Some properties for convex cones are discussed, which are used to obtain an equivalent condition and another important property for nearly cone-subconvexlike set-valued functions. Under the nearly cone-subconvexlikene...Some properties for convex cones are discussed, which are used to obtain an equivalent condition and another important property for nearly cone-subconvexlike set-valued functions. Under the nearly cone-subconvexlikeness, some characterizations of the super efficiency are given in terms of scalarization and Lagrangian multipliers. Related results are generalized.展开更多
Some properties of convex cones are obtained and are used to derive several equivalent conditions as well as another important property for nearly cone-subconvexlike set-valued functions. Under the assumption of nearl...Some properties of convex cones are obtained and are used to derive several equivalent conditions as well as another important property for nearly cone-subconvexlike set-valued functions. Under the assumption of nearly cone-subconvexlikeness,a Lagrangian multiplier theorem on Benson proper efficiency is presented. Related results are generalized.展开更多
The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a gene...The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.展开更多
In this paper,we prove existence results of soutions for the nonlinear implicit complementarity problems NICP(T,S,K) where K is a closed weakly locally compact convex cone in a reflexive Banach space E,T is a nonlinea...In this paper,we prove existence results of soutions for the nonlinear implicit complementarity problems NICP(T,S,K) where K is a closed weakly locally compact convex cone in a reflexive Banach space E,T is a nonlinear operator from K into E* (i. e.,the dual space of E) and S is a nonlinear operator from K into E. Our results are the essential improvements and extension of the results obtained previously by several authors including Thera,Ding,and Zeng.展开更多
In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 4...In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.展开更多
In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epideriva...In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.展开更多
Nonlinear convex cone programming(NCCP)models have found many practical applications.In this paper,we introduce a flexible first-order primal-dual algorithm,called the variant auxiliary problem principle(VAPP),for sol...Nonlinear convex cone programming(NCCP)models have found many practical applications.In this paper,we introduce a flexible first-order primal-dual algorithm,called the variant auxiliary problem principle(VAPP),for solving NCCP problems when the objective function and constraints are convex but may be nonsmooth.At each iteration,VAPP generates a nonlinear approximation of the primal augmented Lagrangian model.The approximation incorporates both linearization and a distance-like proximal term,and then the iterations of VAPP are shown to possess a decomposition property for NCCP.Motivated by recent applications in big data analytics,there has been a growing interest in the convergence rate analysis of algorithms with parallel computing capabilities for large scale optimization problems.We establish O(1/t)convergence rate towards primal optimality,feasibility and dual optimality.By adaptively setting parameters at different iterations,we show an O(1/t2)rate for the strongly convex case.Finally,we discuss some issues in the implementation of VAPP.展开更多
This paper studies the known density theorem of Arrow- Barankin-Blackwell. T he following main result is obtained: If X is a Hausdorff locally convex topological space and C X is a closed convex cone with bounded bas...This paper studies the known density theorem of Arrow- Barankin-Blackwell. T he following main result is obtained: If X is a Hausdorff locally convex topological space and C X is a closed convex cone with bounded base, then for every nonempty weakly compact convex subset A, the set of positive proper efficient points of A is dense in the set of efficient points of A.展开更多
This article proposes a method for fitting models subject to a convex and log-convex constraint on the probability vector of a product multinomial (binomial) distribution. We present an iterative algorithm for findi...This article proposes a method for fitting models subject to a convex and log-convex constraint on the probability vector of a product multinomial (binomial) distribution. We present an iterative algorithm for finding the restricted maximum likelihood estimates (MLEs) of the probability vector and show that the algorithm converges to the true solution. Some examples are discussed to illustrate the method.展开更多
Within the context of cone-ordered topological vector spaces, this paper introduces the concepts of cone bounded point and cone bounded set for vector set. With their aid, a class of new cone quasiconvex mappings in t...Within the context of cone-ordered topological vector spaces, this paper introduces the concepts of cone bounded point and cone bounded set for vector set. With their aid, a class of new cone quasiconvex mappings in topological vector spaces is dened, and their fundamental properties are presented. The relationships between the cone bounded quasiconvex mapping defined in this paper and cone convex mapping, and other known cone quasiconvex mapping are also discussed.展开更多
In this paper we study the connection between the metric projection operator PK : B →K, where B is a reflexive Banach space with dual space B^* and K is a non-empty closed convex subset of B, and the generalized pr...In this paper we study the connection between the metric projection operator PK : B →K, where B is a reflexive Banach space with dual space B^* and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K : B → K and πK : B^* → K. We also present some results in non-reflexive Banach spaces.展开更多
In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order pr...In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle.J. Math. Anal. Appl., 419, 904–937(2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for-efficient solutions in the sense of N′emeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.展开更多
文摘Some properties for convex cones are discussed, which are used to obtain an equivalent condition and another important property for nearly cone-subconvexlike set-valued functions. Under the nearly cone-subconvexlikeness, some characterizations of the super efficiency are given in terms of scalarization and Lagrangian multipliers. Related results are generalized.
