Uncomputability of Global Optima for Nonconvex Functions in the Oracle Model

Abstract

While it is well known that finding approximate optima of non-convex functions is computationally intractable, we show that the problem is, in fact, uncomputable in the oracle model. Specifically, we prove that no algorithm with access only to a function oracle can compute the global minimum or even an $\epsilon$-approximation of the minimizer or minimal value. We then characterize a necessary and sufficient condition under which global optima become computable, based on the existence of a computable predicate that subsumes the global optimality condition. As an illustrative example, we consider the basin of attraction around a global minimizer as such a property and propose a simple algorithm that converges to the global minimum when a bound on the basin is known. Finally, we provide numerical experiments on standard benchmark functions to demonstrate the algorithm's practical performance.