Iterative Methods For Nonlinear Optimization Problems In Hilbert Spaces

By using the technique in fixed point theory, the thesis aims to study and investigate some nonlinear optimization problems in Hilbert spaces, such as zero problem of maximal monotone operators, monotone variational inequality and fixed point problems, split feasibility problem and split common fixed point problem. The thesis consists of five chapters.In Chapter 1, we first recall some preliminary concepts and present the research back-ground. The concept of zero problem of maximal monotone operators and proximal point al-gorithm is presented in Section 1. Also the theory on the existence and approximation of fixed points for nonexpansive operators is recalled. Then we respectively collect some important properties of monotone and maximal monotone operators in Section 2, and of nonexpansive, averaged and firmly nonexpansive operators in Section 3.Chapter 2 is devoted to investigating the problem for finding zeros of the sum of two max-imal monotone operators. In Section 1 we recall the forward-backward splitting method for solving such problem. In Section 2, Combettes'relaxed forward-backward splitting method with errors:xn+1=(1-αn)xn+αnJrn(xn-rnAxn)+en, is considered by improving the sufficient conditions to guarantee the weak convergence of the algorithm:∑│rn+1-rn│<∞,∑‖en‖<∞,0<limrn≤2k,0≤αn≤4k/2k+rn,∑αn(4k/2k+rn-αn)=∞. Based on the Halpern iteration, we introduce in Section 3, a modified forward-backward splitting method:xn+1=αnx0+(1-αn)Jrn(xn-rnAxn)+en, and prove its convergence provided that lim an=0,∑αn=∞,lim│rn+1-rn│=0,∑‖en‖<∞,0<lim inf rn≤lim sup rn<2k. On the other hand, based on the Haugazeau iteration, we introduce another modified forward-backward splitting method to approximate zeros of the sum of two maximal monotone opera-tors. In Section 4, applications in the proximal point algorithm, monotone variational inequality problem and convexly constrained minimum problem are also considered.Chapter 3 is devoted to solving common solutions for monotone variational inequality and fixed point problems. In Section 1, we recall the existing algorithms for solving common so-lutions. Section 2 deals with the case whenever the operator A is inverse strongly monotone operator. In this case, we introduce an algorithm:xn+1=(1-αn)xn+αnSPC(xn-rnAxn) and prove its weak convergence provided that∑αn(1-αn)=∞;∑│rn+1-rn│<∞,0< lim rn<2k. Section 3 deals with the case whenever the operator A is Lipschitz continu-ous monotone operator. In this case, we introduce a new extragradient method, based on Tseng's algorithm, to solve common solutions for monotone variational inequality and fixed point problems. The choice of the stepsize is according to Armijo rule and thus the calculation for the Lipschitz coefficient is avoided. In Section 4, applications in common fixed points for strict pseudo-contractions and nonexpansive operators, and common fixed points for Lipschitz pseudo-contractions and nonexpansive operators are also considered. Chapter 4 is devoted to solving the split common fixed point problem and the split feasibil-ity problem in Hilbert spaces. Section 1 mainly recalls the definition of various split problems and Byrne's CQ algorithm. Because of the weak convergence of the CQ algorithm, we present in Section 2 a modified CQ algorithm according to the damped projection method, and prove its strong convergence to the minimal norm solution of the split feasibility problem. In Section 3, we propose a new method for solving the split feasibility problem. The advantage of this proposed algorithm is that the step dose not rely on the norm of the matrix A. Moreover, to guarantee the convergence of the proposed algorithm, it suffices to assume that the solution set of the problem is nonempty. Section 4 deals with the case whenever the convex subsets of the split feasibility problem are level sets. In view of the algorithm in the previous sec-tion and Tseng's extragradient method, we construct two relaxed projection algorithms. To solve the split common fixed point problem, we introduce in the last section a cyclic algorithm: xn+1=U[n][xn+rA*(T[n]-I)Axn],[n]=nmodp. The advantage of such algorithm is that it can be reduced to the CQ algorithm for the split feasibility problem whenever p=s=1.Chapter 5 is devoted to numerical experiments for various algorithms presented in the pre-vious chapter. We first consider the relation between the relax factor and the CQ algorithm. The experiments indicate that for the over-relax factor the algorithm converges fast. Also we study the converge rate between the fixed and variable steps for the CQ algorithm. The experiments imply that the CQ algorithm with variable step converges faster than that with fixed step.