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1.
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). 相似文献
2.
X.Q. Yang 《Journal of Optimization Theory and Applications》2003,116(2):437-452
Gap functions play a crucial role in transforming a variational inequality problem into an optimization problem. Then, methods solving an optimization problem can be exploited for finding a solution of a variational inequality problem. It is known that the so-called prevariational inequality is closely related to some generalized convex functions, such as linear fractional functions. In this paper, gap functions for several kinds of prevariational inequalities are investigated. More specifically, prevariational inequalities, extended prevariational inequalities, and extended weak vector prevariational inequalities are considered. Furthermore, a class of gap functions for inequality constrained prevariational inequalities is investigated via a nonlinear Lagrangian. 相似文献
3.
XQ Yang 《Mathematical Programming》1998,81(3):327-347
In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. 相似文献
4.
Solution of Finite-Dimensional Variational Inequalities Using Smooth Optimization with Simple Bounds
R. Andreani A. Friedlander J. M. Martínez 《Journal of Optimization Theory and Applications》1997,94(3):635-657
The variational inequality problem is reduced to an optimization problem with a differentiable objective function and simple bounds. Theoretical results are proved, relating stationary points of the minimization problem to solutions of the variational inequality problem. Perturbations of the original problem are studied and an algorithm that uses the smooth minimization approach for solving monotone problems is defined. 相似文献
5.
《Optimization》2012,61(3):333-351
The paper considers two cases of variational inequality problems. The first case involves an affine monotone operator over a convex set defined by a separation oracle. Aninterior-point algorithm that mixes an interior cutting plane method and a short-step path-following method will be presented. Its complexity will be established. The second case is an extension of the first and involves a nonlinear monotone operator defined over the same type of convex set. The algorithm for the latter case is different from the former one only in the path-following stage 相似文献
6.
General algorithm for variational inequalities 总被引:7,自引:0,他引:7
M. A. Noor 《Journal of Optimization Theory and Applications》1992,73(2):409-413
In this paper, we consider a general auxiliary principle technique to suggest and analyze a novel and innovative iterative algorithm for solving variational inequalities and optimization problems. We also discuss the convergence criteria. 相似文献
7.
Liqun Ban Boris S. Mordukhovich 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(2):441-461
This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis. 相似文献
8.
We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operatorF, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author. 相似文献
9.
Qinghai He 《Journal of Mathematical Analysis and Applications》2007,333(2):1070-1078
In this paper, variational conclusions of set-valued bifunctions on convex subsets of Banach spaces are investigated and several new results are obtained. The results are applied to the fixed point theory and variational inequalities. We obtain two fixed point theorems and existence theorems of solutions of variational inequalities. 相似文献
10.
《Optimization》2012,61(3):233-238
A recent theorem Of W.Takahashi will be pointed out as an equivalent formulation of Ekelands variational principle. This gives rise to study functions having sets of weak sharp minimas in a generalized sense. Connections to the proximal point algorithm in the convex case leads to the basic and still open question: How to use Ekelands principle numerically 相似文献
11.
12.
In this paper, an entropy-like proximal method for the minimization of a convex function subject to positivity constraints is extended to an interior algorithm in two directions. First, to general linearly constrained convex minimization problems and second, to variational inequalities on polyhedra. For linear programming, numerical results are presented and quadratic convergence is established.Corresponding author. His research has been supported by C.E.E grants: CI1* CT 92-0046. 相似文献
13.
《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
14.
A relaxed projection method for variational inequalities 总被引:4,自引:0,他引:4
Masao Fukushima 《Mathematical Programming》1986,35(1):58-70
This paper presents a modification of the projection methods for solving variational inequality problems. Each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself. The algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions.This work was supported in part by Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan. 相似文献
15.
We present some results about Lipschitzian behavior of solutions to variational conditions when the sets over which the conditions
are posed, as well as the functions appearing in them, may vary. These results rely on calmness and inner semicontinuity,
and we describe some conditions under which those conditions hold, especially when the sets involved in the variational conditions
are convex and polyhedral. We then apply the results to find error bounds for solutions of a strongly monotone variational
inequality in which both the constraining polyhedral multifunction and the monotone operator are perturbed.
相似文献
16.
A combined relaxation method for variational inequalities with nonlinear constraints 总被引:1,自引:0,他引:1
Igor V. Konnov 《Mathematical Programming》1998,80(2):239-252
A simple iterative method for solving variational inequalities with a set-valued, nonmonotone mapping and a convex feasible set is proposed. This set can be defined by nonlinear functions. The method is based on combining and extending ideas contained in various relaxation methods of nonsmooth optimization. Also a modification of the averaging method for the problem under consideration is proposed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.This research was supported in part by RFFI grant No. 95-01-00061. 相似文献
17.
In this article, we provide a general iterative method for solving an equilibrium and a constrained convex minimization problem. By using the idea of regularized gradient-projection algorithm (RGPA), we find a common element, which is also a solution of a variational inequality problem. Then the strong convergence theorems are obtained under suitable conditions. 相似文献
18.
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
19.
本文将单位映射变分不等式和互补问题的解的存在性定理推广到集值映射上.讨论了集值映射互补问题的解与Kakutani不动点之间关系,以及集值映射互补问题的解的计算方法.最后给出了它们在不可微规划中的应用. 相似文献
20.
This paper is devoted to the study of a new necessary condition in variational inequality problems: approximated gradient
projection (AGP). A feasible point satisfies such condition if it is the limit of a sequence of the approximated solutions
of approximations of the variational problem. This condition comes from optimization where the error in the approximated solution
is measured by the projected gradient onto the approximated feasible set, which is obtained from a linearization of the constraints
with slack variables to make the current point feasible.
We state the AGP condition for variational inequality problems and show that it is necessary for a point being a solution
even without constraint qualifications (e.g., Abadie’s). Moreover, the AGP condition is sufficient in convex variational inequalities.
Sufficiency also holds for variational inequalities involving maximal monotone operators subject to the boundedness of the
vectors in the image of the operator (playing the role of the gradients). Since AGP is a condition verified by a sequence,
it is particularly interesting for iterative methods.
Research of R. Gárciga Otero was partially supported by CNPq, FAPERJ/Cientistas do Nosso Estado, and PRONEX Optimization.
Research of B.F. Svaiter was partially supported by CNPq Grants 300755/2005-8 and 475647/2006-8 and by PRONEX Optimization. 相似文献