Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants

TL;DR

This paper introduces a novel deterministic global optimization algorithm that reduces high-dimensional problems to univariate ones using space-filling curves and employs multiple estimates of Lipschitz and Holder constants, demonstrating promising numerical performance.

Contribution

The paper proposes a new global optimization algorithm combining space-filling curves with multiple Holder constant estimates, extending ideas from the DIRECT method to more complex functions.

Findings

Algorithm effectively reduces dimensionality to univariate problems.

The method converges under established conditions.

Numerical tests show competitive performance against existing methods.

Abstract

In this paper, the global optimization problem $\min_{y\in S} F(y)$ with $S$ being a hyperinterval in $\Re^N$ and $F(y)$ satisfying the Lipschitz condition with an unknown Lipschitz constant is considered. It is supposed that the function $F(y)$ can be multiextremal, non-differentiable, and given as a `black-box'. To attack the problem, a new global optimization algorithm based on the following two ideas is proposed and studied both theoretically and numerically. First, the new algorithm uses numerical approximations to space-filling curves to reduce the original Lipschitz multi-dimensional problem to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm at each iteration applies a new geometric technique working with a number of possible H\"{o}lder constants chosen from a set of values varying from zero to infinity showing so that ideas introduced in a popular DIRECT method can be used in the H\"{o}lder global optimization. Convergence conditions of the resulting deterministic global optimization method are established. Numerical experiments carried out on several hundreds of test functions show quite a promising performance of the new algorithm in comparison with its direct competitors.