A computational study of primal heuristics inside an MI(NL)P solver

Primal heuristics are a fundamental component of state-of-the-art global solvers for mixed integer linear programming (MIP) and mixed integer nonlinear programming (MINLP). In this paper, we investigate the impact of primal heuristics on the overall solution process. We present a computational study, in which we compare the performance of the MIP and MINLP solver SCIP with and without primal heuristics on six test sets with altogether 983 instances from academic and industrial sources. We analyze how primal heuristics affect the solver regarding seven different measures of performance and show that the impact differs by orders of magnitude. We further argue that the harder a problem is to solve to global optimality, the more important the deployment of primal heuristics becomes.

[1]  Timo Berthold.  Heuristic algorithms in global MINLP solvers , 2014 .

[2]  Timo Berthold,et al.  Rounding and Propagation Heuristics for Mixed Integer Programming , 2011, OR.

[3]  Christodoulos A. Floudas,et al.  GloMIQO: Global mixed-integer quadratic optimizer , 2012, Journal of Global Optimization.

[4]  Andrea Lodi,et al.  MIPLIB 2010 , 2011, Math. Program. Comput..

[5]  Nenad Mladenovic,et al.  A recipe for finding good solutions to MINLPs , 2011, Math. Program. Comput..

[6]  F. Wilcoxon.  Individual Comparisons by Ranking Methods , 1945 .

[7]  Thorsten Koch,et al.  Progress in presolving for mixed integer programming , 2015, Math. Program. Comput..

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

[9]  Timo Berthold,et al.  Extending a CIP framework to solve MIQCPs , 2012 .

[10]  Thorsten Koch,et al.  Solving Hard MIPLIB2003 Problems with ParaSCIP on Supercomputers: An Update , 2014, 2014 IEEE International Parallel & Distributed Processing Symposium Workshops.

[11]  Fred W. Glover,et al.  The feasibility pump , 2005, Math. Program..

[12]  Thorsten Koch,et al.  Could we use a million cores to solve an integer program? , 2012, Mathematical Methods of Operations Research.

[13]  Tobias Achterberg,et al.  Mixed Integer Programming: Analyzing 12 Years of Progress , 2013 .

[14]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[15]  Alexander Shapiro,et al.  On complexity of multistage stochastic programs , 2006, Oper. Res. Lett..

[16]  Tobias Achterberg,et al.  SCIP: solving constraint integer programs , 2009, Math. Program. Comput..

[17]  Andrea Lodi,et al.  A storm of feasibility pumps for nonconvex MINLP , 2012, Mathematical Programming.

[18]  Thorsten Koch,et al.  ParaSCIP: A Parallel Extension of SCIP , 2010, CHPC.

[19]  Thorsten Koch,et al.  Constraint Integer Programming: A New Approach to Integrate CP and MIP , 2008, CPAIOR.

[20]  Timo Berthold.  RENS: The optimal rounding , 2014 .

[21]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[22]  Q. Mcnemar.  Note on the sampling error of the difference between correlated proportions or percentages , 1947, Psychometrika.

[23]  Timo Berthold.  Primal Heuristics for Mixed Integer Programs , 2006 .

[24]  Timo Berthold,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Improving the Feasibility Pump Improving the Feasibility Pump , 2022 .

[25]  im Fach Mathematik , .

[26]  Thorsten Koch,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Miplib 2003 , 2022 .

[27]  Roland Wunderling,et al.  Paralleler und objektorientierter Simplex-Algorithmus , 1996 .

[28]  Matteo Fischetti,et al.  Exploiting Erraticism in Search , 2014, Oper. Res..

[29]  Pietro Belotti,et al.  Rounding-based heuristics for nonconvex MINLPs , 2011, Mathematical Programming Computation.

[30]  Thorsten Koch,et al.  Evaluating Gas Network Capacities , 2015, MOS-SIAM Series on Optimization.

[31]  Timo Berthold,et al.  Measuring the impact of primal heuristics , 2013, Oper. Res. Lett..

[32]  Nikolaos V. Sahinidis,et al.  BARON: A general purpose global optimization software package , 1996, J. Glob. Optim..

[33]  Timo Berthold,et al.  Undercover: a primal MINLP heuristic exploring a largest sub-MIP , 2014, Math. Program..

[34]  Matteo Fischetti,et al.  Feasibility pump 2.0 , 2009, Math. Program. Comput..

[35]  Armin Fügenschuh,et al.  Validation of nominations in gas network optimization: models, methods, and solutions , 2015, Optim. Methods Softw..

[36]  Gérard Cornuéjols,et al.  A Feasibility Pump for mixed integer nonlinear programs , 2009, Math. Program..

[37]  Stephen G. Nash,et al.  Guidelines for reporting results of computational experiments. Report of the ad hoc committee , 1991, Math. Program..

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

[39]  Pierre Bonami,et al.  Heuristics for convex mixed integer nonlinear programs , 2012, Comput. Optim. Appl..

[40]  Christodoulos A. Floudas,et al.  A review of recent advances in global optimization , 2009, J. Glob. Optim..