Algorithms for some minimax problems

摘要

The problem of minimizing differentiable functions on an entire vector space and on bounded subsets thereof has been studied by many authors. In this paper, we consider the problem of minimizing a nondifferentiable functionof the form ϕ(z)=max⁡i∈{1,...,N}fi(z) on the entire space En, or on a bounded set Ω in En, where the fi are continuously differentiable functions. In Section 1 an expansion ofis found, and continuous and discrete algorithms for finding a stationary point (that is, a point satisfying the necessary condition) are given. Some special cases are discussed.Most of the results obtained here can be applied to minimax problems in function spaces and in particular to some time optimal control problems, optimal control problems in the presence of constraints on the phase coordinates, and some pursuit problems.