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.

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.

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

KVASOV, Dmitry;SERGEEV, Yaroslav
2012-01-01

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.
2012
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.
Global optimization; Lipschitz gradients; Geometric algorithms
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/139688
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 49
  • ???jsp.display-item.citation.isi??? 46
social impact