The Study Of Minimax Problem And Its Application

Minimax problem is not only one of the most important problems in the classical optimization field,but also the basic mathematical problem originated from the game theory.Many scholars put forward deep research and innovation improvement,it has been widely used in different fields,such as variational inequality,countermeasures,economic mathematics,etc.The sparse optimization problem refers to the process in which the original signal can be recovered totally from a set of incomplete measurements.However,this end is not easy to achieve for that the usual algorithms for the large sparse optimization problem may cause too complexity computational cost.Aiming at the Dantzig selector and?₁-minimizition problems,this article studies the transformation methods based on the minimax problems,and discusses the related algorithm according to the transformed models.The main contributions of this paper are summarized as follows:1.A minimax solving algorithm of the Dantzig selector has been proposed.Dantzig selector is a restoring model,which can estimate the related sparse signals from the linear measurement data.The transformation from the Dantzig selector problem to the minimax model is discussed in detail,the equivalence between them is proved,the properties of the minimax problem is proposed,the obtained solutions are exactly the maximal non-negative root of the corresponding function,and the corresponding Newton-based methods are discussed.2.A minimax solving algorithm of the?₁-minimization problem is studied.The difficulties of?₁-problem to the large-scale sparse signal reconstruction process lie in that the objective functions are not differentiable and too complexity computational costs.This paper alters?₁-minimization into minimax problem by dual approaches and the saddle point algorithm discussed in the first part of the paper is used to solve the problem.The efficiency of the method is demonstrated by the numerical experimentation.3.For the?₁-minimization problem,due to the absence of analytical solution,we need to design a verification model that to verify convergence and precision of the algorithm,therefore,the generated method of the ₁?-minimization model is discussed.A novel generated method of the weighted?₁-minimization verification model is proposed to balance the influence of all the coefficients,for that the big coefficients in the sparse problem may make significant effects to the solving procedure.