Acceleration of univariate global optimization algorithms working with Lipschitz functions and Lipschitz first derivatives

TL;DR

This paper introduces and analyzes six new algorithms for one-dimensional global optimization problems involving Lipschitz functions and their derivatives, incorporating adaptive estimation and local improvement techniques to enhance search efficiency.

Contribution

The paper presents a unified framework for six algorithms that accelerate univariate Lipschitz optimization using adaptive estimation and local improvement methods.

Findings

Algorithms demonstrate promising performance on benchmark problems.

Adaptive techniques improve convergence speed.

Local improvement accelerates the search process.

Abstract

This paper deals with two kinds of the one-dimensional global optimization problems over a closed finite interval: (i) the objective function $f(x)$ satisfies the Lipschitz condition with a constant $L$; (ii) the first derivative of $f(x)$ satisfies the Lipschitz condition with a constant $M$. In the paper, six algorithms are presented for the case (i) and six algorithms for the case (ii). In both cases, auxiliary functions are constructed and adaptively improved during the search. In the case (i), piece-wise linear functions are constructed and in the case (ii) smooth piece-wise quadratic functions are used. The constants $L$ and $M$ either are taken as values known a priori or are dynamically estimated during the search. A recent technique that adaptively estimates the local Lipschitz constants over different zones of the search region is used to accelerate the search. A new technique called the \emph{local improvement} is introduced in order to accelerate the search in both cases (i) and (ii). The algorithms are described in a unique framework, their properties are studied from a general viewpoint, and convergence conditions of the proposed algorithms are given. Numerical experiments executed on 120 test problems taken from the literature show quite a promising performance of the new accelerating techniques.