Lipschitz global optimization - February 14, 2017

Lipschitz global optimization

Yaroslav D. Sergeyev

University of Calabria, Rende (CS), Italy

Lobachevsky State University, Nizhni Novgorod, Russia

Tuesday, February 14, 2017, 12:30 -13:30

University of Tehran

School of Mathematics, Statistics and Computer Science

Hashtroodi hall

 

Abstract: Global optimization is a thriving branch of applied mathematics and an extensive literature is dedicated to it. In this lecture, the global optimization problem of a multidimensional function satisfying the Lipschitz condition over a hyper-interval with an unknown Lipschitz constant is considered. It is supposed that the objective function can be black box, multi-extremal, and non-differentiable. It is also assumed that evaluation of the objective function at a point is a time-consuming operation. Algorithms for solving this problem discussed in the literature can be distinguished, for example, by the way of obtaining information about the Lipschitz constant and by the strategy of exploration of the search domain. Different exploration techniques based on various adaptive partition strategies are analyzed. The main attention is dedicated to two types of algorithms. The first of them is based on using space-filling curves in global optimization. A family of derivative-free numerical algorithms applying space-filling curves to reduce the dimensionality of the global optimization problem is discussed. A number of unconventional ideas, such as adaptive strategies for estimating Lipschitz constant, balancing global and local information to accelerate the search, etc. are presented. Diagonal global optimization algorithms are the second type of methods under consideration. They have a number of attractive theoretical properties and have proved to be efficient in solving applied problems.