An Efficlent Method For Solving Nenlinear Unconstrained Min-max Problem

Min-max problem is a typical non-differential optimization in math program. It wants toget the objection function's minimum under the maximum of the functions. For thenon-differentiability of the maximum function, this paper presents a new differentialapproximation function so that we can use some outstanding unconstrained optimizationalgorithms to solve the nonlinear unconstrained min-max problem.Firstly, the background and signification were introduced. Then, we reviewed themin-max problem. The main work and content arrangements of this paper were presented.And then, we introduced the basic knowledge and basic theory of min-max problem,including fundamental concepts, theorems, optimization conditions and some previousunconstrained optimization algorithms.Secondly, we constructed a new differential approximation function for the nondifferentiabilityof the objection function. Then, the property of the new differentialapproximation function is discussed and the processes of proves show that using the newapproximation function to solve the nonlinear unconstrained min-max problem is feasible andeffective. On the basis of that, we constructed some new algorithms among which someoutstanding unconstrained optimization algorithms were used to solve the unconstrainedmin-max problems. The convergence of the new algorithms and some relative results wereproved and given.In the end, some numerical experiment were used to testify it and compared with someprevious algorithms. The results showed that using the new algorithms with some previousunconstrained optimization algorithms to solve the min-max problem had a fasterconvergence speed and less iteration.