近似邻近点算法收敛性的一个注记

近似邻近点算法在最优化理论与方法研究中具有重要作用.在不同误差准则下,近似邻近点算法具有不同的收敛性.利用极大单调算子等工具给出了一个具体的例子,解释了在一些误差准则下近似邻近点算法的收敛性.     

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.     

一种基于邻近点算法的变步长原始-对偶算法

本文考虑求解一种源于信号及图像处理问题的鞍点问题.基于邻近点算法的思想,我们对原始-对偶算法进行改进,构造一种对称正定且可变的邻近项矩阵,得到一种新的原始-对偶算法.新算法可以看成一种邻近点算法,因此它的收敛性易于分析,且无需较强的假设条件.初步实验结果表明,当新算法被应用于求解图像去模糊问题时,和其他几种主流的高效算法相比,新算法能得到较高质量的结果,且计算时间也是有竞争力的.     

An Accelerated Inexact Proximal Point Algorithm for Convex Minimization

The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact proximal point algorithm with a new inexact criterion for solving convex minimization, and show its O(1/k) iteration-complexity. Then we show that this inexact proximal point algorithm is eligible for being accelerated by some influential acceleration schemes proposed by Nesterov. Accordingly, an accelerated inexact proximal point algorithm with an iteration-complexity of O(1/k ²) is proposed.     

An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds

In this paper we present an extension of the proximal point algorithm with Bregman distances to solve constrained minimization problems with quasiconvex and convex objective function on Hadamard manifolds. The proposed algorithm is a modified and extended version of the one presented in Papa Quiroz and Oliveira (J Convex Anal 16(1): 49–69, 2009). An advantage of the proposed algorithm, for the nonconvex case, is that in each iteration the algorithm only needs to find a stationary point of the proximal function and not a global minimum. For that reason, from the computational point of view, the proposed algorithm is more practical than the earlier proximal method. Another advantage, for the convex case, is that using minimal condition on the problem data as well as on the proximal parameters we get the same convergence results of the Euclidean proximal algorithm using Bregman distances.     

一般单调变分不等式的近似邻近外梯度算法

近似邻近点算法是求解单调变分不等式的一个有效方法,该算法通过解决一系列强单调子问题,产生近似邻近点序列来逼近变分不等式的解,而外梯度算法则通过每次迭代中增加一个投影来克服一般投影算法限制太强的缺点,但它们均未能改变迭代步骤中不规则闭凸区域上投影难计算的问题.于是,本文结合外梯度算法的迭代格式,构造包含原投影区域的半空间,将投影建立在半空间上,简化了投影的求解过程,并对新的邻近点序列作相应限制,使得改进的算法具有较好的收敛性.     

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 Hybrid Approximate Extragradient – Proximal Point Algorithm Using the Enlargement of a Maximal Monotone Operator

We propose a modification of the classical extragradient and proximal point algorithms for finding a zero of a maximal monotone operator in a Hilbert space. At each iteration of the method, an approximate extragradient-type step is performed using information obtained from an approximate solution of a proximal point subproblem. The algorithm is of a hybrid type, as it combines steps of the extragradient and proximal methods. Furthermore, the algorithm uses elements in the enlargement (proposed by Burachik, Iusem and Svaiter) of the operator defining the problem. One of the important features of our approach is that it allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems. This yields a more practical proximal-algorithm-based framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. It is further demonstrated that the modified forward-backward splitting algorithm of Tseng falls within the presented general framework.     

Proximal point algorithm with Schur decomposition on the cone of symmetric semidefinite positive matrices

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function −lndet(X). Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric.     

Proximal Point Algorithm for Quasi-Convex Minimization Problems in Metric Spaces

In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove Δ-convergence of the generated sequence to a critical point (which is defined in the text) of an objective quasi-convex, proper and lower semicontinuous function with at least a minimum point as well as some strong convergence results to a minimum point with some additional conditions. The results extend the recent results of the proximal point algorithm in Hadamard manifolds and CAT(0) spaces.     

A proximal point algorithm for the monotone second-order cone complementarity problem

This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173–1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.     

Descentwise inexact proximal algorithms for smooth optimization

The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i)?an algorithm to compute the proximal point, (ii)?a rule to stop this algorithm, (iii)?an update formula for the proximal parameter. In this work we focus on?(ii), when smoothness is present??so that Newton-like methods can be used for?(i): we aim at giving adequate stopping rules to reach overall efficiency of the method. Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a ??sufficient?? decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems.     

On general over-relaxed proximal point algorithm and applications

A general framework to the over-relaxed proximal point algorithm based on H-maximal monotonicity is introduced, and then it is applied to the approximation solvability of a general class of nonlinear inclusion problems based on the generalized resolvent operator technique. This form of the proximal point algorithm seems to be more application-oriented to inclusion problems.     

求解凸规划及鞍点问题定制的PPA 算法及其收敛速率

线性约束的凸优化问题和鞍点问题的一阶最优性条件是一个单调变分不等式. 在变分不等式框架下求解这些问题, 选取适当的矩阵G, 采用G- 模下的PPA 算法, 会使迭代过程中的子问题求解变得相当容易. 本文证明这类定制的PPA 算法的误差界有1/k 的收敛速率.     

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.     

Finite termination of the proximal point algorithm

This paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm. We give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm.This material is based on research supported by National Science Foundation Grants DCR-8521228 and CCR-8723091, and Air Force Office of Scientific Research Grant AFOSR-86-0172.     

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.     

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.     

AN ITERATIVE ALGORITHM FOR MAXIMAL MONOTONE MULTIVALUED OPERATOR EQUATIONS

1 IntroductionThe multivalued operator equations occur in various applications, e.g., mecha11ical systeimwith dry and viscous damping, electrical networks with switches, oscil1ations in viscoelastic-ity, optimization probIems with uonsmooth data, dynanilcal systems with nondifferentiablepotential, and optimal colltroI problellls. There have been a number of results, for instance,[1l-[6l, oll the solutions of multivallled operator equations. Amoug theln, R.T.Rockafellar[1]gave a prorimal poin…     

复合凸优化的快速邻近点算法

在大数据时代,随着数据采集手段的不断提升,大规模复合凸优化问题大量的出现在包括统计数据分析,机器与统计学习以及信号与图像处理等应用中.本文针对大规模复合凸优化问题介绍了一类快速邻近点算法.在易计算的近似准则和较弱的平稳性条件下,本文给出了该算法的全局收敛与局部渐近超线性收敛结果.同时,我们设计了基于对偶原理的半光滑牛顿法来高效稳定求解邻近点算法所涉及的重要子问题.最后,本文还讨论了如何通过深入挖掘并利用复合凸优化问题中由非光滑正则函数所诱导的非光滑二阶信息来极大减少半光滑牛顿算法中求解牛顿线性系统所需的工作量,从而进一步加速邻近点算法.