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arXiv · 1307.4302

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

Dmitri E. Kvasov·Yaroslav D. Sergeyev
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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.

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Dmitri E. Kvasov, Yaroslav D. Sergeyev. 2013-07-15. Lipschitz gradients for global optimization in a one-point-based partitioning scheme. https://doi.org/10.1016/j.cam.2012.02.020

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