Multidimensional Lipschitz global optimization based on efficient diagonal partitions

Abstract

This is a summary of the author's PhD thesis, supervised by Yaroslav D. Sergeyev and defended on May 5, 2006, at the University of Rome "La Sapienza". The thesis is written in English and is available from the author upon request. In this work, the global optimization problem of a multidimensional "black-box" function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered. The objective function is assumed hard to evaluate. A new efficient diagonal scheme for constructing fast algorithms for solving this problem is examined and illustrated by developing several powerful global optimization methods. A deep theoretical study is performed which highlights the benefit of the approach introduced over traditionally used diagonal algorithms. Theoretical conclusions are confirmed by results of extensive numerical experiments.Keywords Global optimization · Black-box functions · Derivative-free methods · Partition strategies · Diagonal approach MSC classification (2000) 65K05 · 90C26 · 90C56 1 Problem statement Many decision-making problems arising in various fields of human activity (technological processes, engineering design, economic models, biology studies, etc.) can be

Global Search Based on Efficient Diagonal Partitions and a Set of Lipschitz Constants

“…Note also that the number of generated intervals (equal to 21) is greater than the number of trial points (equal to 17). Such a difference becomes more pronounced in the course of further subdivisions (see [21]). …”

“…Thus, diagonal algorithms obtain a more complete information about the objective function than central-sampling methods. Second, many efficient one-dimensional global optimization algorithms can be easily extended to the multivariate case by means of the diagonal scheme (see, e.g, [20,21,24,25,26]). …”

“…Since the time-consuming operation of evaluation of the function is replaced by a significantly faster operation of reading (up to 2 N times) the function values from the database, the new partition strategy considerably speeds up the search and also leads to saving computer memory. It is particularly important that the advantage of the new scheme increases with the growth of the problem dimension (see [21,31]). …”

“…Numerous algorithms have been proposed (see, e.g., [5,14,15,18,21,23,26,27,29,32,33]) for solving problem (1.1)-(1.3). One of the main questions to be considered in this occasion is: How can the Lipschitz constant L be specified?…”

“…The more promising and practical approaches are based on an adaptive estimation of L in the course of the search. In such a way, algorithms can use either a global estimate of the Lipschitz constant (see, e.g., [21,26,33]) valid for the whole region D from (1.3), or local estimates L i valid only for some subregions D i ⊆ D (see, e.g., [20,24,29,30,33]). …”