The Mixed Trust Region Algorithm For Unconstrained Optimization

In this paper, we mainly study the mixed trust region algorithm for solvingunconstrained optimization.The trust region algorithm get more and more attention forit’s good stability and strong convergence, and become one of the important methodfor solving the nonlinear programming. But the traditional trust region methodincrease the amount of calculation for that the trial step is solving repeatedly, howeverthe line search method can more easily find the search direction. In1991, Yuan andNocedal[44]propose the idea of constructing the algorithm that combines the trust regionmethod and the line search method, the algorithm gains the advantages of the twomethods, the theory and the numerical results are the ideal. This paper mainly study thefeasibility and numerical results of new algorithm which with various techniques inthe framework of the trust region and the line search’s combination.The main research results are as follows:1、A non-monotone adaptive trust region algorithm based on quadratic model waspresented. The algorithm combines the non-monotone technique and seek out the trialstep though the trust region method, if the trial step is not accepted, the filter technologyis used. If the trial step is not accepted by the filter set, then the Newton direction ischoosed, and along the direction non-monotone Armijo line search rule is used to getthe step size, so as to get a new iterative point. This algorithm guarantees that thetrust region subproblem solved only once in each iteration,and reduced the amount ofcalculation. Numerical experiments show that the new algorithm is effsctive.2、A non-monotone adaptive trust region algorithm based on conic model waspresented. For strongly non-quadratic form and more dramatic change of curvaturefunction, the trust region method of conic model will be better than the quadratic modelto approach the objective function. So the algorithm which presented in the secondchapter is generalized in this chapter. Firstly, a simple method for solving conic modelsubproblems, then the new algorithm is proposed. The global convergence andQ-superlinear convergence is proved, And the new algorithm is compared with the basictrust region algorithm,the filter trust region algorithm and the conic model filter trustregion algorithm with backtracking in the numerical experiments, the results shows thenew algorithm is quite effective.