Trust-region Algorithm Using Two-dimen-sional Subspace Technique With New Conic Model

The trust-region methods are effective for solving nonlinear optimization problems, they are blessed with both strong convergence properties in theory and a good result in computing. Two-dimensional subspace technique means that we can search a direction or get a trial step in some two-dimension space, its applications have the advantage of reducing both computation cost and memory size. So it has very good properties to combine two-dimensional subspace technique with trust-region methods, this idea has been applied to quadratic model, and good results have been achieved. But if the objective function has strong non-quadratic behavior or its curvature changes severely, the quadratic model methods often produce a poor prediction of the minimization of the function while the conic model one may serve better. Consequently, two-dimensional subspace algorithm with quadratic model is extended to conic model, and a good result is obtained.This paper applies two-dimensional subspace technique to trust-region subproblem involving a conic model, and proposes two-dimensional subspace trust-region subproblem with conic model. In addition, the two-dimension subspace trust-region algorithm with new conic model is established for solving unconstrained optimization problems. The dissertation consists of five chapters. Chapters 1 and 2 are introductory. They contains some preparatory materials of trust region method, the conic model ,subspace technique and the dogleg method. In Chapter 3, we first introduce two-dimensional subspace trust-region subproblem with quadratic model ,then we convert trust-region subproblem with conic model into a two dimension case, and give the detailed process and the algorithm. Moreover, we have proved the decreasing property of this algorithm. In Chapter 4 we solve the unconstrained optimization problem by the result got in Chapter 3, and propose the two-dimensional subspace trust-region algorithm with conic model and prove its global convergence. Finally, in Chapter 5, numerical experiment results are given.