The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex...The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.展开更多
In this paper, we introduce some new systems of generalized vector quasi-variational inclusion problems and system of generalized vector ideal （resp., proper, Pareto, weak） quasi-optimization problems in locally FC-...In this paper, we introduce some new systems of generalized vector quasi-variational inclusion problems and system of generalized vector ideal （resp., proper, Pareto, weak） quasi-optimization problems in locally FC-uniform spaces without convexity structure. By using the KKM type theorem and Himmelberg type fixed point theorem proposed by the author, some new existence theorems of solutions for the systems of generalized vector quasi-variational inclusion problems are proved. As to its applications, we obtain some existence results of solutions for systems of generalized vector quasi-optimization problems.展开更多
In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-...In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness. Moreover, after a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type.展开更多
Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions...Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.展开更多
We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three t...We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.展开更多
With the help of a level mapping,this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems.First,we introduce a kind of level mapping which generali...With the help of a level mapping,this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems.First,we introduce a kind of level mapping which generalizes one given in Han and Gong(Optimization 65:1337–1347,2016).Then,we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping.Finally,in terms of the semicontinuity of the level mapping,we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric setvalued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.展开更多
The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of...The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of the marginal map and the approximate value in VOPEC (ε) for vector-valued mapping.展开更多
Support vector machine has become an increasingly popular tool for machine learning tasks involving classification, regression or novelty detection. Training a support vector machine requires the solution of a very la...Support vector machine has become an increasingly popular tool for machine learning tasks involving classification, regression or novelty detection. Training a support vector machine requires the solution of a very large quadratic programming problem. Traditional optimization methods cannot be directly applied due to memory restrictions. Up to now, several approaches exist for circumventing the above shortcomings and work well. Another learning algorithm, particle swarm optimization, for training SVM is introduted. The method is tested on UCI datasets.展开更多
The purpose of this paper is to devise exact l_(1) exponential penalty function method to solve multiobjective optimization problems with exponentialtype invexity.The conditions governing the equivalence of the(weak)...The purpose of this paper is to devise exact l_(1) exponential penalty function method to solve multiobjective optimization problems with exponentialtype invexity.The conditions governing the equivalence of the(weak)efficient solutions to the vector optimization problem and the(weak)efficient solutions to associated unconstrained exponential penalized multiobjective optimization problem are studied.Examples are given to illustrate the obtained results.展开更多
The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of mini- mizing a vector function whose each component is the sum of a differentiable function and a convex function, subjcct to a...The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of mini- mizing a vector function whose each component is the sum of a differentiable function and a convex function, subjcct to a set of differentiable nonlinear inequalities on a convex subset C of R^n, under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification.展开更多
This paper establishes some suffcient conditions for the lower semicontinuity of the effcient solution mapping for the semi-infinite vector optimization problem with perturbations of both the objective function and th...This paper establishes some suffcient conditions for the lower semicontinuity of the effcient solution mapping for the semi-infinite vector optimization problem with perturbations of both the objective function and the constraint set in normed linear spaces. The constraint set is the set of weakly effcient solutions of vector equilibrium problem, and perturbed by the perturbation of the criterion mapping to the vector equilibrium problem.展开更多
In this paper, we introduce a new class of generalized convex function, namely, a-pseudounivex function, by combining the concepts of pseudo-univex and α-invex functions. Further, we establish some relationships betw...In this paper, we introduce a new class of generalized convex function, namely, a-pseudounivex function, by combining the concepts of pseudo-univex and α-invex functions. Further, we establish some relationships between vector variational-like inequality problems and vector optimization problems under the assumptions of α-pseudo-univex functions. Results obtained in this paper present a refinement and improvement of previously known results.展开更多
在不需要紧性假设下,利用拟C-凸函数及回收锥的性质,建立了向量优化问题有效点集的稳定性,获得了一列目标函数和可行集均扰动情形下的向量优化问题与对应的向量优化问题有效点集的Painlevé-Kuratowski内收敛性结果。所得结果推广...在不需要紧性假设下,利用拟C-凸函数及回收锥的性质,建立了向量优化问题有效点集的稳定性,获得了一列目标函数和可行集均扰动情形下的向量优化问题与对应的向量优化问题有效点集的Painlevé-Kuratowski内收敛性结果。所得结果推广和改进了相关文献(Attouch H,RiahiH.Stability results for Ekeland’s-variational principle and cone extremal solution;Huang X X.Stabilityin vector-valued and set-valued optimization)中的相应结果,并给出例子说明了所得结果的正确性。展开更多
文摘The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.
