We present a new spatial branch-and-bound approach for treating optimization problems with nonconvex inequality constraints. It is able to approximate the set of all global minimal points in case of solvability, and else to detect infeasibility. The new technique covers the nonconvex constraints by means of an improvement function which, although nonsmooth, can be treated by standard bounding operations.
The method is shown to be successful under a weak regularity condition, and we also give a transparent interpretation of the output in case that this condition is violated. Numerical tests illustrate the performance of the algorithm.