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Stability Analysis of the Solution Set for Convex Optimization Problem and the Application

Author: WangXing
Tutor: FanJiangHua
School: Guangxi Normal University
Course: Applied Mathematics
Keywords: convex optimization problem convex vector optimization problem stability mixed variational inequality vector variational inequality
CLC: O177.91
Type: Master's thesis
Year: 2010
Downloads: 130
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Abstract


The theory of convex optimization has well developed in economics variational theory, mechanics and other sciences since the 1960s. In this thesis, we focus on the stability of solution set for convex optimization problem and the application. The organization of the paper is as follows:In chapter 1, we make a brief review of the relevant research in the area and summarize our main conclusions, also we introduce some basic notations and preliminaries which are used in this thesis.In chapter 2, we study the stability of the solution set for convex scalax optimization in reflexive Banach space when both the objective function and constrained set are perturbed, and we established the stability of the solution set on nonemptiness and boundedness. The main conclusions in Chapter 2 are following:Theorem 2.2.1 Let (Z1,d1),(Z2,d2) be two metric space, u0∈Z1, v0∈Z2 be given points. Let L:Z1→2X be a continuous set-valued mapping with nonempty closed convex values and int(barr(L(u0)))≠(?). Suppose that there exists a neighborhood U×V of (u0, v0) and M=∪u∈U L(u) such that (?)f:M×V→? 2X* is a lower semicontinuous set-valued mapping with nonempty values, and the subdifferential of f(·,v) is denoted by (?)f(·,v) fv(x) is proper lower semicontinuous on M with respect to x for any v∈V. If f∞v0(d)> 0, (?)d∈(L(u))∞\{0}, then there exists a neighborhood U’×V’ of (u0, v0) with U’×V’ (?) U×V, such that f∞v0(d)> 0,(?)(u,v)∈U’×V’,(?)∈(L(u))∞\{0}.Theorem 2.2.2 Let (Z1, d1), (Z2, d2) be two metric space, u0∈E Z1, v0∈Z2 be given points. Let L:Z1→2X be a continuous set-valued mapping with nonempty closed convex values and int(barr(L(u0)))≠(?). Suppose that there exists a neighborhood U x V of (u0,v0) and M=∪u∈U(u) such that (?)f:M x V→> 2X* is a lower semicontinuous set-valued mapping with nonempty values. If the objective function f:M×V→R is strongly continuous and S(uo,vo) is nonempty and bounded. Then:(i) There exists a neighborhood U’×V’ of (u0, v0) with U’×V’∈U×V, such that S(u, u) is nonempty and bounded for every (u, v)∈U’x V’.(ii)ω-limsup(u,u)→(u0,v0) S(u,v) (?)S(u0,v0). In chapter 3, we study the stability of the solution set of the mixed variational inequality in reflective Banach space by use of the conclusions in convex optimization, we obtained the stability of the solution set of the mixed variational inequality. In the process, we also established various characterizations of the nonemptiness and boundedness of the solution set. The main conclusions in Chapter 3 are following:Theorem 3.2.3 Let (Z1,d1),(Z2,d2) be two metric space, u0∈Z1, v0∈Z2 be given points. Let L:Z1→2X be a continuous set-valued mapping with nonempty closed convex values and int(barr(L(u0)))≠0. Suppose that there exists a neighborhood U x V of (u0, v0) and M=∪i∈U L(u) such that F:M×V→2X* is a lower semicontinuous set-valued mapping with nonempty values, and (?)φis lower semicontinuous set-valued mapping on M with nonempty values. If there exists y*0∈F(L(u0),v0) such that (y*0,d)+φ∞(d)> 0, (?)d∈(L(u0))∞\{0}, then there exists a neighborhood U’x V of (u0, v0) with U’x V (?) U×V and y*∈F(L(u),v) such that (y*,d)+φ∞(d)> 0,(?)(u,v)∈U’×V’,(?)d∈(L(u))∞\{0}.Theorem 3.2.4 Let (Z1, d1), (Z2,d2) be two metric space,u0∈Zi, v0∈Z2 be given points. Let L:Z1→2X be a continuous set-valued mapping with nonempty closed convex values and int(barr(L(u0)))≠0. Suppose that there exists a neighborhood U x V of (u0,v0) and M=∪u∈U L(u) such that F:M x V→2X* is a lower semicontinuous set-valued mapping with nonempty weakly compact convex values,φis strong continuous on M and (?)φis lower semicontinuous set-valued mapping on M with nonempty values.Suppose that (i) F:M×V→2X* is a weakly upper semicontinuous set-valued mapping and for each v∈V, F(x, v) is pseumonotone on M.(ii) The solution set of GMVI(F(·,vo), L(u0)) is nonempty and bounded.Then (i) there exists a neighborhood U’x V of (u0,v0) with U’x V (?) U×V, such that for every (u,v)∈U’x V’, the solution set of GMVI(F(·,vo), L(u0)) is nonempty and bounded,(ii)ω-lim sup(u,v)→(u0,v0 S(u, v) (?) S(uo,vo), where S(u, v) and S(u0, v0) are the solution of GMVI(F(·, v), L(u)) and GMVI(F(·,v0), L(u0)) respectively.In chapter 4, we study the stability of the weakly efficient solution set for convex vec-tor optimization problem in reflective Banach space when both the objective function and constrained set are perturbed, we established the stability of the solution set on nonempti-ness and boundedness. As an application, we obtain stability results for vector variational inequality in real reflexive Banach space by use of the conclusions in convex vector optimiza-tion. The main conclusions in Chapter 4 are following:Theorem 4.2.1 If the following assumptions hold:(i) (?)fi:X×Z2→2X* (i = 1,2,...,p) is a lower semicontinuous set-valued mapping with nonempty compact convex values. (ii) L:Z1→2X is a continuous set-valued mapping with nonempty closed convex values.(iii) f(·,v) is proper lower semicontinuous on X, and there exists x∈X such that x∈∩pi=1 int(domfi(·, v)) for each i∈{1,2,...,p}(iv) there exists (u0,v0)∈Z1 x Z2 such that Sw(u0,v0) is nonempty and bounded.Then there exists a neighborhood U x V of (u0,v0) such that Sw(u,v) is nonempty on U×V.Theorem 4.2.2 If the following assumptions hold:(i) (?)fi:X x Z2→2x*(i= 1,2,...,p) is a lower semicontinuous set-valued mapping with nonempty compact convex values.(ii) L:Z1→2X is a continuous set-valued mapping with nonempty closed convex values.(iii) f(·,v) is proper lower semicontinuous on X, and there exists x∈X such that x∈∩pi=1 int(domfi(·, v)) for each i∈{1,2,...,p}(iv) there exists (u0,v0)∈Z1x Z2 such that Sw(u0,v0) is nonempty and bounded, Si(u0, v0) (i= 1,2,...,p) is nonempty and Sw(u, v)=∪pi=1 Si(u,v), (?)(u,v)∈Z1 x Z2.Then there exists a neighborhood U x V of (u0,v0) such that Sw(v,v) is nonempty on U×V.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Functional Analysis > Nonlinear Functional Analysis
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