Strategies To Address The Ill-conditioning Problems Of Smoothing Algorithms For Unconstrained Minimax Problems

Unconstrained minimax problem is a class of typical non-differential optimization problem.It is widely used in the fields of engineering optimal design,structural optimization and data fitting,etc.Smoothing algorithm based on Newton direction or negative gradient direction and Armijo line search is an effective method to solve this kind of problem.This kind of method obtains a series of smooth approximating problems of original problem by taking a family of smoothing parameters which tend to infinity.When the smoothing parameters tend to infinity,the solutions of the series of smooth approximating problems tend to the solution of the original problem.However,in numerical experiments,the smoothing algorithm has the following two ill-conditioning problems: for smoothing parameter with small value,it is difficult to obtain high precision solutions for smoothing approximating problems.In addition,when the smoothing parameter gets more and more large,the ill-conditioning problem of smooth approximating problem become more and more serious and it is difficult to obtain a general accurate solution.For these two ill-conditioning problems,this paper gives two corresponding strategies.For the first kind of ill-conditioning problems,a line search strategy is presented based on the reduction of the norm of gradient and combine with the Newton direction and original smoothing algorithm.It effectively improves the accuracy of the solution and proves the convergence of the algorithm under certain conditions.For the second kind of ill-conditioning problems,we use the approximate solution of the smooth approximating problem,an equivalent smooth equations at the stable point is obtained by selecting active set which satisfies certain conditions.The equations are not ill-conditioned under general conditions,so the traditional Newton method can be used to solve the equations.Under general conditions,the convergence of Newton method for solving equivalent smooth equations is proved.Preliminary numerical experiments show the feasibility and effectiveness of the two improved strategies.