Stability And Connectedness Analysis Of Solution Set For Variational Inequalities

Variational inequality theory has wide applications in finance, economics, trans-portation, optimization, operations research, and engineering sciences. In recentyears, the stability and connectedness of variational inequality (vector variationalinequality included) have been extensively studied by many researchers. In thisthesis, we focus on stability and connectedness of vector variational inequality. Thethesis consists of four chapters.In chapter 1, we make a brief review of the current research in stability andconnectedness of variational inequality; and introduce some basic notations andpreliminaries which are used in this thesis.In chapter 2, we study the stability of a variational inequality in re?exive Ba-nach space, when both the mapping and the set are perturbed. We established thestability of the solution set of the Minty variational inequality when the mappingis properly quasimonotone; and the stability of the variational inequality as themapping is pseudomonotone. The main conclusions in chapter 2 are following:Theorem 2.2.2 Let (Z1,d1), (Z2,d2) be metric spaces, u0∈Z1, v0∈Z2 be givenpoints. Let L : Z1→2X be a continuous set-valued mapping with nonempty closedconvex values and int(barr(L(u0))) = ?. Suppose that there exists a neighborhoodU×V of (u0,v0) and M = u∈U L(u) such that F : M×V→2X? is a lowersemicontinuous set-valued mapping with nonempty values. Suppose that(i) For each v∈V , the mapping x→F(x,v) is properly quasimonotone on M.(ii) The solution set of MV IP(F(·,v0),L(u0)) is nonempty and bounded. Then (i) there exists a neighborhood U×V of (u0,v0) with U×V ? U×V , suchthat for every (u,v)∈U×V , the solution set of MV IP(F(·,v),L(u)) is nonemptyand bounded; (ii)ω? limsup(u,v)→(u0,v0) SM(u,v) ? SM(u0,v0), where SM(u,v) andSM(u0,v0) are the solution sets of MV IP(F(·,v),L(u)) and MV IP(F(·,v0),L(u0))respectively.Theorem 2.2.3 Let (Z1,d1), (Z2,d2) be metric spaces, u0∈Z1, v0∈Z2 be givenpoints. Let L : Z1→2X be a continuous set-valued mapping with nonempty closedconvex values and int(barr(L(u0))) = ?. Suppose that there exists a neighborhoodU×V of (u0,v0) and M = u∈U L(u) such that F : M×V→2X? is a lowersemicontinuous set-valued mapping with nonempty weakly compact convex values.Suppose that(i) For each v∈V , the mapping x→F(x,v) is upper hemicontinuous andpseudomonotone on M.(ii) The solution set of GV IP(F(·,v0),L(u0)) is nonempty and bounded.Then (i) there exists a neighborhood U×V of (u0,v0) with U×V ? U×V ,such that for every (u,v)∈U×V , the solution set of GV IP(F(·,v),L(u)) isnonempty and bounded; (ii)ω?limsup(u,v)→(u0,v0) S(u,v) ? S(u0,v0), where S(u,v)and S(u0,v0) are the solution sets of GV IP(F(·,v),L(u)) and GV IP(F(·,v0),L(u0))respectively.In chapter 3, we study the stability of a set-valued weak vector variational in-equality (WV V I) in Rn. We established the upper semi-continuity and the lowersemi-continuity of the solution set mapping for the set-valued WV V I, when boththe mapping and the set involved in the weak vector variational inequality are per-turbed, without putting the compactness assumption on the mapping or the set. Asan application, we obtain a stability result for a class of parametric vector optimiza-tion problems in Rn. The main conclusions in chapter 3 are following:Theorem 3.2.3 If the following assumptions hold,(i)fi : Rn×Z2→2Rn is a lower semicontinuous set-valued mapping withnonempty compact convex values for every i = 1,2,···,p.(ii) L : Z1→2Rn is a continuous set-valued mapping with nonempty closedconvex values. (iii) f(·,v) is pseudomonotone on Rn and fi(·,v) is upper semicontinuous on Rnfor every i = 1,2,···,p.(iv) There exists some (u0,v0)∈Z1×Z2, such that Sw(u0,v0) is bounded andSi(u0,v0) is nonempty, for every i = 1,2,···,p.Then Sw(·,·) is upper semicontinuous at (u0,v0)∈Z1×Z2.Theorem 3.2.5 If the following assumptions hold,(i)fi : Rn×Z2→2Rn is a lower semicontinuous set-valued mapping withnonempty compact convex values for every i = 1,2,···,p.(ii) L : Z1→2Rn is a continuous set-valued mapping with nonempty closedconvex values.(iii) f(·,v) is strictly pseudomonotone on Rn and fi(·,v) is upper semicontinuouson Rn for every i = 1,2,···,p.(iv) There exists some (u0,v0)∈Z1×Z2, such that Sw(u0,v0) is nonempty andbounded.Then Sw(·,·) is lower semicontinuous at (u0,v0).In chapter 4, we study the connectedness of the solution set for a set-valuedWV V I in Rn when the mapping is pseudomonotone and strictly pseudomonotone,which is weaker than monotone mapping and strongly monotone mapping. We canobtain our results without putting the compactness assumption on the set. More-over, we show that the connectedness result in vector variational inequality problemcan be applied to a convex vector optimization problem and a strictly convex vectoroptimization problem. The main conclusions in chapter 4 are following:Theorem 4.2.1 Let K be a unbounded closed convex subset of Rn; let fi : K→2Rn,i = 1,2,···,p, be upper semicontinuous set-valued mappings with nonemptycompact convex values, let the mapping f = (f1,f2,···,fp) be pseudomonotone onK. If for any d∈K∞\ {0}, there exist y∈K and ui∈fi(y) such thatui,d > 0, for all i = 1,2,···,p. (1)Then Sw(f,K) is nonempty, compact and connected.Theorem 4.2.2 Let K be a unbounded closed convex subset of Rn; let fi : K→2Rn,i = 1,2,···,p, be upper semicontinuous set-valued mappings with nonempty compact convex values, let the mapping f = (f1,f2,···,fp) be strictly pseudomono-tone on K. If for any d∈K∞\ {0}, there exist y∈K and ui∈fi(y) such thatui,d > 0, for all i = 1,2,···,p. (2)Then(i) The solution set Sw(f,K) is compact, path-connected.(ii) The solution set S(f,K) is bounded, path-connected.