A deterministic global optimization using smooth diagonal auxiliary functions

TL;DR

This paper introduces a deterministic global optimization algorithm for black-box functions with unknown Lipschitz constants, utilizing smooth diagonal auxiliary functions and efficient partitioning, validated through extensive numerical testing.

Contribution

It proposes a novel 'Divide-the-Best' algorithm that handles black-box functions with unknown Lipschitz constants using smooth auxiliary functions and diagonal partitions.

Findings

Algorithm converges under specified conditions

Performed on 800 test functions with positive results

Effective for functions with unknown Lipschitz constants

Abstract

In many practical decision-making problems it happens that functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. In this paper, a global optimization problem is considered where both the goal function~$f(x)$ and its gradient $f'(x)$ are black-box functions. It is supposed that $f'(x)$ satisfies the Lipschitz condition over the search hyperinterval with an unknown Lipschitz constant~$K$. A new deterministic `Divide-the-Best' algorithm based on efficient diagonal partitions and smooth auxiliary functions is proposed in its basic version, its convergence conditions are studied and numerical experiments executed on eight hundred test functions are presented.