文摘Some properties of convex cones are obtained and are used to derive several equivalent conditions as well as another important property for nearly cone-subconvexlike set-valued functions. Under the assumption of nearly cone-subconvexlikeness,a Lagrangian multiplier theorem on Benson proper efficiency is presented. Related results are generalized.
基金the National Natural Science Foundation of China(10571035)
文摘The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.
文摘In this paper,we prove existence results of soutions for the nonlinear implicit complementarity problems NICP(T,S,K) where K is a closed weakly locally compact convex cone in a reflexive Banach space E,T is a nonlinear operator from K into E* (i. e.,the dual space of E) and S is a nonlinear operator from K into E. Our results are the essential improvements and extension of the results obtained previously by several authors including Thera,Ding,and Zeng.
文摘In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.
基金supported by the National Nature Science Foundation of China(11431004,11471291)the General Project of Chongqing Frontier and Applied Foundation Research(cstc2015jcyj A00050)the Key Project of Chongqing Frontier and Applied Foundation Research(cstc2017jcyj BX0055,cstc2015jcyj BX0113)
文摘In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.
基金This research was supported by the National Natural Science Foundation of China(Nos.71471112 and 71871140).
文摘Nonlinear convex cone programming(NCCP)models have found many practical applications.In this paper,we introduce a flexible first-order primal-dual algorithm,called the variant auxiliary problem principle(VAPP),for solving NCCP problems when the objective function and constraints are convex but may be nonsmooth.At each iteration,VAPP generates a nonlinear approximation of the primal augmented Lagrangian model.The approximation incorporates both linearization and a distance-like proximal term,and then the iterations of VAPP are shown to possess a decomposition property for NCCP.Motivated by recent applications in big data analytics,there has been a growing interest in the convergence rate analysis of algorithms with parallel computing capabilities for large scale optimization problems.We establish O(1/t)convergence rate towards primal optimality,feasibility and dual optimality.By adaptively setting parameters at different iterations,we show an O(1/t2)rate for the strongly convex case.Finally,we discuss some issues in the implementation of VAPP.
文摘This paper studies the known density theorem of Arrow- Barankin-Blackwell. T he following main result is obtained: If X is a Hausdorff locally convex topological space and C X is a closed convex cone with bounded base, then for every nonempty weakly compact convex subset A, the set of positive proper efficient points of A is dense in the set of efficient points of A.
基金Supported by the National Natural Science Foundation of China(No.11071008)Scientific Foundations of Beijing Jiaotong University(No.2012JBM105)
文摘This article proposes a method for fitting models subject to a convex and log-convex constraint on the probability vector of a product multinomial (binomial) distribution. We present an iterative algorithm for finding the restricted maximum likelihood estimates (MLEs) of the probability vector and show that the algorithm converges to the true solution. Some examples are discussed to illustrate the method.
基金Supported by the National Natural Science Foundation of China (No.70071026).
文摘Within the context of cone-ordered topological vector spaces, this paper introduces the concepts of cone bounded point and cone bounded set for vector set. With their aid, a class of new cone quasiconvex mappings in topological vector spaces is dened, and their fundamental properties are presented. The relationships between the cone bounded quasiconvex mapping defined in this paper and cone convex mapping, and other known cone quasiconvex mapping are also discussed.
文摘In this paper we study the connection between the metric projection operator PK : B →K, where B is a reflexive Banach space with dual space B^* and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K : B → K and πK : B^* → K. We also present some results in non-reflexive Banach spaces.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11471236 and 11561049)
文摘In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle.J. Math. Anal. Appl., 419, 904–937(2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for-efficient solutions in the sense of N′emeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.