Global Optima is not Limit Computable

TL;DR

This paper investigates the limit computability of finding global optima in continuous functions, proving certain problems are not limit computable and proposing an algorithm with convergence guarantees under specific conditions.

Contribution

It provides a proof that checking and finding global minima are not limit computable and introduces a convergent algorithm assuming known basin size bounds.

Findings

Checking if a point is a global minimum is not limit computable.

An algorithm converges to global minima given a lower bound on basin size.

Numerical experiments support the convergence results.

Abstract

We study the limit computability of finding a global optimum of a continuous function. We give a short proof to show that the problem of checking whether a point is a global minimum is not limit computable. Thereby showing the same for the problem of finding a global minimum. In the next part, we give an algorithm that converges to the global minima when a lower bound on the size of the basin of attraction of the global minima is known. We prove the convergence of this algorithm and provide some numerical experiments.