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1.
Lingchen Kong Levent Tunçel Naihua Xiu 《Journal of Optimization Theory and Applications》2012,153(2):357-376
We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem. Employing an algebraic
characterization of homogeneous cones due to Vinberg from the 1960s, we generalize the properties of existence and uniqueness
of solutions for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of homogeneous
cone complementarity problem. We provide sufficient conditions for a continuous function so that the associated homogeneous
cone complementarity problems have solutions. In particular, we give sufficient conditions for a monotone continuous function
so that the associated homogeneous cone complementarity problem has a unique solution (if any). Moreover, we establish a global
error bound for the homogeneous cone complementarity problem under some conditions. 相似文献
2.
By considering the notion of regular exceptional family of elements (REFE), we define the class of REFE-acceptable mappings.
By definition, a complementarity problem on a Hilbert space defined by a REFE-acceptable mapping and a closed convex cone
has either a solution or a REFE. We present several classes of REFE-acceptable mappings. For this, neither the topological
degree nor the Leray-Schauder alternative is necessary. By using the concept of REFE-acceptable mappings, we present necessary
and sufficient conditions for the nonexistence of regular exceptional family of elements. These conditions are used for generating
several existence theorems and existence and uniqueness theorems for complementarity problems.
The authors are grateful to Prof. A.B. Németh for many helpful conversations.
The research of S.Z. Németh was supported by Hungarian Research Grants OTKA T043276 and OTKA K60480. 相似文献
3.
Exceptional Family of Elements for a Variational Inequality Problem and its Applications 总被引:1,自引:0,他引:1
This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P
* -mappings is also obtained. 相似文献
4.
M. Bianchi N. Hadjisavvas S. Schaible 《Journal of Optimization Theory and Applications》2006,129(1):23-31
In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality
problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued
variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities
corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in
more general spaces and under considerably weaker assumptions on the defining map. 相似文献
5.
This paper introduces a new concept of exceptional family for variational inequality problems with a general convex constrained
set. By using this new concept, the authors establish a general sufficient condition for the existence of a solution to the
problem. This condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational
inequalities. Suffi-cient solution conditions for a class of nonlinear complementarity problems with Po mappings are also
obtained. 相似文献
6.
Nagurney (1999) used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney (1999). It is a simpler model than that of Nagurney (1999). We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney (]999). Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results. 相似文献
7.
Ren-you Zhong Huan-xia Lian Jiang-hua Fan 《Journal of Optimization Theory and Applications》2013,159(2):341-359
In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained. 相似文献
8.
Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities 总被引:3,自引:0,他引:3
This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle. 相似文献
9.
Qing-Jie Hu Zi-Sheng Ouyang Zhong-Mei Wang 《Journal of Mathematical Analysis and Applications》2012,388(1):519-524
In this paper, by extending the concept of exceptional family to complementarity problems over the cone of symmetric copositive real matrices, we propose an existence theorem of a solution to the copositive complementarity problem. Extensions of Isac–Carbone?s condition, Karamardian?s condition, weakly properness and coercivity are also introduced. Several applications of these results are presented, and we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone copositive complementarity problems. 相似文献
10.
J. S. Pang 《Journal of Optimization Theory and Applications》1986,49(1):107-134
In an earlier paper, the author has given some necessary and sufficient conditions for the convergence of iterative methods for solving the linear complementarity problem. These conditions may be viewed as global in the sense that they apply to the methods regardless of the constant vector in the linear complementarity problem. More precisely, the conditions characterize a certain class of matrices for which the iterative methods will converge, in a certain sense, to a solution of the linear complementarity problem for all constant vectors. In this paper, we improve on our previous results and establish necessary and sufficient conditions for the convergence of iterative methods for solving each individual linear complementarity problem with a fixed constant vector. Unlike the earlier paper, our present analysis applies only to the symmetric linear complementarity problem. Various applications to a strictly convex quadratic program are also given.The author gratefully acknowledges several stimulating conversations with Professor O. Mangasarian on the subject of this paper. He is also grateful to a referee, who has suggested Lemma 2.2 and the present (stronger) version of Theorem 2.1 as well as several other constructive comments.This research was based on work supported by the National Science Foundation under Grant No. ECS-81-14571, sponsored by the United States Army under Contract No. DAAG29-80-C-0041, and was completed while the author was visiting the Mathematics Research Center at the University of Wisconsin, Madison, Wisconsin. 相似文献
11.
G. Isac 《Journal of Global Optimization》2005,31(3):405-420
In several recent papers we obtained existence theorems for complementarity problems and variational inequalities using for each of them a particular notion of exceptional family of elements. Now, in this paper we introduce a new notion of exceptional family of elements. This notion is based on an Implicit Leray-Schauder Alternative. By this new notion we obtain a unification of the study of solvability of complementarity problems and of variational inequalities. The paper is finished with a section dedicated to variational inequalities with δ-pseudomonotone operators. 相似文献
12.
In this paper, we introduce a novel sufficient existence condition that has no exceptional family of elements for semidefinite complementarity problem. In addition, we give a particular example to show that the new condition is not stronger than Isac–Carbone’s condition. 相似文献
13.
14.
By employing the notion of exceptional family of elements, we establish some existence results for generalized variational
inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence
of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For
quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the
pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization
of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the
pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution
if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is
strictly feasible. 相似文献
15.
16.
Z.H. Huang 《Journal of Optimization Theory and Applications》2003,118(3):567-585
By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result. 相似文献
17.
Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints
N. Movahedian 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(5):2694-2705
In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Then, we derive a necessary optimality result for nonsmooth MPEC on any Asplund space. Also, under generalized convexity assumptions, we establish sufficient optimality conditions for this program in Banach spaces. 相似文献
18.
李永昆 《数学物理学报(A辑)》1997,(Z1)
该文建立了周期时滞Logistic方程N'(t)=r(t)N(t)[1-N(t-τ)/k(t)]的正周期解的存在性,并获得了正周期解的唯一性和全局吸引性的充分条件.所得结果推广和改进了[1]的结果. 相似文献
19.
In this paper, on the base of the methodology of the new modulus-based matrix splitting method in [Optim. Lett., (2022) 16:1427-1443], we establish a two-step matrix splitting (TMS) method for solving the mixed linear complementarity problem (MLCP). Two sufficient conditions to ensure the convergence of the proposed method are presented. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed method. 相似文献
20.
V.Yu. Protasov 《Linear algebra and its applications》2010,433(4):781-789
We establish a criterion for a finite family of matrices to possess a common invariant cone. The criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family. In spite of theoretical simplicity, the practical use of the criterion may be difficult. We show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable. Corollaries of the criterion, which give sufficient and necessary conditions, are derived. Finally, we introduce a “co-directional number” of several matrices. We prove that this parameter is close to zero iff there is a small perturbation of matrices, after which they get an invariant cone. An algorithm for its computation is presented. 相似文献