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

摘要

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.