Novel local tuning techniques for speeding up one-dimensional algorithms in expensive global optimization using Lipschitz derivatives

Lipschitz global optimization is an important research field with numerous applications in engineering, electronics, machine learning, optimal decision making, etc. In many of these applications, even in the univariate case, evaluations of the objective functions and derivatives are often time consuming and the number of function evaluations executed by algorithms is extremely high due to the presence of multiple local extrema. As a result, the problem of an acceleration of the global search arises inevitably. In this paper, ideas allowing one to speed up the global search in cases where the objective function has the first derivative satisfying the Lipschitz condition are proposed. Several methods using a priori known global Lipschitz constants, their global and local dynamic estimates are presented. Two new algorithms using smooth piece-wise quadratic support functions are introduced and their convergence conditions are established. All the methods are implemented both in the traditional floating-point arithmetic and in the Infinity Computing framework allowing one to efficiently compute exact derivatives in the case where the optimized function is given as a black box. Numerical experiments executed on two classes of randomly generated test functions show a promising behavior of global optimization methods using the introduced local tuning techniques for speeding up the process of the global search.