Research On Trust Region Methods For Nonsmooth Problems

Nonsmooth optimization is widely used in medicine,economics,engineering design,optimal control and other fields.At present,most of the nonsmooth optimization methods require the objective function to be convex,while the problems encountered in practical applications are often non-convex and nonsmooth.Trust region method is easier to obtain global convergence than line search method,and it can solve non-convex and ill-conditioned problems.Moreover,trust region method combined with nonmonotone technology and adaptive technology has good performance in numerical calculation.Therefore,the purpose of this thesis is to extend the trust region method from smooth to nonsmooth optimization problem,which only requires the objective function to be local Lipschitz.The main research work of this thesis is as follows:1.A nonsmooth trust region method based on quasisecant is proposed.A new trust region subproblem is established based on quasisecant.The modified BFGS formula is used to update the trust region subproblem.The numerical experiments show that the algorithm is effective.2.A nonmonotone trust region method based on quasisecant is presented.The global convergence of the algorithm is proved under certain conditions.The numerical results show that the algorithm can overcome the Marotos effect to some extent.3.An adaptive trust region method based on quasisecant is proposed,which combines line search to generate new iteration points.Under appropriate assumptions,the global convergence of the algorithm is proved.Finally,the effectiveness of the algorithm is verified by numerical experiments.