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The Subdifferential for Set-valued Mappings and Optimality Conditions

Author: GuoXiaoLe
Tutor: LiShengJie
School: Chongqing University
Course: Computational Mathematics
Keywords: Vector optimization Subdifferential Calculus rules D.C. optimizationproblem Optimality condition
CLC: O174
Type: PhD thesis
Year: 2012
Downloads: 29
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Abstract


In this thesis, the existence theorems, some properties and some calculus rules forcalculating the subdifferential of the sum, composition and intersection of twoset-valued mappings for the several kinds of subdiferential for set-valued mappings arestudied. Optimality conditions of cone-convex vector optimization problems and specialnonconvex vector optimization problems, which are D.C. vector optimization problemsare established. A generalized ε subdifferential is introduced for a nonconvex vectorvalued mapping. And the existence theorems, the properties and the calculus rules of thegeneralized ε subdifferential for the sum and the difference of two vector valuedmappings were disscussed. And, as applications, optimality conditions are establishedfor vector optimization problems. It is organized as follows:In Chapter1, the development and current researches on the topic of vectoroptimization problems are firstly recalled. Then, the development and current researchesfor subdifferential and D.C. multiobjective optimization problems are reviewed,respectively. Finally, the motivations and the main research work are also given.In Chapter2, some basic notions and definitions of vector optimization problemsare recalled. These definitions mainly refer to contingent derivative, contingentepiderivative, different kinds of subdifferential and kinds of efficient solutions.In Chapter3, firstly, the existence theorems of two kinds of weak subgradients forset-valued mappings introduced in (Yang,1992) and (Chen and Jahn,1998), which arethe generalizations of Theorem7in (Chen and Jahn,1998) and Theorem4.1in (Peng etal,2005) respectively, are proven. Then, an existence theorem of the subgradients forset-valued mappings, which introduced by Borwein in (Borwein,1981), and therelations between this subdifferential and the subdifferential introduced by Baier andJahn in (Baier and Jahn,1998), are obtained.In Chapter4, some properties of the CJ-weak subdifferential of set-valuedmappings introduced in (Chen and Jahn,1998) and the calculus rules of the CJ-weaksubdifferential for the sum of two set-valued mappings are obtained by using a so-calledSandwich theorem. Moreover, in virtue of the property of the contingent derivative,some properties and some exact calculus rules for calculating the subdifferential of thesum, composition and intersection of two set-valued mappings are given.In Chapter5, by using the concepts of the CJ-weak subdifferential of set-valued mappings and the L-subdifferential introduced by the contingent derivative in (Li,1998),necessary and sufficient optimality conditions are discussed of cone-convex set-valuedoptimization problems, whose constraint sets are determined by a fixed set and aset-valued mapping, respectively. And some results are compared with the relatedresults in the literature.In Chapter6, by using the concept of the strong subdifferential for set-valuedmappings, the sufficient and necessary optimality conditions for generalized D.C.multiobjective optimization problems are established. And the necessary optimalityconditions are compared with the related results in the literature. Moreover, by using aspecial scalarization function, a real set-valued optimization problem is introduced andthe equivalent relations between the solutions are proved for the real set-valuedoptimization problem and a generalized D.C. multiobjective optimization problem.Then, by virtue of the concepts of the strong subdifferential and the ε-subdifferentialof vector valued mappings, sufficient and necessary optimality conditions areestablished for an ε-weak Pareto minimal point and an ε-proper Pareto minimal pointof a D.C. vector optimization problem. As an application, sufficient and necessaryoptimality conditions are also given for an ε-weak Pareto minimal point and anε-proper Pareto minimal point of a vector fractional mathematical programming.In Chapter7, a generalized ε subdifferential, which was defined by a norm, isfirst introduced for a nonconvex vector valued mapping. Some existence theorems andthe properties of the generalized ε subdifferential are discussed. A relationshipbetween the generalized ε subdifferential and a directional derivative is investigatedfor a vector valued mapping. Then, the calculus rules of the generalizedε subdifferential for the sum and the difference of two vector valued mappings weregiven. The positive homogeneity of the generalized ε subdifferential is also provided.Finally, as applications, necessary and sufficient optimality conditions are establishedfor vector optimization problems.In Chapter8, the results of this thesis are briefly summarized. And some problemswhich are remained and thought over in future are put forward.

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