Strong-branching inequalities for convex mixed integer nonlinear programs

Strong branching is an effective branching technique that can significantly reduce the size of the branch-and-bound tree for solving mixed integer nonlinear programming (MINLP) problems. The focus of this paper is to demonstrate how to effectively use “discarded” information from strong branching to strengthen relaxations of MINLP problems. Valid inequalities such as branching-based linearizations, various forms of disjunctive inequalities, and mixing-type inequalities are all discussed. The inequalities span a spectrum from those that require almost no extra effort to compute to those that require the solution of an additional linear program. In the end, we perform an extensive computational study to measure the impact of each of our proposed techniques. Computational results reveal that existing algorithms can be significantly improved by leveraging the information generated as a byproduct of strong branching in the form of valid inequalities.

[1]  Martin W. P. Savelsbergh,et al.  MINTO, a mixed INTeger optimizer , 1994, Oper. Res. Lett..

[2]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[3]  Martin W. P. Savelsbergh,et al.  A Computational Study of Search Strategies for Mixed Integer Programming , 1999, INFORMS J. Comput..

[4]  Iiro Harjunkoski,et al.  MINLP: Trim-loss Problem , 2009, Encyclopedia of Optimization.

[5]  Christian Kirches,et al.  Mixed-integer nonlinear optimization*† , 2013, Acta Numerica.

[6]  Matteo Fischetti,et al.  On the separation of disjunctive cuts , 2011, Math. Program..

[7]  Jeff T. Linderoth,et al.  Algorithms and Software for Convex Mixed Integer Nonlinear Programs , 2012 .

[8]  R. Boorstyn,et al.  Large-Scale Network Topological Optimization , 1977, IEEE Trans. Commun..

[9]  E. Balas,et al.  Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework , 1996 .

[10]  William J. Cook,et al.  The Traveling Salesman Problem: A Computational Study , 2007 .

[11]  Thorsten Koch,et al.  Branching rules revisited , 2005, Oper. Res. Lett..

[12]  Oktay Günlük,et al.  Mixing mixed-integer inequalities , 2001, Math. Program..

[13]  Omprakash K. Gupta,et al.  Branch and Bound Experiments in Convex Nonlinear Integer Programming , 1985 .

[14]  E. Balas,et al.  Strengthening cuts for mixed integer programs , 1980 .

[15]  I. Grossmann.  Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques , 2002 .

[16]  Jeff T. Linderoth,et al.  FilMINT: An Outer-Approximation-Based Solver for Nonlinear Mixed Integer Programs , 2008 .

[17]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1987, Math. Program..

[18]  Sven Leyffer,et al.  User manual for filterSQP , 1998 .

[19]  I. Grossmann,et al.  Logic-based MINLP algorithms for the optimal synthesis of process networks , 1996 .

[20]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[21]  Egon Balas,et al.  programming: Properties of the convex hull of feasible points * , 1998 .

[22]  Egon Balas.  A modified lift-and-project procedure , 1997, Math. Program..

[23]  Laurence A. Wolsey,et al.  Lot-Sizing with Constant Batches: Formulation and Valid Inequalities , 1993, Math. Oper. Res..

[24]  Tapio Westerlund,et al.  Optimization of block layout design problems with unequal areas: A comparison of MILP and MINLP optimization methods , 2005, Comput. Chem. Eng..

[25]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[26]  Jon Lee,et al.  Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations , 2010, Math. Program..

[27]  Carlo A. Furia,et al.  User manual , 2023, International Transport Forum Policy Papers.

[28]  S. Leyffer,et al.  More Branch-and-Bound Experiments in Convex Nonlinear Integer Programming , 2011 .

[29]  I. Grossmann,et al.  An LP/NLP based branch and bound algorithm for convex MINLP optimization problems , 1992 .

[30]  William J. Cook,et al.  The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics) , 2007 .

[31]  Samir Elhedhli,et al.  Service System Design with Immobile Servers, Stochastic Demand, and Congestion , 2006, Manuf. Serv. Oper. Manag..

[32]  Ignacio E. Grossmann,et al.  LOGMIP: a disjunctive 0–1 nonlinear optimizer for process systems models , 1997 .

[33]  Oktay Günlük,et al.  IBM Research Report MINLP Strengthening for Separable Convex Quadratic Transportation-Cost UFL , 2007 .

[34]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[35]  Sven Leyffer,et al.  FilMINT: An Outer Approximation-Based Solver for Convex Mixed-Integer Nonlinear Programs , 2010, INFORMS J. Comput..

[36]  Sanjay Mehrotra,et al.  A branch-and-cut method for 0-1 mixed convex programming , 1999, Math. Program..

[37]  David W.T. Rippin,et al.  Optimal design of a multi-product batch plant , 1998 .

[38]  Martin W. P. Savelsbergh,et al.  Conflict graphs in solving integer programming problems , 2000, Eur. J. Oper. Res..

[39]  S. Leyffer.  Experiments with MINLP Branching Techniques , 2010 .

[40]  Michael R. Bussieck,et al.  MINLPLib - A Collection of Test Models for Mixed-Integer Nonlinear Programming , 2003, INFORMS J. Comput..