In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the mult...In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.展开更多
This article is concerned with second-order necessary and sufficient optimality conditions for optimal control problems governed by 3-dimensional Navier-Stokes equations. The periodic state constraint is considered.
New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebr...New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebraic operation. The examples of (V, ρ)_(h,φ)-type I functions are given. The sufficient optimality conditions are obtained for multi-objective programming problem involving above new generalized convexity.展开更多
In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These co...In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.展开更多
There are two approaches of defining the solutions of a set-valued optimization problem: vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions...There are two approaches of defining the solutions of a set-valued optimization problem: vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives. In the case of vector criterion, some optimality conditions are derived for isolated (weak) minimizers. With set criterion, necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.展开更多
The definition of generalized unified (C, α, ρ, d)-convex function is given. The concepts of generalized unified (C, α, ρ, d)-quasiconvexity, generalized unified (C, α, ρ, d)-pseudoconvexity and generalized unif...The definition of generalized unified (C, α, ρ, d)-convex function is given. The concepts of generalized unified (C, α, ρ, d)-quasiconvexity, generalized unified (C, α, ρ, d)-pseudoconvexity and generalized unified (C, α, ρ, d)-strictly pseudoconvex functions are presented. The sufficient optimality conditions for multiobjective nonsmooth semi-infinite programming are obtained involving these generalized convexity lastly.展开更多
This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for...This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for Henig effcient solutions of set-valued optimization problems whose constraint condition is determined by a fixed set.展开更多
In this paper, by using the notion of convexificator, we introduce the generalized standard Abadie constraint qualification and the generalized MPVC Abadie constraint qualification, and define the generalized stationa...In this paper, by using the notion of convexificator, we introduce the generalized standard Abadie constraint qualification and the generalized MPVC Abadie constraint qualification, and define the generalized stationary conditions for the nonsmooth mathematical program with vanishing constraints (MPVC for short). We show that the generalized strong stationary is the first order necessary optimality condition for nonsmooth MPVC under the generalized standard Abadie constraint qualification. Sufficient conditions for global or local optimality for nonsmooth MPVC are also derived under some generalized convexity assumptions.展开更多
The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz Joh...The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.展开更多
Quadratic 0-1 problems with linear inequality constraints are briefly considered in this paper.Global optimality conditions for these problems,including a necessary condition and some sufficient conditions,are present...Quadratic 0-1 problems with linear inequality constraints are briefly considered in this paper.Global optimality conditions for these problems,including a necessary condition and some sufficient conditions,are presented.The necessary condition is expressed without dual variables.The relations between the global optimal solutions of nonconvex quadratic 0-1 problems and the associated relaxed convex problems are also studied.展开更多
This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators ...This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of (F, α, ρ, d)-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.展开更多
In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective...In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective programming and its saddle point theorem about cone efficient solution. We set up Mond-Weir type duality and Craven type duality for nonsmooth multiobjective programming with generalized essentially convex functions, and prove them.展开更多
This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex func...This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are given.Sufficient optimality conditions are obtained under pseudo-quasi-type-I generalized convexity assumption.Furthermore,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle points.The main results in the present paper are verified by concrete examples.展开更多
This paper obtains sufficient optimality conditions for a nonlinear nondifferentiable multiobjective semi-infinite programming problem involving generalized(C,α,ρ,d)-convex functions.The authors formulate Mond-Weir-...This paper obtains sufficient optimality conditions for a nonlinear nondifferentiable multiobjective semi-infinite programming problem involving generalized(C,α,ρ,d)-convex functions.The authors formulate Mond-Weir-type dual model for the nonlinear nondifferentiable multiobjective semiinfinite programming problem and establish weak,strong and strict converse duality theorems relating the primal and the dual problems.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
We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presentin...We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.展开更多
This paper is devoted to developing first-order necessary,second-order necessary,and second-order sufficient optimality conditions for a multiobjective optimization problem whose order is induced by a finite product o...This paper is devoted to developing first-order necessary,second-order necessary,and second-order sufficient optimality conditions for a multiobjective optimization problem whose order is induced by a finite product of second-order cones(here named as Q-multiobjective optimization problem).For an abstract-constrained Q-multiobjective optimization problem,we derive two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions.For Q-multiobjective optimization problem with explicit constraints,we demonstrate first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as second-order sufficient optimality conditions under upper second-order regularity for the explicit constraints.As applications,we obtain optimality conditions for polyhedral conic,second-order conic,and semi-definite conic Q-multiobjective optimization problems.展开更多
In this paper, we discuss a large number of sets of global parametric sufficient optimality conditions under various generalized (η,ρ)-invexity assumptions for a semi-infinite minmax fractional programming problem.
In this paper, we investigate the optimality conditions of a class of special nonsmoothprogramming min F(x) = which arises from Li-norm optimization,where ci ∈ R is constant and fi ∈ C1,i = 1,2, ...,m. These conditi...In this paper, we investigate the optimality conditions of a class of special nonsmoothprogramming min F(x) = which arises from Li-norm optimization,where ci ∈ R is constant and fi ∈ C1,i = 1,2, ...,m. These conditions can easily be tested by computer.展开更多
The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qu...The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker (KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.展开更多
文摘In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.
基金This work was supported by National Natural Science Foundation of China (10401041)Natural Science Foundation of Hubei Province (2004ABA009)
文摘This article is concerned with second-order necessary and sufficient optimality conditions for optimal control problems governed by 3-dimensional Navier-Stokes equations. The periodic state constraint is considered.
