Convergence analysis of Taylor models and McCormick-Taylor models

This article presents an analysis of the convergence order of Taylor models and McCormick-Taylor models, namely Taylor models with McCormick relaxations as the remainder bounder, for factorable functions. Building upon the analysis of McCormick relaxations by Bompadre and Mitsos (J Glob Optim 52(1):1–28, 2012), convergence bounds are established for the addition, multiplication and composition operations. It is proved that the convergence orders of both qth-order Taylor models and qth-order McCormick-Taylor models are at least q + 1, under relatively mild assumptions. Moreover, it is verified through simple numerical examples that these bounds are sharp. A consequence of this analysis is that, unlike McCormick relaxations over natural interval extensions, McCormick-Taylor models do not result in increased order of convergence over Taylor models in general. As demonstrated by the numerical case studies however, McCormick-Taylor models can provide tighter bounds or even result in a higher convergence rate.

[1]  J. Eckmann,et al.  A computer-assisted proof of universality for area-preserving maps , 1984 .

[2]  M. Berz,et al.  TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS , 2003 .

[3]  Christodoulos A Floudas,et al.  Global minimum potential energy conformations of small molecules , 1994, J. Glob. Optim..

[4]  Martin Berz,et al.  SUPPRESSION OF THE WRAPPING EFFECT BY TAYLOR MODEL- BASED VERIFIED INTEGRATORS: LONG-TERM STABILIZATION BY PRECONDITIONING , 2011 .

[5]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[6]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[7]  Christodoulos A. Floudas,et al.  Tight convex underestimators for $${{\mathcal C}^2}$$-continuous problems: I. univariate functions , 2008, J. Glob. Optim..

[8]  Christodoulos A. Floudas,et al.  A New Class of Improved Convex Underestimators for Twice Continuously Differentiable Constrained NLPs , 2004, J. Glob. Optim..

[9]  Stefan Friedrich,et al.  Topology , 2019, Arch. Formal Proofs.

[10]  G. Alefeld,et al.  Interval analysis: theory and applications , 2000 .

[11]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[12]  Daniel Scholz,et al.  Theoretical rate of convergence for interval inclusion functions , 2012, J. Glob. Optim..

[13]  Chrysanthos E. Gounaris,et al.  Tight convex underestimators for C 2 -continuous problems: II. multivariate functions. , 2008 .

[14]  Christodoulos A. Floudas,et al.  Tight convex underestimators for $${\mathcal{C}^2}$$ -continuous problems: II. multivariate functions , 2008, J. Glob. Optim..

[15]  S. Yau.  Mathematics and its applications , 2002 .

[16]  Leo Liberti,et al.  Branching and bounds tighteningtechniques for non-convex MINLP , 2009, Optim. Methods Softw..

[17]  Alexander Mitsos,et al.  Convergence rate of McCormick relaxations , 2012, J. Glob. Optim..

[18]  Paul I. Barton,et al.  McCormick-Based Relaxations of Algorithms , 2009, SIAM J. Optim..

[19]  A. Neumaier.  Complete search in continuous global optimization and constraint satisfaction , 2004, Acta Numerica.

[20]  R. Baker Kearfott,et al.  The cluster problem in multivariate global optimization , 1994, J. Glob. Optim..

[21]  Martin Berz,et al.  5. Remainder Differential Algebras and Their Applications , 1996 .

[22]  Kyoko Makino,et al.  Rigorous analysis of nonlinear motion in particle accelerators , 1998 .

[23]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[24]  Martin Berz,et al.  Efficient Control of the Dependency Problem Based on Taylor Model Methods , 1999, Reliab. Comput..

[25]  Martin Berz,et al.  SUPPRESSION OF THE WRAPPING EFFECT BY TAYLOR MODEL-BASED VERIFIED INTEGRATORS : LONG-TERM STABILIZATION BY SHRINK WRAPPING , 2006 .

[26]  Martin Berz.  From Taylor series to Taylor models , 1997 .

[27]  Jon G. Rokne,et al.  Computer Methods for the Range of Functions , 1984 .

[28]  Edward M. B. Smith,et al.  A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs , 1999 .

[29]  Nikolaos V. Sahinidis,et al.  Global optimization of mixed-integer nonlinear programs: A theoretical and computational study , 2004, Math. Program..

[30]  A. M. Sahlodin,et al.  Convex/concave relaxations of parametric ODEs using Taylor models , 2011, Comput. Chem. Eng..

[31]  Martin Berz,et al.  Computational differentiation : techniques, applications, and tools , 1996 .

[32]  Martin Berz,et al.  Computation and Application of Taylor Polynomials with Interval Remainder Bounds , 1998, Reliab. Comput..

[33]  Kaisheng Du,et al.  The Cluster Problem in Global Optimization: the Univariate Case , 1993 .

[34]  M. Stadtherr,et al.  Validated solutions of initial value problems for parametric ODEs , 2007 .

[35]  Daniel Scholz,et al.  The theoretical and empirical rate of convergence for geometric branch-and-bound methods , 2010, J. Glob. Optim..

[36]  Arnold Neumaier,et al.  Taylor Forms—Use and Limits , 2003, Reliab. Comput..

[37]  Nedialko S. Nedialkov,et al.  On Taylor Model Based Integration of ODEs , 2007, SIAM J. Numer. Anal..