共查询到20条相似文献,搜索用时 31 毫秒
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In this paper, we consider the monotone affine variational inequality problem (AVIP for short). Based on a smooth reformulation of the AVIP, we propose a Newton-type method to solve the monotone AVIP, where a testing procedure is embedded into our algorithm. Under mild assumptions, we show that the proposed algorithm may find a maximally complementary solution to the monotone AVIP in a finite number of iterations. Preliminary numerical results are reported. 相似文献
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P. Tossings 《Applied Mathematics and Optimization》1994,29(2):125-159
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. 相似文献
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We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem. 相似文献
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Dang Van Hieu 《Optimization》2017,66(12):2291-2307
The paper proposes a new shrinking gradient-like projection method for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method with the monotone hybrid method. Only one optimization program is solved onto the feasible set at each iteration in our algorithm without any extra-step dealing with the feasible set. The absence of an optimization problem in the algorithm is explained by constructing slightly different cutting-halfspace in the monotone hybrid method. Theorem of strong convergence is established under standard assumptions imposed on equilibrium bifunctions. An application of the proposed algorithm to multivalued variational inequality problems (MVIP) is presented. Finally, another algorithm is introduced for MVIPs in which we only use a value of main operator at the current approximation to construct the next approximation. Some preliminary numerical experiments are implemented to illustrate the convergence and computational performance of our algorithms over others. 相似文献
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Error bounds for proximal point subproblems and associated inexact proximal point algorithms 总被引:1,自引:0,他引:1
We study various error measures for approximate solution of proximal point regularizations of the variational inequality problem,
and of the closely related problem of finding a zero of a maximal monotone operator. A new merit function is proposed for
proximal point subproblems associated with the latter. This merit function is based on Burachik-Iusem-Svaiter’s concept of
ε-enlargement of a maximal monotone operator. For variational inequalities, we establish a precise relationship between the
regularized gap function, which is a natural error measure in this context, and our new merit function. Some error bounds
are derived using both merit functions for the corresponding formulations of the proximal subproblem. We further use the regularized
gap function to devise a new inexact proximal point algorithm for solving monotone variational inequalities. This inexact
proximal point method preserves all the desirable global and local convergence properties of the classical exact/inexact method,
while providing a constructive error tolerance criterion, suitable for further practical applications. The use of other tolerance
rules is also discussed.
Received: April 28, 1999 / Accepted: March 24, 2000?Published online July 20, 2000 相似文献
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Hideaki Iiduka 《Journal of Optimization Theory and Applications》2011,148(3):580-592
Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational
inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms to solve these problems have been
proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point
set of a nonexpansive mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative
algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain
assumptions. 相似文献
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《Optimization》2012,61(6):873-885
Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section. 相似文献
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D. Han 《Applied Mathematics and Optimization》2002,45(1):63-74
The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever
the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as
structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating
direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just
makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results
to illustrate the efficiency of the method.
Accepted 4 May 2001. Online publication 19 October, 2001. 相似文献
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Na Zhao 《Applied mathematics and computation》2010,217(7):3368-3378
Many optimization problems can be reformulated as a system of equations. One may use the generalized Newton method or the smoothing Newton method to solve the reformulated equations so that a solution of the original problem can be found. Such methods have been powerful tools to solve many optimization problems in the literature. In this paper, we propose a Newton-type algorithm for solving a class of monotone affine variational inequality problems (AVIPs for short). In the proposed algorithm, the techniques based on both the generalized Newton method and the smoothing Newton method are used. In particular, we show that the algorithm can find an exact solution of the AVIP in a finite number of iterations under an assumption that the solution set of the AVIP is nonempty. Preliminary numerical results are reported. 相似文献
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《Optimization》2012,61(9):1841-1854
We introduce a new iteration method for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of strict pseudocontractions in a real Hilbert space. The weak convergence of the iterative sequences generated by the method is obtained thanks to improve and extend some recent results under the assumptions that the cost mapping associated with the variational inequality problem only is pseudomonotone and not necessarily inverse strongly monotone. Finally, we present some numerical examples to illustrate the behaviour of the proposed algorithm. 相似文献
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《Optimization》2012,61(11):2099-2124
ABSTRACTIn this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others. 相似文献
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Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space. 相似文献
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LQP交替方向法是求解可分离结构型单调变分不等式问题的一种非常有效的方法.它不仅可以充分地利用目标函数的可分结构,将原问题分解为多个更易求解的子问题,还更适合求解大规模问题.对于带有三个可分离算子的单调变分不等式问题,结合增广拉格朗日算法和LQP交替方向法提出了一种部分并行分裂LQP交替方向法,构造了新算法的两个下降方向,结合这两个下降方向得到了一个新的下降方向,沿着这个新的下降方向给出了最优步长.并在较弱的假设条件下,证明了新算法的全局收敛性. 相似文献
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Numerical Algorithms - In this paper, we introduce a new algorithm which combines the inertial projection and contraction method and the viscosity method for solving monotone variational inequality... 相似文献
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In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities. 相似文献
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Xiuyun Zheng Hongwei Liu Jianguang Zhu 《Journal of Applied Mathematics and Computing》2012,40(1-2):341-355
The variational inequality problem can be reformulated as a system of equations. One can solve the reformulated equations to obtain a solution of the original problem. In this paper, based on a symmetric perturbed min function, we propose a new smoothing function, which has some nice properties. By which we propose a new non-interior smoothing algorithm for solving the variational inequality problem, which is based on both the non-interior continuation method and the smoothing Newton method. The proposed algorithm only needs to solve at most one system of equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results are reported. 相似文献
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An explicit hierarchical fixed point algorithm is introduced to solve monotone variational inequalities, which are governed
by a pair of nonexpansive mappings, one of which is used to define the governing operator and the other to define the feasible
set. These kinds of variational inequalities include monotone inclusions and convex optimization problems to be solved over
the fixed point sets of nonexpansive mappings. Strong convergence of the algorithm is proved under different circumstances
of parameter selections. Applications in hierarchical minimization problems are also included. 相似文献