The Mixed Trust Region Algorithm For Unconstrained Optimization Problems

Trust region (TR) method is an important class of numerical method for solving nonlinear optimization problems. This Algorithm have been proposed for solving systems of nonlinear equations, unconstrained and constrained optimization, non-differentiable optimization. TR method have been proven to be very effective and robust techniques with excellent global conver -gence properties for solving non-linear optimization problems. So this method has attracted many scholars' attention . Especially, this method has been a research hot spot for non-linear optimization in the recent decade.To the current, the mathematical workers at home and abroad have made the following several major trust region methods, such as interior point TR algorithm, nonmonotone TR algorithm, self-adaptive TR algorithm, conic model TR algorithm, memory TR algorithm and filter TR algorithm and so on.However, It is very slow about research of combining several TR methods. Almost all the algorithms share a common feature: it is necessary to solving a quadratic subproblem with a trust region bound at each iteration in these methods, which in general give rise to great computational effort. In addition, in recent years, the particle swarm optimization (PSO) algorithm has been widely concerned. But the PSO algorithm is easy "premature" . TR algorithm has good convergence, and the study of combining TR algorithm with PSO algorithm does not begin .For the above reasons, the following two problems have been chosen as the main goal in the dissertation.(1) Construct a few algorithms for solving systems of nonlinear equations and nonlinear optimization problems by combining non-monotone technical with adaptive technologies, as well as line search technology.(2) Develop a mixed search method by combining TR algorithm with PSO algorithm.The content of this paper is organized into five chapters.In chapter I, we introduce the history and status about trust region methods, the creativities and the practical importance of this paper. In chapter II, we construct a trust region method for solving nonlinear systems. This method has more extensive scope of application, easy to implement . The problem is transformed into a nonlinear optimization with nonnegative constraints by introducing slack variables.In chapter III, An nonmonotone adaptive trust region method with fixed stepsize technique for solving unconstrained optimization problems is presented . This method may not only avoid Maratos effect, but also overcome ills in which the trust region radius of the regulation is too mechanical. Numerical results show that the new method is also effective to unconstrained optimization questions of high dimension.In chapter IV, A new nonmonotone adaptive trust-region method for solving nonlinear equations is proposed. The convergence of the presented method is proofed . Limited numerical experiments show that the algorithm is effective.In chapter V a mixed search method is proposed by combining trust region algorithm with and particle swarm optimization algorithm for solving unconstrained optimization problems. This method may avoid solving subproblem with a trust region bound in the iterative process, and reduce the cost of the calculation. Numerical results show that the new method is effective.