An approximate proximal-extragradient type method for monotone variational inequalities

Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter's APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter's APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method.     

A modified alternating projection based prediction–correction method for structured variational inequalities

In this paper, we propose a novel alternating projection based prediction–correction method for solving the monotone variational inequalities with separable structures. At each iteration, we adopt the weak requirements for the step sizes to derive the predictors, which affords fewer trial and error steps to accomplish the prediction phase. Moreover, we design a new descent direction for the merit function in correction phase. Under some mild assumptions, we prove the global convergence of the modified method. Some preliminary computational results are reported to demonstrate the promising and attractive performance of the modified method compared to some state-of-the-art prediction–contraction methods.     

New extragradient-type methods for solving variational inequalities

In this paper, we propose new methods for solving variational inequalities. The proposed methods can be viewed as a refinement and improvement of the method of He et al. [B.S. He, X.M. Yuan, J.J. Zhang, Comparison of two kinds of prediction–correction methods for monotone variational inequalities, Comp. Opt. Appl. 27 (2004) 247–267] by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. Under certain conditions, the global convergence of the both methods is proved. Preliminary numerical experiments are included to illustrate the efficiency of the proposed methods.     

A COMPARISON OF DIFFERENT CONTRACTION METHODS FOR MONOTONE VARIATIONAL INEQUALITIES

It is interesting to compare the efficiency of two methods when their computational loads in each iteration are equal. In this paper, two classes of contraction methods for monotone variational inequalities are studied in a unified framework. The methods of both classes can be viewed as prediction-correction methods, which generate the same test vector in the prediction step and adopt the same step-size rule in the correction step. The only difference is that they use different search directions. The computational loads of each iteration of the different classes are equal. Our analysis explains theoretically why one class of the contraction methods usually outperforms the other class. It is demonstrated that many known methods belong to these two classes of methods. Finally, the presented numerical results demonstrate the validity of our analysis.     

The interior proximal extragradient method for solving equilibrium problems

In this article we present a new and efficient method for solving equilibrium problems on polyhedra. The method is based on an interior-quadratic proximal term which replaces the usual quadratic proximal term. This leads to an interior proximal type algorithm. Each iteration consists in a prediction step followed by a correction step as in the extragradient method. In a first algorithm each of these steps is obtained by solving an unconstrained minimization problem, while in a second algorithm the correction step is replaced by an Armijo-backtracking linesearch followed by an hyperplane projection step. We prove that our algorithms are convergent under mild assumptions: pseudomonotonicity for the two algorithms and a Lipschitz property for the first one. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms.     

The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems

Summary The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an inner iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.This work was supported in part by National Science Foundation Grants DCR-8412314 and DCR-8502014The work of this author was completed while he was on sabbatical leave at the Centre for Mathematical Analysis and Mathematical Sciences Research Institute at the Australian National University, Canberra, Australia     

A Logarithmic-Quadratic Proximal Prediction-Correction Method for Structured Monotone Variational Inequalities

Inspired by the Logarithmic-Quadratic Proximal (LQP) method for variational inequalities, we present a prediction-correction method for structured monotone variational inequalities. Each iteration of the new method consists of a prediction and a correction. Both the predictor and the corrector are obtained easily with tiny computational load. In particular, the LQP system that appears in the prediction is approximately solved under significantly relaxed inexactness restriction. Global convergence of the new method is proved under mild assumptions. In addition, we present a self-adaptive version of the new method that leads to easier implementations. Preliminary numerical experiments for traffic equilibrium problems indicate that the new method is effectively applicable in practice. Presented at the 6th International conference on Optimization: Techniques and Applications, Ballarat Australia, December 9–11, 2004. This author was supported by NSFC Grant 10571083, the MOEC grant 20020284027 and Jiangsu NSF grant BK2002075     

A New Inexactness Criterion for Approximate Logarithmic-Quadratic Proximal Methods

Recently,a class of logarithmic-quadratic proximal(LQP)methods was intro- duced by Auslender,Teboulle and Ben-Tiba.The inexact versions of these methods solve the sub-problems in each iteration approximately.In this paper,we present a practical inexactness criterion for the inexact version of these methods.     

Prediction-correction method with BB step sizes

In the prediction-correction method for variational inequality (VI) problems, the step size selection plays an important role for its performance. In this paper, we employ the Barzilai-Borwein (BB) strategy in the prediction step, which is efficient for many optimization problems from a computational point of view. To guarantee the convergence, we adopt the line search technique, and relax the conditions to accept the BB step sizes as large as possible. In the correction step, we utilize a longer step length to calculate the next iteration point. Finally, we present some preliminary numerical results to show the efficiency of the algorithms.     

Local properties of inexact methods for minimizing nonsmooth composite functions

Methods for minimization of composite functions with a nondifferentiable polyhedral convex part are considered. This class includes problems involving minimax functions and norms. Local convergence results are given for “active set” methods, in which an equality-constrained quadratic programming subproblem is solved at each iteration. The active set consists of components of the polyhedral convex function which are active or near-active at the current iteration. The effects of solving the subproblem inexactly at each iteration are discussed; rate-of-convergence results which depend on the degree of inexactness are given.     

Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

In this paper, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of stochastic differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.     

Stability and Accuracy of Inexact Interior Point Methods for Convex Quadratic Programming

We consider primal–dual interior point methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different levels of accuracy in the inexact computation and different processes for retrieving the step after block elimination. Our analysis is general and includes as special cases sources of inexactness due either to roundoff and computational errors or to the iterative solution of the augmented system using typical procedures. In the roundoff case, we recover and extend some known results.     

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.     

A hybrid algorithm for nonlinear minimax problems

In this paper, a hybrid algorithm for solving finite minimax problem is presented. In the algorithm, we combine the trust-region methods with the line-search methods and curve-search methods. By means of this hybrid technique, the algorithm, according to the specific situation at each iteration, can adaptively performs the trust-region step, line-search step or curve-search step, so as to avoid possibly solving the trust-region subproblems many times, and make better use of the advantages of different methods. Moreover, we use second-order correction step to circumvent the difficulties of the Maratos effect occurred in the nonsmooth optimization. Under mild conditions, we prove that the new algorithm is of global convergence and locally superlinear convergence. The preliminary experiments show that the new algorithm performs efficiently.     

On the O(1/t) convergence rate of the LQP prediction–correction method

The logarithmic-quadratic proximal (abbreviated by LQP) prediction–correction method is attractive for structured monotone variational inequalities, and it ensures the global convergence under some suitable conditions. In this paper, we are interested in investigating the convergence rate of the LQP prediction–correction method. Motivated by the research work about the convergence rate or iteration complexity for various first-order algorithms in the literature, we provide a simple proof to show the $O(1/t)$ convergence rate for the LQP prediction–correction method.     

Inexact Alternating-Direction-Based Contraction Methods for Separable Linearly Constrained Convex Optimization

Alternating direction method of multipliers has been well studied in the context of linearly constrained convex optimization. In the last few years, we have witnessed a number of novel applications arising from image processing, compressive sensing and statistics, etc., where the approach is surprisingly efficient. In the early applications, the objective function of the linearly constrained convex optimization problem is separable into two parts. Recently, the alternating direction method of multipliers has been extended to the case where the number of the separable parts in the objective function is finite. However, in each iteration, the subproblems are required to be solved exactly. In this paper, by introducing some reasonable inexactness criteria, we propose two inexact alternating-direction-based contraction methods, which substantially broaden the applicable scope of the approach. The convergence and complexity results for both methods are derived in the framework of variational inequalities.     

An improved two-step method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is a noncooperative game in which the strategy set of each player, as well as his payoff function, depend on the rival players strategies. As a generalization of the standard Nash equilibrium problem (NEP), the GNEP has recently drawn much attention due to its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. In this paper, we propose an improved two-step (a prediction step and a correction step) method for solving the quasi-variational inequality (QVI) formulation of the GNEP. Per iteration, we first do a projection onto the feasible set defined by the current iterate (prediction) to get a trial point; then, we perform another projection step (correction) to obtain the new iterate. Under certain assumptions, we prove the global convergence of the new algorithm. We also present some numerical results to illustrate the ability of our method, which indicate that our method outperforms the most recent projection-like methods of Zhang et al. (2010).     

On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems

In this paper, we first present a local Hermitian and skew-Hermitian splitting (LHSS) iteration method for solving a class of generalized saddle point problems. The new method converges to the solution under suitable restrictions on the preconditioning matrix. Then we give a modified LHSS (MLHSS) iteration method, and further extend it to the generalized saddle point problems, obtaining the so-called generalized MLHSS (GMLHSS) iteration method. Numerical experiments for a model Navier-Stokes problem are given, and the results show that the new methods outperform the classical Uzawa method and the inexact parameterized Uzawa method.     

Inexact non-interior continuation method for monotone semidefinite complementarity problems

Chen and Tseng (Math Program 95:431?C474, 2003) extended non-interior continuation methods for solving linear and nonlinear complementarity problems to semidefinite complementarity problems (SDCP), in which a system of linear equations is exactly solved at each iteration. However, for problems of large size, solving the linear system of equations exactly can be very expensive. In this paper, we propose a version of one of the non-interior continuation methods for monotone SDCP presented by Chen and Tseng that incorporates inexactness into the linear system solves. Only one system of linear equations is inexactly solved at each iteration. The global convergence and local superlinear convergence properties of the method are given under mild conditions.     

一种求解序列二次规划结合信赖域的多维滤子算法

求解非线性规划问题最有效的方法之一为序列二次规划。但是,由于序列二次规划结合信赖域时,会出现可能无解的情况(即不相容性)。而本文针对不相容性提出了一类序列二次规划结合信赖域的多维相容滤子算法。首先,本文根据一般文献中提及的方法对其约束条件引进参数变量,对其目标函数加以惩罚,即实行了可行化处理(也就是无需可行性恢复阶段),从而克服了不相容性。其次,本文提出了多维滤子条件来对迭代步进行选择性的接受,从而避免了传统二维滤子算法的严格条件,使得对迭代步的接受程度大大的放松。最后针对可能出现的maratos效应,我们通过二阶校正策略提出了一种修改后的多维滤子算法。同时,在一定的假设条件下算法具有全局收敛性。