real Hilbert space. The considered method, well-known in the convex case, unifies proximal method, relaxation and inertial-type extrapolation. The convergence theorems established in this new setting improve recent ones.     

Strong Convergence of an Inexact Proximal Point Algorithm for Equilibrium Problems in Banach Spaces

We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we solve an inexact regularized equilibrium problem with a flexible error criterion to obtain an axillary point. Using this axillary point and the inexact solution of the previous iterate, we construct two appropriate hyperplanes which separate the current iterate from the solution set of the given problem. Then the next iterate is defined as the Bregman projection of the initial point onto the intersection of two halfspaces obtained from the two constructed hyperplanes containing the solution set of the original problem. Assuming standard hypotheses, we present a convergence analysis for our algorithm, establishing that the generated sequence strongly and globally converges to a solution of the problem which is the closest one to the starting point of the algorithm.     

Rate of convergence for proximal point algorithms on Hadamard manifolds

In this paper, an estimate of convergence rate concerned with an inexact proximal point algorithm for the singularity of maximal monotone vector fields on Hadamard manifolds is discussed. We introduce a weaker growth condition, which is an extension of that of Luque from Euclidean spaces to Hadamard manifolds. Under the growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. The main results presented in this paper generalize and improve some corresponding known results.     

The relaxed inexact projection methods for the split feasibility problem

In this paper we present several relaxed inexact projection methods for the split feasibility problem (SFP). Each iteration of the first proposed algorithm consists of a projection onto a halfspace containing the given closed convex set. The algorithm can be implemented easily and its global convergence to the solution can be established under suitable conditions. Moreover,we present some modifications of the relaxed inexact projection method with constant stepsize by adopting Armijo-like search. We furthermore present a variable-step relaxed inexact projection method which does not require the computation of the matrix inverses and the largest eigenvalue of the matrix A^TA, and the objective function can decrease sufficiently at each iteration. We show convergence of these modified algorithms under mild conditions. Finally, we perform some numerical experiments, which show the behavior of the algorithms proposed.     

A Projection-Proximal Point Algorithm for Solving Generalized Variational Inequalities

In this paper, a projection-proximal point method for solving a class of generalized variational inequalities is considered in Hilbert spaces. We investigate a general iterative algorithm, which consists of an inexact proximal point step followed by a suitable orthogonal projection onto a hyperplane. We prove the convergence of the algorithm for a pseudomonotone mapping with weakly upper semicontinuity and weakly compact and convex values. We also analyze the convergence rate of the iterative sequence under some suitable conditions.     

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.     

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

We present the convergence analysis of the inexact infeasible path-following (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of the interior-point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis. Communicated by Y. Zhang.     

A proximal augmented Lagrangian method for equilibrium problems

Considering a recently proposed proximal point method for equilibrium problems, we construct an augmented Lagrangian method for solving the same problem in reflexive Banach spaces with cone constraints generating a strongly convergent sequence to a certain solution of the problem. This is an inexact hybrid method meaning that at a certain iterate, a solution of an unconstrained equilibrium problem is found, allowing a proper error bound, followed by a Bregman projection of the initial iterate onto the intersection of two appropriate halfspaces. Assuming a set of reasonable hypotheses, we provide a full convergence analysis.     

Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii–Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii–Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas–Rachford splitting method, the Fermat–Weber location problem as well as the alternating projection method by John von Neumann.     

Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators

For variational inequalities characterizing saddle points of Lagrangians associated with convex programming problems in Hilbert spaces, the convergence of an interior proximal method based on Bregman distance functionals is studied. The convergence results admit a successive approximation of the variational inequality and an inexact treatment of the proximal iterations.An analogous analysis is performed for finite-dimensional complementarity problems with multi-valued monotone operators.     

A Proximal Point Method in Nonreflexive Banach Spaces

We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.     

An inertial-like proximal algorithm for equilibrium problems

The paper concerns with an inertial-like algorithm for approximating solutions of equilibrium problems in Hilbert spaces. The algorithm is a combination around the relaxed proximal point method, inertial effect and the Krasnoselski–Mann iteration. The using of the proximal point method with relaxations has allowed us a more flexibility in practical computations. The inertial extrapolation term incorporated in the resulting algorithm is intended to speed up convergence properties. The main convergence result is established under mild conditions imposed on bifunctions and control parameters. Several numerical examples are implemented to support the established convergence result and also to show the computational advantage of our proposed algorithm over other well known algorithms.     

A NOTE ON THE CONVERGENCE OF THE DFP ALGORITHM ON QUADRATIC UNIFORMLY CONVEX FUNCTIONS

Abstract

This paper analyzes the open question of the convergence of the DFP algorithm with inexact line searches. We prove that the DFP algorithm is convergent for quadratic uniformly convex function with commonly used inexact line searches.     

求解一般约束优化问题的一个全局收敛的混合不精确SQP算法

对于一般约束优化问题,本文通过一种特殊的耦合策略,把一个局邵超线性收敛的不精确SQP算法与广义梯度投影法相结合,从而给出了一个混合算法.该算法无需计算拉格朗日函数的海色矩阵,并且在适当的假设下,算法具有全局和局部超线性收敛性.     

New results on the proximal point algorithm in non-positive curvature metric spaces

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.     

一类B可微方程的非精确阻尼牛顿法

本文提出了求解一类基于双障碍问题的Ｂ可微方程的非精确阻尼牛顿法,并在一定条件下,证明了该算法的全局收敛性和二阶收敛性。     

Another version of the proximal point algorithm in a Banach space

In this paper, we study some non-traditional schemes of proximal point algorithm for nonsmooth convex functionals in a Banach space. The proximal approximations to their minimal points and/or their minimal values are considered separately for unconstrained and constrained minimization problems on convex closed sets. For the latter we use proximal point algorithms with the metric projection operators and first establish the estimates of the convergence rate with respect to functionals. We also investigate the perturbed projection proximal point algorithms and prove their stability. Some results concerning the classical proximal point method for minimization problems in a Banach space is also presented in this paper.     

Forcing strong convergence of proximal point iterations in a Hilbert space

This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Güler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns. Received January 6, 1998 / Revised version received August 9, 1999?Published online November 30, 1999     

A line search filter algorithm with inexact step computations for equality constrained optimization

In this paper, a new line search filter algorithm for equality constrained optimization is presented. The approach belongs to the class of inexact Newton-like methods. It can also be regarded as an inexact version of generic sequential quadratic programming (SQP) methods. The trial step is obtained by truncatedly solving the primal-dual system based on any robust and efficient linear system solver. Practical termination tests for the linear system solver are established to ensure global convergence. Preliminary numerical results demonstrate the approach is potentially useful.