Convergence rate of McCormick relaxations

Theory for the convergence order of the convex relaxations by McCormick (Math Program 10(1):147–175, 1976) for factorable functions is developed. Convergence rules are established for the addition, multiplication and composition operations. The convergence order is considered both in terms of pointwise convergence and of convergence in the Hausdorff metric. The convergence order of the composite function depends on the convergence order of the relaxations of the factors. No improvement in the order of convergence compared to that of the underlying bound calculation, e.g., via interval extensions, can be guaranteed unless the relaxations of the factors have pointwise convergence of high order. The McCormick relaxations are compared with the αBB relaxations by Floudas and coworkers (J Chem Phys, 1992, J Glob Optim, 1995, 1996), which guarantee quadratic convergence. Illustrative and numerical examples are given.

[1]  Christodoulos A. Floudas,et al.  Finding all solutions of nonlinearly constrained systems of equations , 1995, J. Glob. Optim..

[2]  Nikolaos V. Sahinidis,et al.  A polyhedral branch-and-cut approach to global optimization , 2005, Math. Program..

[3]  C. Floudas,et al.  A global optimization approach for Lennard‐Jones microclusters , 1992 .

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

[5]  Paul I. Barton,et al.  The cluster problem revisited , 2013, Journal of Global Optimization.

[6]  Christodoulos A. Floudas,et al.  αBB: A global optimization method for general constrained nonconvex problems , 1995, J. Glob. Optim..

[7]  Nils Tönshoff,et al.  Implementation and Computational Results , 1997 .

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

[9]  Paul I. Barton,et al.  Global solution of semi-infinite programs , 2004 .

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

[11]  G. McCormick.  Nonlinear Programming: Theory, Algorithms and Applications , 1983 .

[12]  N. Sahinidis,et al.  Convexification and Global Optimization in Continuous And , 2002 .

[13]  S. K. Mishra,et al.  Nonconvex Optimization and Its Applications , 2008 .

[14]  Christodoulos A. Floudas,et al.  Global optimization for molecular conformation problems , 1993, Ann. Oper. Res..

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

[16]  Paul I. Barton,et al.  Relaxation-Based Bounds for Semi-Infinite Programs , 2008, SIAM J. Optim..

[17]  Christodoulos A. Floudas,et al.  Rigorous convex underestimators for general twice-differentiable problems , 1996, J. Glob. Optim..

[18]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[19]  C. Adjiman,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results , 1998 .

[20]  Paul I. Barton,et al.  Interval Methods for Semi-Infinite Programs , 2005, Comput. Optim. Appl..

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

[22]  J. F. Price,et al.  On descent from local minima , 1971 .

[23]  P. I. Barton,et al.  Construction of Convex Relaxations Using Automated Code Generation Techniques , 2002 .

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

[25]  Nikolaos V. Sahinidis,et al.  Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming , 2002 .

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

[27]  Sanjo Zlobec,et al.  On the Liu–Floudas Convexification of Smooth Programs , 2005, J. Glob. Optim..

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

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

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

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

[32]  Christodoulos A. Floudas,et al.  A global optimization method, αBB, for process design , 1996 .

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

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

[35]  Christodoulos A. Floudas,et al.  Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems , 2004, J. Glob. Optim..

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

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

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

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

[40]  Nikolaos V. Sahinidis,et al.  Analysis of Bounds for Multilinear Functions , 2001, J. Glob. Optim..