Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a uniform approach

This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddle-point problem. We model these two classes of problems uniformly by a mixed variational inequality, and show how PPA with … Read more

A relaxed customized proximal point algorithm for separable convex programming

The alternating direction method (ADM) is classical for solving a linearly constrained separable convex programming problem (primal problem), and it is well known that ADM is essentially the application of a concrete form of the proximal point algorithm (PPA) (more precisely, the Douglas-Rachford splitting method) to the corresponding dual problem. This paper shows that an … Read more

A proximal point algorithm for sequential feature extraction applications

We propose a proximal point algorithm to solve LAROS problem, that is the problem of finding a “large approximately rank-one submatrix”. This LAROS problem is used to sequentially extract features in data. We also develop a new stopping criterion for the proximal point algorithm, which is based on the duality conditions of \eps-optimal solutions of … Read more

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods

This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) method. Iteration-complexity results are established for the A-HPE method, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE method are … Read more

Iteration-Complexity of a Newton Proximal Extragradient Method for Monotone Variational Inequalities and Inclusion Problems

In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient framework has been used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth … Read more

A contraction method with implementable proximal regularization for linearly constrained convex programming

The proximal point algorithm (PPA) is classical, and it is implicit in the sense that the resulting proximal subproblems may be as difficult as the original problem. In this paper, we show that with appropriate choices of proximal parameters, the application of PPA to the linearly constrained convex programming can result in easy proximal subproblems. … Read more

Convergence analysis of primal-dual algorithms for total variation image restoration

Recently, some attractive primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primal-dual algorithms and shows that the involved parameters of those primal-dual algorithms (including the step sizes) can be significantly enlarged if … Read more

An accelerated inexact proximal point algorithm for convex minimization

The proximal point algorithm (PPA) is classical and popular in the community of Optimization. In practice, inexact PPAs which solves the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact PPA with a new inexact criterion for solving convex minimization, and show that the … Read more

The unified framework of some proximal-based decomposition methods for monotone variational inequalities with separable structure

Some existing decomposition methods for solving a class of variational inequalities (VI) with separable structures are closely related to the classical proximal point algorithm, as their decomposed sub-VIs are regularized by proximal terms. Differing in whether the generated sub-VIs are suitable for parallel computation, these proximal-based methods can be categorized into the parallel decomposition methods … Read more

Convergence to the optimal value for barrier methods combined with Hessian Riemannian gradient flows and generalized proximal algorithms

We consider the problem $\min_{x\in\R^n}\{f(x)\mid Ax=b, \ x\in\overline{C},\ g_j(x)\le0,\ j=1,\ldots,s\}$, where $b\in\R^m$, $A\in\R^{m\times n}$ is a full rank matrix, $\overline{C}$ is the closure of a nonempty, open and convex subset $C$ of $\R^n$, and $g_j(\cdot)$, $ j=1,\ldots,s$, are nonlinear convex functions. Our strategy consists firstly in to introduce a barrier-type penalty for the constraints $g_j(x)\le0$, … Read more