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optimality conditions of the Set-valued Optimization Problem with Generalized Cone Convex Set-valued Maps Characterized by Contingent Epiderivative

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Acta Mathematicae Applicatae Sinica 2018年第1期34卷 11-18页

作者： Zhi-ang ZHOU Xin-min YANG Qiu-sheng QIU College of Sciences Chongqing University of Technology Chongqing 400054 China College of Mathematical Sciences Chongqing Normal University Chongqing 400047 China Department of Mathematics Zhejiang Normal University Jinhua 321004 China

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.

关键词： set-valued maps generalized cone convexity Henig proper efficiency contingent epiderivative optimality conditions

optimality conditions in set optimization employing higher-order radial derivatives

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Applied Mathematics(A Journal of Chinese Universities) 2017年第2期32卷 225-236页

作者： YU Guo-lin Department of Mathematics Beifang University of NationalitiesYinchuan 750021China.

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.

关键词： higher-order radial derivative optimality conditions set-valued optimization vector criterion,set criterion.

optimality conditions OF A CLASS OF SPECIAL NONSMOOTH PROGRAMMING

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Journal of Computational Mathematics 2003年第6期21卷 791-800页

作者： Song-baiSheng Hui-fuXu CollegesofScience NanjingUniversityofAeronauticsandAstronauticsNanjing210016China AustralianGraduateSchoolofManagement theUniversityofNewSouthWalesSydneyAustralia

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... 详细信息

关键词： Generalized gradient Directional derivative optimality conditions Nons-mooth programming.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

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Open Journal of Optimization 2024年第1期13卷 1-20页

作者： Twum B. Stephen Avoka John Christian J. Etwire School of Mathematical Science C. K. Tedam University of Technology and Applied Sciences Navrongo Ghana Department of Mathematics University for Development Studies Tamale Ghana

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

关键词： Quadratic Programming Lagrangian Function Lagrange Multipliers optimality conditions Subsidiary Equations Modified Lagrange Method

Second-Order optimality conditions for Multiobjective Optimization Whose Order Induced by Second-Order Cone

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Journal of the Operations Research Society of China 2018年第2期6卷 267-288页

作者： Li-Wei Zhang Ji-Hong Zhang Yu-Le Zhang School of Mathematical Sciences Dalian University of TechnologyDalian 116024LiaoningChina

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.

关键词： Second-order cone-induced multiobjective optimization optimality conditions Polyhedral cone Second-order cone Semi-definite cone

THE CHECK OF optimality conditions IN NONSMOOTH PROGRAMMING

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Applied Mathematics(A Journal of Chinese Universities) 1998年第3期13卷 347-350页

作者： Yang Qinghi Department of mathematics the Graduate School Chinese University of Science and Technology Beijing

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关键词： 90C30 Nonsmooth programming optimality conditions semismooth regularity

The optimality conditions for Multiobjective Semi-infinite Programming Involving Generalized Unified （C, α, p, d）-convexity

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Chinese Quarterly Journal of Mathematics 2013年第2期28卷 241-249页

作者： ZHANG Qing-xiang ZHANG Yong-zhan College of Mathematics and Computer Science Yan'an University Yan'an 716000 China

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.

关键词： generalized convexity multiobjective semi-infinite programming efficient solution optimality conditions

Medium-term scheduling of operating states in electric power systems

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Global Energy Interconnection 2022年第5期5卷 472-483页

作者： Aizenberg N. Palamarchuk S. Melentiev Energy Systems Institute Siberian Branch of the Russian Academy of Sciences664033130 Lermontov st.IrkutskRussia

This article presents a mathematical model for the medium-term scheduling of the operating states of electric power systems.The scheduling period is divided into several time intervals.The model can be used to determine the equilibrium state in which each supplier earns maximum profit from supplying electricity to the wholesale market.We estimated the maximum value of public welfare,which indicates the total financial gains of suppliers and consumers,to determine the prices at the nodes of the power system.This was done by considering the balance constraints at the nodes of the power system and constraints on the allowable values of generation,power flows,and volumes of energy resources consumed over several time intervals.This problem belongs to the class of bi-level Stackelberg game-theoretic models with several leaders.The market equilibrium is modeled simultaneously in several intervals,given the multiplicity and duration of interactions.We considered two approaches for solving the multi-interval equilibrium state problem.The first approach involved directly solving a system of joint optimality conditions for electricity suppliers and consumers.The second approach involved iterative searches until the equilibrium state was reached.This article presents the results of medium-term scheduling using a case study of a simplified real-world power system.

关键词： Scheduling Mathematical model optimality conditions Imperfect competition Equilibrium state Nodal prices.

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

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Journal of the Operations Research Society of China 2023年第3期11卷 673-692页

作者： S.Khoshkhabar-amiranloo School of Mathematics Institute for Research in Fundamental SciencesTehranIran

This paper deals with approximate weak minimal solutions of set-valued optimization problems under vector and set optimality criteria.The relationships between various concepts of approximate weak minimal solutions are investigated.Some topological properties and existence theorems of these solutions are given.It is shown that for set-valued optimization problems with upper(outer)cone-semicontinuous objective values or closed objective maps the approximate weak minimal and strictly approximate lower weak minimal solution sets are closed.By using the polar cone and two scalarization processes,some necessary and sufficient optimality conditions in the sense of vector and set criteria are provided.

关键词： Set-valued optimization Approximate weak minimal solutions Existence theorems optimality conditions Scalarization functions