On the approximation of solution regions for nonlinear inequalities

Abstract

In this paper, we present an algorithm for solving nonlinear inequalities involving a Lipschitz function f. The algorithm builds upon the covering principles of global optimization, drawing from existing works, which are based on adaptively constructing lower and upper bounds for f over a given interval. Focusing on these bounds, the interval is classified: discarded if f is strictly positive, accepted as part of the solution region if f is non-positive, or subdivided if the sign of f is indeterminate. For the third case, we suggest a partition of the interval into three equal sub-intervals. Numerical experiments demonstrate promising results compared with existing works employing different techniques.