求解单调变分不等式的近似邻近点算法的收敛性分析

为了求解单调变分不等式,建立了一个新的误差准则,并且在不需要增加诸如投影,外梯度等步骤的情况下证明了邻近点算法的收敛性.     

Weak sharp solutions for variational inequalities in Banach spaces

In this paper, we introduce the notion of a weak sharp set of solutions to a variational inequality problem (VIP) in a reflexive, strictly convex and smooth Banach space, and present its several equivalent conditions. We also prove, under some continuity and monotonicity assumptions, that if any sequence generated by an algorithm for solving (VIP) converges to a weak sharp solution, then we can obtain solutions for (VIP) by solving a finite number of convex optimization subproblems with linear objective. Moreover, in order to characterize finite convergence of an iterative algorithm, we introduce the notion of a weak subsharp set of solutions to a variational inequality problem (VIP), which is more general than that of weak sharp solutions in Hilbert spaces. We establish a sufficient and necessary condition for the finite convergence of an algorithm for solving (VIP) which satisfies that the sequence generated by which converges to a weak subsharp solution of (VIP), and show that the proximal point algorithm satisfies this condition. As a consequence, we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of (VIP).     

About the relaxed cocoercivity and the convergence of the proximal point algorithm

The aim of this paper is to study the convergence of two proximal algorithms via the notion of (α, r)-relaxed cocoercivity without Lipschitzian continuity. We will show that this notion is enough to obtain some interesting convergence theorems without any Lipschitz-continuity assumption. The relaxed cocoercivity case is also investigated.     

Inexact proximal point method for general variational inequalities

In this paper, we suggest and analyze a new inexact proximal point method for solving general variational inequalities, which can be considered as an implicit predictor-corrector method. An easily measurable error term is proposed with further relaxed error bound and an optimal step length is obtained by maximizing the profit-function and is dependent on the previous points. Our results include several known and new techniques for solving variational inequalities and related optimization problems. Results obtained in this paper can be viewed as an important improvement and refinement of the previously known results. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.     

An APPA-based descent method with optimal step-sizes for monotone variational inequalities

To solve monotone variational inequalities, some existing APPA-based descent methods utilize the iterates generated by the well-known approximate proximal point algorithms (APPA) to construct descent directions. This paper aims at improving these APPA-based descent methods by incorporating optimal step-sizes in both the extra-gradient steps and the descent steps. Global convergence is proved under mild assumptions. The superiority to existing methods is verified both theoretically and computationally.     

Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.     

Some proximal algorithms for linearly constrained general variational inequalities

Based on the classical proximal point algorithm (PPA), some PPA-based numerical algorithms for general variational inequalities (GVIs) have been developed recently. Inspired by these algorithms, in this article we propose some proximal algorithms for solving linearly constrained GVIs (LCGVIs). The resulted subproblems are regularized proximally, and they are allowed to be solved either exactly or approximately.     

On convergence criteria of generalized proximal point algorithms

We analyze some generalized proximal point algorithms which include the previously known proximal point algorithms as special cases. Weak and strong convergence of the proposed proximal point algorithms are proved under some mild conditions.     

Coupling proximal methods and variational convergence

An approximation method which combines a data perturbation by variational convergence with the proximal point algorithm, is presented. Conditions which guarantee convergence, are provided and an application to the partial inverse method is given.     

一个新的BUNDLE近似点算法及其收敛性质

提出了一类修正的近似点算法并讨论了算法的收敛性质及其Ｂｕｄｌｅ变形的收敛性质。     

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.     

Weak sharpness for set-valued variational inequalities and applications to finite termination of iterative algorithms

In this paper, we introduce the notion of weak sharpness for set-valued variational inequalities in the n-dimensional Euclidean space and then present some characterizations of weak sharpness. We also give some examples to illustrate this notion. Under the assumption of weak sharpness, by using the inner limit of a set sequence we establish a sufficient and necessary condition to guarantee the finite termination of an arbitrary algorithm for solving a set-valued variational inequality involving maximal monotone mappings. As an application, we show that the sequence generated by the hybrid projection-proximal point algorithm proposed by Solodov and Svaiter terminates at solutions in a finite number of iterations. These obtained results extend some known results of classical variational inequalities.     

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.     

The perturbed proximal point algorithm and some of its applications

Following the works of R. T. Rockafellar, to search for a zero of a maximal monotone operator, and of B. Lemaire, to solve convex optimization problems, we present a perturbed version of the proximal point algorithm. We apply this new algorithm to convex optimization and to variational inclusions or, more particularly, to variational inequalities.     

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.     

Modified subgradient extragradient algorithms for solving monotone variational inequalities

ABSTRACT

In this paper, we introduce two new algorithms for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a new step size, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are presented to show the efficiency and advantage of the proposed algorithms.     

An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.     

Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational inequalities

This paper presents an acceleration step for the linearly convergent diagonalization and projection algorithms for finite-dimensional variational inequalities which is reminiscent of a PARTAN step in nonlinear programming. After establishing the convergence of this technique for both algorithms, several numerical examples are presented to illustrate the sometimes dramatic savings in computation time which this simple acceleration step yields.     

On Gabay’s algorithms for mixed variational inequalities

Iterative numerical algorithms for variational inequalities are systematically constructed from fixed-point problem characterizations in terms of resolvent operators. The applied method is the one introduced by Gabay in [Ga], used here in the context of discrete variational inequalities, and with the emphasis on mixed finite element models. The algorithms apply to nonnecessarily potential problems, generalizing primal and mixed Uzawa and augmented Lagrangian-type algorithms. They are also identified with Euler and operator splitting methods for the time discretization of evolution first-order problems.     

A partial inexact alternating direction method for structured variational inequalities

In this article, a new method is proposed for solving a class of structured variational inequalities (SVIs). The proposed method is referred to as the partial inexact proximal alternating direction (piPAD) method. In the method, two subproblems are solved independently. One is handled by an inexact proximal point method and the other is solved directly. This feature is the major difference between the proposed method and some existing alternating direction-like methods. The convergence of the piPAD method is proved. Two examples of the modern convex optimization problem arising from engineering and information sciences, which can be reformulated into the encountered SVIs, are presented to demonstrate the applicability of the piPAD method. Also, some preliminary numerical results are reported to validate the feasibility and efficiency of the piPAD method.