基金supported by the Natural Science Foundation of Sichuan Education Department of China(No. 07ZA092)the Sichuan Province Leading Academic Discipline Project (No. SZD0406)
文摘In this paper, we introduce some new systems of generalized vector quasi-variational inclusion problems and system of generalized vector ideal （resp., proper, Pareto, weak） quasi-optimization problems in locally FC-uniform spaces without convexity structure. By using the KKM type theorem and Himmelberg type fixed point theorem proposed by the author, some new existence theorems of solutions for the systems of generalized vector quasi-variational inclusion problems are proved. As to its applications, we obtain some existence results of solutions for systems of generalized vector quasi-optimization problems.
文摘In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness. Moreover, after a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type.
文摘Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.
文摘We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.
基金This research was supported by the National Natural Science Foundation of China(No.11801051).
文摘With the help of a level mapping,this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems.First,we introduce a kind of level mapping which generalizes one given in Han and Gong(Optimization 65:1337–1347,2016).Then,we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping.Finally,in terms of the semicontinuity of the level mapping,we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric setvalued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.
文摘The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of the marginal map and the approximate value in VOPEC (ε) for vector-valued mapping.
文摘Support vector machine has become an increasingly popular tool for machine learning tasks involving classification, regression or novelty detection. Training a support vector machine requires the solution of a very large quadratic programming problem. Traditional optimization methods cannot be directly applied due to memory restrictions. Up to now, several approaches exist for circumventing the above shortcomings and work well. Another learning algorithm, particle swarm optimization, for training SVM is introduted. The method is tested on UCI datasets.
基金The research of the first author is financially supported by the University Grant Commission,New Delhi,India(No.41-801/2012(SR)).
文摘The purpose of this paper is to devise exact l_(1) exponential penalty function method to solve multiobjective optimization problems with exponentialtype invexity.The conditions governing the equivalence of the(weak)efficient solutions to the vector optimization problem and the(weak)efficient solutions to associated unconstrained exponential penalized multiobjective optimization problem are studied.Examples are given to illustrate the obtained results.
基金Supported by the National Natural Science Foundation of China(No.70671064,No.60673177)the Province Natural Science Foundation of Zhejiang(No.Y7080184)the Education Department Foundation of Zhejiang Province(No.20070306)
文摘The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of mini- mizing a vector function whose each component is the sum of a differentiable function and a convex function, subjcct to a set of differentiable nonlinear inequalities on a convex subset C of R^n, under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification.
基金supported by the National Natural Science Foundation of China under Grant Nos.1106102311201216and 11471291
文摘This paper establishes some suffcient conditions for the lower semicontinuity of the effcient solution mapping for the semi-infinite vector optimization problem with perturbations of both the objective function and the constraint set in normed linear spaces. The constraint set is the set of weakly effcient solutions of vector equilibrium problem, and perturbed by the perturbation of the criterion mapping to the vector equilibrium problem.
基金The research is supported by the Department of Science and Technology,Ministry of Science and Technology,Government of India under the SERC Fast Track Scheme for Young Scientists 2001-2002(No.SR/FTP/MS-22/2001).
文摘In this paper, we introduce a new class of generalized convex function, namely, a-pseudounivex function, by combining the concepts of pseudo-univex and α-invex functions. Further, we establish some relationships between vector variational-like inequality problems and vector optimization problems under the assumptions of α-pseudo-univex functions. Results obtained in this paper present a refinement and improvement of previously known results.
文摘在不需要紧性假设下,利用拟C-凸函数及回收锥的性质,建立了向量优化问题有效点集的稳定性,获得了一列目标函数和可行集均扰动情形下的向量优化问题与对应的向量优化问题有效点集的Painlevé-Kuratowski内收敛性结果。所得结果推广和改进了相关文献(Attouch H,RiahiH.Stability results for Ekeland’s-variational principle and cone extremal solution;Huang X X.Stabilityin vector-valued and set-valued optimization)中的相应结果,并给出例子说明了所得结果的正确性。
