A method of feasible directions using function approximations, with applications to min max problems☆

Abstract This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gi(ξ)⩽0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when φ(x) = max {f(x, y) ¦ y ϵ Ω y } , a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms. It is outlined how this new algorithm can be used for solving problems of the form min y ϵ Ω x max y ϵ Ω y f(x, y) under the assumption that Ωmξ={x|gi(x)⩽0, j=1,…,s} ∩ R n, Ωy={y|ζi(y)⩽0, i-1,…,t} ∩ R m, with f, gj, ζi continuously differentiable, f(x, ·) concave, ζi convex for i = 1,…, t, and Ω x , Ω y compact.