Trust Region Subproblem Algorithm

Trust region (TR) method, whose idea is novel, and whose algorithm is areliable. It has a strong convergence. It not only can solve some good stateproblems quickly, but also can solve some pathological optimization problemseffectively. However, solving a subproblem in TR algorithm is the key toachieve the algorithm.In this paper, studies were done in two aspects of TR subproblem's solvingand construction. In the aspect of solving TR, we first proposed a double secantdogleg method for solving TR subproblem, which will be closer to the Newtondirection than the single dogleg path and the tangent single dogleg path, whosenumerical effect is better than the tangent single dogleg method and the hybriddogleg method. Secondly, we create a kind of self-adjusting technology andadaptive technology. Combining with the double secant dogleg method, wepropose a new nonmonotone adaptive TR algorithm with self-adjusting step forunconstrained optimization problems. Numerical experiments show that itgreatly improves the computational efficiency; Finally, in the aspect ofsubproblem's construction, we propose a rangeability function model for the TRsubproblem. The model not only contains some formation about function value,gradient,Hessian matrix of an objective function, but also it can easily solvesome non-quadratic functions.The content of this paper includes the following sections:In chapter I, we introduce research and hotspots about TR methods, theinnovation and breakthrough of this paper.In chapter II, we construct a double secant dogleg method for solving TRsubproblem. This chapter is focused on the positive definite case, and isdescribed by construction, algorithm and convergence properties of the doublesecant dogleg method.In chapter III, a double secant dogleg method for solving indefinite TRsubproblem is presented. This chapter is focused on the indefinite case, and isdescribed by construction, algorithm and convergence properties of the double secant dogleg method.In chapter IV, a new nonmonotone adaptive TR algorithm withself-adjusting step for unconstrained optimization problems is proposed. And theconvergence analysis of algorithms and numerical results are given.In chapter V, we build a rangeability function model for solving TRsubproblem, and give the convergence analysis of algorithms.