Algorithms for solving non-linear inequalities in a finite number of iterations using approximate derivatives

Abstract

We present two new algorithms, based on Newton's method, for solving a finite set of non-linear inequalities (under mild assumptions) in a finite number of iterations using approximate derivatives. The algorithms utilize function values at each iteration to evaluate an approximation to the gradient. The method is efficient when gradient evaluation is costly relative to function evaluations. The algorithms use a technique for systematic expansion of the linearized feasible set to ensure existence of a Newton-type search direction.