Lipschitz gradients for global optimization in a one-point-based partitioning scheme

TL;DR

This paper introduces a new multidimensional geometric optimization method that leverages Lipschitz gradients and a one-point-based partitioning strategy, showing promising results against established methods.

Contribution

It extends a one-dimensional Lipschitz gradient algorithm to a multidimensional setting using a novel geometric approach and partitioning strategy.

Findings

Outperforms popular DIRECT-based methods in numerical tests.

Demonstrates effectiveness on 800 multidimensional test functions.

Provides a new approach for global optimization with Lipschitz gradients.

Abstract

A global optimization problem is studied where the objective function $f(x)$ is a multidimensional black-box function and its gradient $f'(x)$ satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant $K$. Different methods for solving this problem by using an a priori given estimate of $K$, its adaptive estimates, and adaptive estimates of local Lipschitz constants are known in the literature. Recently, the authors have proposed a one-dimensional algorithm working with multiple estimates of the Lipschitz constant for $f'(x)$ (the existence of such an algorithm was a challenge for 15 years). In this paper, a new multidimensional geometric method evolving the ideas of this one-dimensional scheme and using an efficient one-point-based partitioning strategy is proposed. Numerical experiments executed on 800 multidimensional test functions demonstrate quite a promising performance in comparison with popular DIRECT-based methods.