基金Supported by the NSF of Shaanxi Provincial Educational Department(06JK152)
文摘New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebraic operation. The examples of (V, ρ)_(h,φ)-type I functions are given. The sufficient optimality conditions are obtained for multi-objective programming problem involving above new generalized convexity.
文摘In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.
基金Supported by the National Natural Science Foundation of China(11361001)Natural Science Foundation of Ningxia(NZ14101)
文摘There are two approaches of defining the solutions of a set-valued optimization problem: vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives. In the case of vector criterion, some optimality conditions are derived for isolated (weak) minimizers. With set criterion, necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.
基金Supported by the Science Foundation of Shaanxi Provincial Educational Department Natural Science Foundation of China(06JK152) Supported by the Graduate Innovation Project of Yanan uni- versity(YCX201003)
文摘The definition of generalized unified (C, α, ρ, d)-convex function is given. The concepts of generalized unified (C, α, ρ, d)-quasiconvexity, generalized unified (C, α, ρ, d)-pseudoconvexity and generalized unified (C, α, ρ, d)-strictly pseudoconvex functions are presented. The sufficient optimality conditions for multiobjective nonsmooth semi-infinite programming are obtained involving these generalized convexity lastly.
基金Supported by the National Natural Science Foundation of China(10871216) Supported by the Science and Technology Research Project of Chongqing Municipal Education Commission(KJ100419) Supported by the Natural Science Foundation Project of CQ CSTC(cstcjjA00019)
文摘This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for Henig effcient solutions of set-valued optimization problems whose constraint condition is determined by a fixed set.
文摘In this paper, by using the notion of convexificator, we introduce the generalized standard Abadie constraint qualification and the generalized MPVC Abadie constraint qualification, and define the generalized stationary conditions for the nonsmooth mathematical program with vanishing constraints (MPVC for short). We show that the generalized strong stationary is the first order necessary optimality condition for nonsmooth MPVC under the generalized standard Abadie constraint qualification. Sufficient conditions for global or local optimality for nonsmooth MPVC are also derived under some generalized convexity assumptions.
基金the National Natural Science Foundation(69972036) and the Natural Science Foundation of Shanxi province(995L02)
文摘The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.
文摘Quadratic 0-1 problems with linear inequality constraints are briefly considered in this paper.Global optimality conditions for these problems,including a necessary condition and some sufficient conditions,are presented.The necessary condition is expressed without dual variables.The relations between the global optimal solutions of nonconvex quadratic 0-1 problems and the associated relaxed convex problems are also studied.
基金Supported by Chongqing Key Lab. of Operations Research and System Engineering
文摘This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of (F, α, ρ, d)-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.
文摘In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective programming and its saddle point theorem about cone efficient solution. We set up Mond-Weir type duality and Craven type duality for nonsmooth multiobjective programming with generalized essentially convex functions, and prove them.
基金supported by Natural Science Foundation of China(Nos.11861002 and 12171601)the Key Project of North Minzu University(No.ZDZX201804)+1 种基金the Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)the Postgraduate Innovation Project of North Minzu Universit(No.YCX21157)..
文摘This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are given.Sufficient optimality conditions are obtained under pseudo-quasi-type-I generalized convexity assumption.Furthermore,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle points.The main results in the present paper are verified by concrete examples.
文摘This paper obtains sufficient optimality conditions for a nonlinear nondifferentiable multiobjective semi-infinite programming problem involving generalized(C,α,ρ,d)-convex functions.The authors formulate Mond-Weir-type dual model for the nonlinear nondifferentiable multiobjective semiinfinite programming problem and establish weak,strong and strict converse duality theorems relating the primal and the dual problems.
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a post-doctoral fellowship within the Department of Mathematics of the University of Haifa and by FAPERJ (Grant No.E-26/152.107/1990-Bolsa)Partially supported by CNP_q (Grant No.301280/86).Partially supported by CNP_q (Grant No.3002748/2002-4)
文摘We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.
基金This work was supported by the National Natural Science Foundation of China(Nos.11571059,11731013 and 91330206).
文摘This paper is devoted to developing first-order necessary,second-order necessary,and second-order sufficient optimality conditions for a multiobjective optimization problem whose order is induced by a finite product of second-order cones(here named as Q-multiobjective optimization problem).For an abstract-constrained Q-multiobjective optimization problem,we derive two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions.For Q-multiobjective optimization problem with explicit constraints,we demonstrate first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as second-order sufficient optimality conditions under upper second-order regularity for the explicit constraints.As applications,we obtain optimality conditions for polyhedral conic,second-order conic,and semi-definite conic Q-multiobjective optimization problems.
文摘In this paper, we discuss a large number of sets of global parametric sufficient optimality conditions under various generalized (η,ρ)-invexity assumptions for a semi-infinite minmax fractional programming problem.
基金The project was supported by NNSFC(No. 19771047) and NSFJS (BK97059).
文摘In this paper, we investigate the optimality conditions of a class of special nonsmoothprogramming min F(x) = which arises from Li-norm optimization,where ci ∈ R is constant and fi ∈ C1,i = 1,2, ...,m. These conditions can easily be tested by computer.
基金supported by National Natural Science Foundation of China(Grant No.11431002)Shandong Province Natural Science Foundation(Grant No.ZR2016AM07)
文摘The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker (KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.