Solution Methods For Set-valued Pseudomonotone Variational Inequalities

Variational inequality theory which is a powerful tool with its application is an important part of nonlinear analysis. It has wide applications in finance, economics, transportation, opti-mization, operations research and engineering sciences. Solving variational inequality inclusion, as an important aspect of variational theory, has been extensively and depthsively studied by several methods by many researchers. Such as Tikhonov regularization method (TRM) and the proximal point algorithm(PPA), which are both crucial for solving variational inequalities. In this thesis, we focus on TRM and PPA for set-valued pseudomonotone general and mixed variational inequalities. We analysis the existence of the solution and the convergence theorem for TRM and PPA applied to the variational inequalities. The thesis consists of three chapters.In Chapter 1, we make a brief review of the current research in variational inequalities, TRM and PPA. In addition, we introduce some basic notations and preliminaries which are used in this thesis.In Chapter 2, assume that H be a real Hilbert space, let K C H be a nonempty closed convex set, F:H→2H be a pseudomonotone mapping. We investigate the general variational inequality (denoted by GVI(K, F)):find x∈K and x*∈F(x) such that It is well known that this problem can be rewritten in the form 0∈T(x) with T:H→2H is pseudomonotone, denoted by T(x)= F(x)+NK(x). In this chapter, we study the problem of 0∈T(x) by two generalized proximal point algorithms when T is pseudomonotone, thereby get the solution set of GVI(K,F). To our best knowledge, there is no literature about studying the zeros of pseudomonotone operator by the algorithms which we use. The two generalized proximal point algorithms and the main conclusion in chapter 2 are following:Step 1. Set z0∈H be an initial vector;Step 2. Given zk, (1-γk)∈[γ,∞)(γ> 0) and ck∈[c,+∞) (c>0) find zk+1,ek such that where ek is an error and satisfies whereΣk=0∞ηk<+∞.Theory 2.2.1 Let {zk} be generated by the Algorithm 2.1.1. Assume that p∈S, where S denotes the set of all zeros of T andΔ= supk≥0(1-γk)≤2. Then {zk} converges weakly to a zero of T.Step 1. Set z0∈H be an initial vector;Step 2. Given zk,γk∈(0,1) and ck∈[c,+∞) (c> 0) find zk+1, ek such that where ek is an error and satisfiesTheory 2.3.1 Let {zk} be generated by the Algorithm 2.1.2. p∈S, where S denotes the set of all zeros of T. Assume thatΣk=0∞γk<∞, Then {zk} converges weakly to a point of S, where S denotes the set of all zeros of T.In chapter 3, we use TRM and PPA to solve the set-valued pseudomonotone mixed variational inequalities in a real Hilbert space (denoted by MVI(K, F,(?))):Assume that H be a real Hilbert sapce, let K C H be a nonempty closed convex set, find x∈K, x*∈F(x) such that where (?):H→R∪{+∞} be a convex function.Set-valued mapping F:K→2H is pseudomono-tone with respect to (?). In the reference [33], some facts on the convergence of TRM and PPA for pseudomonotone general variational inequality are obtained. Some comments on PPA for mixed pseudomonotone variational inequality are mentioned, but it does not give the proof of the conver-gence theorem. As a byproduct, we show the convergence theorem for TRM and PPA applied to mixed variational inequality can be proved. The methods and the main conclusions in chapter 3 as follows:Step 1. Given a positive real sequence {εk}k∈N, which is satisfying the condition thatεk→0 as k→∞, Solve to obtain the solution x(εk)∈K;Step 2. If (?)x(εk+1)-x(εk)(?)≤θwhereθis a constant, stop;Step 3. Increase k by 1 and go to step 1.Theorem 3.3.1 Suppose that F:K→2H is pseudomonotone with respect to (?) and upper semicontinuous on K. If S(K, F,(?)) is nonempty and (?) is the least-norm element in that solution set, then the following properties are valid:(i) For everyε> 0, S(K, Fε,(?)) is nonempty whenever Fεis pseudomonotone with respect to (?) on K;(ii) The sets S(K,Fε,(?)),ε> 0, are uniformly bounded. Moreover, for eachε> 0, it holds that(iii) If F is upper semicontinuous on K, then any weakly convergent subsequence of {x(ε)} weakly converges to (?).Theorem 3.3.2 Suppose that K (?) Rn is a nonempty closed convex set, F:K→2Rn is pseudomonotone with respect to (?) and upper semicontinuous on K. If the problem MVI(K, F,(?)) has a solution, then:(a) S(K, Fε,(?)) is nonempty and compact for everyε> 0 whenever Fεis pseudomonotone with respect to (?);(b) The sequence {x(εk)},where{x(εk)} is any vector from S(K, Fε,(?)), converges to the least-norm element in S(K, F,(?));(c) limε→0+ diamS(K,Fε,(?))= 0), where diamΩ:= sup{(?)x-y(?):x,y∈Ω} denotes the diameter of a subsetΩ∈Rn.Theorem 3.3.3 Suppose that F:K→2H is monotone and pseudomonotone with respect to (?) and upper semicontinuous on K. If S(K, F,(?)) is nonempty and (?) is the least-norm element in that solution set, then {x(ε)} converges to (?) asε→0+. Here,{x(ε)}denotes the unique element in S(K, Fε,(?)) when Fεis pseudomonotone with respect to (?).Step 1:Set x0∈H be an initial vector;Step 2:Given xk-1 and {βk}k∈N,βk≥β, whereβis a constant, find a vector xk∈K and select an element xk*∈Fk(x), where such thatStep 3:If (?)xk-xk-1(?)≤θ,stop；Step 4:Increase k by 1 and go to step 2.Step 1:Set z0=x0∈H be an initial vector;Step 2:Given zk-1,{βk}k∈N,βk≥β>0 whereβis a constant and {εk}k∈N satisfies the conditionsεk≥0 and∑k=1∞εk<+∞,find a vector zk∈Kwhere (?)(k)(x):=βkF(x)+x-zk-1,S(K,(?)(k),βK(?))denotes the solution set of the inequality (βkx*+x-zk-1,y-x)+(?)(y)-(?)(x)≥0 (?)y∈H,dist(zk,S(K,(?)(k),βK(?)))denote the distance from zk to S(K,(?)(k),βK(?))；Step 3:If (?)zk-zk-1(?)≤θ,stop;Step 4:Increase k by 1 and go to step 2.Theorem 3.4.1Suppose that F:K→2H is pseudomonotone with respect to (?).x0∈H is a given vector,and {xk} is a sequence generated by the algorithm 3.1.2.Then (?)∈S(K,F,(?)),we haveTheorem 3.4.2 Suppose that F:K→2H is pseudomonotone with respect to (?) and upper semicontinuous on K,S(K,F,(?))is nonempty,x0∈H is a given vector,and {zk} is a sequence generated by the algorithm 3.3.2.Then the following assertions are valid:(a){zk} is a bounded sequence and limk→∞(?)zk-zk-1(?)=0;(b)There exists (?)∈S(K,F,(?))and (?)∈[0,+∞) such that limk→∞(?)zk-(?)=(?) and zk(?).