The Proximal Point Algorithm For Pseudo-monotone Operators

We introduce the background of the proximal point algorithm(PPA)in Chapter one. Then in the second chapter of this paper, we discuss strongconvergence of the inexact proximal point algorithm(IPPA) for ?nding solu-tions of pseudomonotone variational inequalities in an in?nite-dimensionalHilbert space. Solodov and Svaiter proposed a proximal type algorithmwhich does converge strongly in the problem of ?nding zeros of maximalmonotone operators in an in?nite-dimensional Hilbert space[1]. Strong con-vergence property which they discovered is forced by combining proximalpoint iterations with simple projection steps onto intersection of two halfs-paces containing the solution set. It was shown by Tam, Yao and Yen thatthe convergence theorem for IPPA applied to in?nite-dimensional monotonevariational inequalities can be proved without using the theory of maximalmonotone operators[3]. Our purpose is to study strong convergence of IPPAfor ?nding solutions of pseudomonotone variational inequalities. Then weturn to chapter three. For a maximal monotone operator, a well-known clas-sical proximal point algorithm is often used to ?nd the zeros of the operator.Combined Rockafellar's analysis in 1976[23] with Gol'shtein and Tret'yakav'sresult in 1979[24], Eckstein and Bertsekas proposed a generalized proximalpoint algorithm to ?nd the zeros of a maximal monotone operator in a Hilbertspace[25]. Yang and He enhanced the algorithm to ?nd zeros of a maximalmonotone operator in a given closed convex subset in ???? space. They gave a new criterion in the inexact case of their modi?ed relaxed proximal point al-gorithm[26]. In chapter three of this paper, based on Yang and He's RelaxedApproximate Proximal Point Algorithm, we weaken maximal monotonicityof the operator by pseudo-monotone property and ?nd zeros of it. Further-more, in chapter four, we discuss the Relaxed Approximate Proximal PointAlgorithm on a in?nite-dimensional Hilbert space and prove that the se-quence generated from the iteration which is given in chapter four convergesstrongly to a zero of the pseudomonotone operator.