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Existence and Well-posedness of Solutions for Vector Optimization and Related Problems

Author: ZengJing
Tutor: LiShengJie
School: Chongqing University
Course: Computational Mathematics
Keywords: Vector optimization problem Well-posedness Nonlinear scalarization function P.K. convergence Error estimates
CLC: O224
Type: PhD thesis
Year: 2011
Downloads: 95
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Abstract


In this thesis, we establish some solution existence results and well-posedness for vector optimization problems and some related problems. This thesis is divided into nine chapters. It is organized as follows:In Chapter 1, we describe the development and current researches on the topic of existence results and well-posedness for vector optimization problems. We also give the motivation and the main research work.In Chapter 2, we introduce some basic notions, definitions and propositions, which will be used in the sequel.In Chapter 3, we obtain a general Ekeland’s variational principle for set-valued mappings in a complete metric space, which is different from those in [3, 4, 10]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex compact and nonconvex noncompact assumptions of its domain, respective-ly.In Chapter 4, some solution relationships between set-valued optimization prob-lems and vector variational-like inequalities are established under generalized invexities. In addition, a generalized Lagrange multiplier rule for a constrained set-valued optimi-zation problem is obtained under C-preinvexity.In Chapter 5, we discuss the wellposedness and stability of the sets of efficient points of vector-valued optimization problems when the data of the approximate prob-lems converges to the data of the original problem in the sense of Painlevé–Kuratowski. Our results improve the corresponding results obtained by Lucchetti and Miglierina [11, Section 3].In Chapter 6, we introduce a kind of extended Hadamard-type well-posedness for set-valued optimization problems. By virtue of a scalarization function, we obtain some solution relationships between the set-valued optimization problem and a scalar optimi-zation problem. Then, we derive a scalarization theorem of P.K. convergence for se-quences of set-valued mappings. Based on these results, we also establish a sufficient condition of extended Hadamard-type well-posedness for the set-valued optimization problems.In Chapter 7, we introduce a kind of extended Hadamard-type well-posedness for vector equilibrium problems. By virtue of a scalarization function, we also establish a sufficient condition of extended Hadamard-type well-posedness for the vector equili-brium problems.In Chapter 8, we obtain some results on error estimates of approximate solutions to parametric vector quasiequilibrium problems in metric spaces. Under some special cas-es, the error estimates are equivalent to H?lder stability or Lipschitz stability of the set-valued solution map at a given point. An application to variational inequalities is al-so presented.In Chapter 9, we summarize the results of this thesis and make some discussions.

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