Chebyshev model arithmetic for factorable functions

Jai Rajyaguru; Mario E Villanueva; Boris Houska; Benoît Chachuat

doi:10.1007/s10898-016-0474-9

Chebyshev model arithmetic for factorable functions

J Glob Optim. 2017;68(2):413-438. doi: 10.1007/s10898-016-0474-9. Epub 2016 Oct 12.

Authors

Jai Rajyaguru¹, Mario E Villanueva¹, Boris Houska², Benoît Chachuat¹

Affiliations

¹ 1Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ UK.
² 2School of Information Science and Technology, ShanghaiTech University, 319 Yueyang Road, Shanghai, 200031 China.

Abstract

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

Keywords: Chebyshev models; Convergence rate; Factorable functions; Global optimization; Interval analysis; Taylor models.