Research Based On Nonmonotone Trust Region Algorithm

In real life,many problems appearing in science,engineering,management,economics and operational research can be transformed into unconstrained optimization problems,trust region algorithm is one of the important methods to solve these problems.In recent years,as nonmonotone techniques are widely used in search algorithms,nonmonotone trust region methods have attracted great attention in optimization field.The proper use of nonmonotone techniques can not only effectively improve the convergence rate of the algorithm,but also make the trust region method easier to find the global optimal solution.Although the present nonmonotone adaptive trust region algorithm has made great progress compared with the traditional trust region algorithm,it still needs to overcome the difficulties of large amount of computation,many iterations and slow running speed etc.when dealing with unconstrained optimization problems.In view of this,this paper effectively combines the adaptive update algorithm of trust region radius and BB algorithm with nonmonotone technology for large-scale unconstrained optimization problem,and proposes two improved nonmonotone trust region algorithms by using trust region model.Under the necessary assumptions,the global convergence and superlinear convergence speed of the new algorithm are established.Numerical experiments show that it has great competitive advantage.The main tasks are as follows:Firstly,an improved nonmonotone trust region algorithm is proposed.After the new algorithm obtains the information of the matrix indefinite by a vector inner product,the matrix value obtained by the previous iteration is no longer used,but the modified BFGS formula update matrix obtained by the multi-step iteration information is used instead,and when the correction formula is used,it is not necessary to assume that the objective function is convex,and it can still ensure that the matrix is positively determined during the calculation process,thereby reducing the number of iterations of the algorithm.In addition,the nonmonotone technique based on function average weight is applied to the efficient adaptive trust region algorithm framework,and a target function drop is different from the traditional trust region algorithm.Under weaker assumptions,the global convergence of the algorithm to the firstorder stable point is proved.Numerical experiments show that the improved nonmonotone trust region algorithm outperforms the AINATR and RNATR algorithms in the test set given by Andrei and the CUTEst test set.Secondly,a modified nonmonotone trust region BB algorithm is proposed.Due to the simplicity,robustness,low memory requirements and global convergence of the BB algorithm,the method makes full use of the idea of the BB algorithm,and uses the multi-step quasiNewton secant equation to derive the corrected BB step size,using the step information,a diagonal scalar matrix is obtained to approximate the Hessian matrix of the objective function,whereby a new trust region subproblem model is constructed.In addition,the eigenvalue analysis of the scaled memoryless Newton update formula BFGS or DFP is carried out.Based on this,an adaptive trust region radius update strategy is proposed.Under the appropriate conditions,the global convergence and superlinear convergence speed of the algorithm are proved.The numerical experiments show that compared with the existing the modified BB algorithms,the new algorithm presents more obvious ladder shape and monotony,so the convergence speed is faster.The last one,we summarize the two new methods proposed in the paper,and make further reflections and prospects on the further continuation and expansion of related topics.