On Theory Of Some Important Algorithms In Nonlinear Optimization Problems

Some important algorithms for nonlinear optimization problems are studied in this dissertation. The aim of this dissertation is to give some theoretical analysis.Chapter 1 is the introduction of this dissertation, which introduces the variational inequality problem and the complementarity problem, the context of this dissertation and the main results obtained in this dissertation.A hybrid Newton-type method for solving the variational inequality problem (VIP) is studied in Chapter 2. It is known that the VIP can be reformulated as an unconstrained minimization problem through the D-gap function. Recently, a hybrid Newton-type method was proposed by Peng and Fukushima (1999) for minimizing the D-gap function. In this chapter, the hybrid Newton-type method is extended to minimize a generalized D-gapfunction gαβ in the general form. It is shown that the algorithm has nice convergenceproperties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, when the parameter β is chosen in a certain interval, it isproved that the generalized D-gap function gαβ has bounded level sets for the stronglymonotone VIP. An error bound estimation of the algorithm is obtained, which partially gives an answer to the question raised by Yamashita (1997) et al.The authors study the convergence properties of the damped Gauss-Newton algorithm which was originally proposed by Subramanian for the complementarity problem in Chapter 4. The aim of this chapter is to give a global convergence result under weaker conditions. The results here improve and generalize those in the literature. In addition, a new step-size rule, which need not a line search, is presented for the damped Gauss-Newton algorithm. A nice global convergence result is obtained.In Chapter 4, a new algorithm for the solution of nonlinear complementarity problems is developed. The algorithm is based on a reformulation of the complementarity problem as an unconstrained optimization. It is proved that the algorithm is globally convergent.In Chapter 5, the authors study the convergence properties of the gradient projection method for the constrained optimization problem. In this chapter, a new step-size rule, which avoids fulfiling the classical line search and includes choosing a constant as the step size as a special case, is presented and analyzed. Some convergence results of the gradient projectioninmethod under milder conditions are obtained.Finally we conclude the dissertation with some remarks in Chapter 6.