Nonlinear Programming Trust Region Methods

The dissertation consists of six chapters. We mainly consider the convergence and the rate of convergence of some algorithms for solving nonlinear programming problems, while the trust region methods are our focus point.In Chapter Ⅰ, we present a general linesearch stepsize model for unconstrained optimization, we hope it can give a unified approach to lots of different linesearch stepsize models. Also, we discuss the relationship between linesearch and superlinear convergence.We introduce some weak conditions to the trust region method for the unconstrained optimization in Chapter Ⅱ . Under them, we prove the global convergence and superlinear convergence. Moreover, we put forward an easy implementation for Newton's method with trust region, which keeps its quadratic convergence. Still more, we extend the idea to the general case, incorporate the linesearch technique with trust region.In Chapter Ⅲ, A successive linear programming based on trust-region is presented for constrained optimization, which can adjust its penalty multipliers automatically. Under a general updating rule for trust-region radius, we prove its global convergence, and discuss its quadratic convergence for special case.Chapter Ⅳ and Chapter Ⅴ are main contribution of our dissertation. Both of them deal with the equality constrained optimization. In Chapter Ⅳ, a trust-region algorithm based on SQP method is given, its global convergence and superlinear convergence are proved. Comparing with previously published results, our algorithm is much more concise. Especially, its subproblem seems much less difficult than previous ones.In Chapter Ⅴ, we restate some local superlinear convergence results for SQP method, and reprove them under slightly weak conditions. But our proof is much more brief. Some of results are presented for the first time. Further, we also discuss the interaction among the convergent rates of variables, multipliers and pairs.Chapter Ⅵ is only a note on quasi—Newton method for nonsmooth nonlinear equations. We present a sufficient and necessary superlinear convergent condition to quasi-New ton method for semi-smoothness function.