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The Study on Duality Theory and Well-posedness for Optimization Problems

Author: WangHaiJun
Tutor: ChengCaoZong
School: Beijing University of Technology
Course: Probability Theory and Mathematical Statistics
Keywords: Optimization problems Fractional programming problems Conjugate duality Second order duality Parametric well-posedness
CLC: O224
Type: PhD thesis
Year: 2012
Downloads: 161
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Abstract

Duality theory is one of the most important contents in optimization prob-lems, which plays an important role in discussing the optimality conditions foroptimization problems, algorithm design and so on. In this thesis, our main aimsare to study the conjugate duality for fractional programming and vector optimiza-tion problems and the second order duality for minimax fractional programming.Besides, we also discuss the parametric well-posedness for quasivariational-likeinequality problems.In Chapter2, we discuss a type of fractional programming problem with aratio of DC functions (diference of convex functions) as objective function andfnitely many inequalities of DC functions as constraints. By using the Dinkelbachtransformation, we convert the fractional programming problem to a DC program-ming problem with the DC objective function and fnitely many DC inequalitiesconstraints. For the DC programming problem, the Fenchel-Lagrange type dualproblem is established, and the weak and strong duality theorems are proved un-der the assumptions of generalized interior point constraint qualifcation. Then wederive the Farkas type results of the fractional programming problem by using theabove duality theorems. Moreover, using the epigraphs of the involved conjugatefunctions, we give an equivalent statement for the Farkas type result.In Chapter3, we study the conjugate duality for vector optimization prob-lems. On the one hand, by using the conjugate duality approach for convex scalaroptimization problems, we construct two dual models for a multiobjective bilevel optimization problem, obtain the weak and strong duality theorems between theprimal problem and its dual problems based on the properly efcient solutionsand weakly efcient solutions of the primal problem, respectively. On the otherhand, we construct a conjugate dual model for a set-valued optimization problemby using the ε-conjugate map of a set-valued function, and prove some weak andstrong duality theorems between the primal problem and its dual problem. Thenwe introduce a new Lagrange function for the primal problem, and obtain somenecessary and sufcient conditions for the existence of the ε-saddle points.In Chapter4, we consider the second order duality for minimax fractionalprogramming problems. First, we introduce a new generalized second order convexfunction, which is called second order (F, α, ρ, d, p)-univex function. Then we givesome examples to illustrate that it is an extension of some known second orderconvex function in the literatures. In section2, we consider two dual problems for adiferentiable minimax fractional programming, and establish second order weak,strong and strictly converse duality theorems between the primal problem andits dual problems under the conditions of above generalized second order convexfunctions. In section3, we consider two dual problems for a nondiferentiableminimax fractional programming, and obtain the related second order dualitytheorems.In Chapter5, we discuss the parametric well-posedness for quasivariational-like inequality problems in Banach spaces. We introduce the notions of the para-metric well-posedness and the parametric well-posedness in generalized sense forquasivariational-like inequality problems. First, we establish some metric char-acterizations of the parametric well-posedness for quasivariational-like inequality problems, and prove that under suitable conditions, the parametric well-posednessis equivalent to the existence and uniqueness of their solutions. Then, we give somenecessary and efcient conditions for the parametric well-posedness in generalizedsense of quasivariational-like inequality problems by using the noncompact mea-sure of the approximating solution sets.

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