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

机译:使用空间填充曲线以及Lipschitz和Holder常数的多个估计进行确定性全局优化

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In this paper, the global optimization problem min(y is an element of S) F(y) with S being a hyperinterval in R-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 Holder condition. Second, the algorithm at each iteration applies a new geometric technique working with a number of possible Holder 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 Holder 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. (C) 2014 Elsevier B.V. All rights reserved.
机译:在本文中,考虑了全局优化问题min(y是S的元素)F(y),其中S是R-N中的超区间,并且考虑了满足Lipschitz条件且未知Lipschitz常数的F(y)。假定函数F(y)可以是多重极值,不可微的,并以“黑匣子”形式给出。为了解决这个问题,提出了一种基于以下两个思想的全局优化算法,并在理论和数值上进行了研究。首先,新算法使用数值逼近法对曲线进行空间填充,以将原始的Lipschitz多维问题简化为满足Holder条件的单变量问题。其次,每次迭代时的算法都会应用一种新的几何技术,该技术可处理从从零到无穷大的一组值中选择的许多可能的Holder常数,从而使流行的DIRECT方法中引入的思想可用于Holder全局优化。建立了确定性全局优化方法的收敛条件。在数百个测试函数上进行的数值实验表明,与直接竞争者相比,该新算法具有相当可观的性能。 (C)2014 Elsevier B.V.保留所有权利。

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