Adaptive Kernel Search: A heuristic for solving Mixed Integer linear Programs

We introduce Adaptive Kernel Search (AKS), a heuristic framework for the solution of (general) Mixed Integer linear Programs (MIPs). AKS extends and enhances Kernel Search, a heuristic framework that has been shown to produce high-quality solutions for a number of specific (combinatorial) optimization problems in a short amount of time. AKS solves a sequence of carefully constructed restricted MIPs (using a commercial MIP solver). The computational effort required to solve the first restricted MIP guides the construction of the subsequent MIPs. The restricted MIPs are constructed around a kernel, which contains the variables that are presumably non-zero in an optimal solution. Computational results, for a set of 137 instances, show that AKS significantly outperforms other state-of-the-art heuristics for solving MIPs. AKS also compares favorably to CPLEX and offers more flexibility to trade-off solution quality and computing time.

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

[2]  Nenad Mladenović,et al.  New variable neighbourhood search based 0-1 MIP heuristics , 2014 .

[3]  Maria Grazia Speranza,et al.  Kernel search for the capacitated facility location problem , 2012, J. Heuristics.

[4]  Nenad Mladenovic,et al.  Variable neighbourhood decomposition search for 0-1 mixed integer programs , 2009, Comput. Oper. Res..

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

[6]  Maria Grazia Speranza,et al.  A heuristic for BILP problems: The Single Source Capacitated Facility Location Problem , 2014, Eur. J. Oper. Res..

[7]  E. Balas,et al.  Pivot and shift - a mixed integer programming heuristic , 2004, Discret. Optim..

[8]  Matteo Fischetti,et al.  Boosting the feasibility pump , 2014, Mathematical Programming Computation.

[9]  Matteo Fischetti,et al.  Proximity search for 0-1 mixed-integer convex programming , 2014, J. Heuristics.

[10]  Stefano Lucidi,et al.  Feasibility Pump-like heuristics for mixed integer problems , 2010, Discret. Appl. Math..

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

[12]  Egon Balas,et al.  Octane: A New Heuristic for Pure 0-1 Programs , 2001, Oper. Res..

[13]  Edward Rothberg,et al.  An Evolutionary Algorithm for Polishing Mixed Integer Programming Solutions , 2007, INFORMS J. Comput..

[14]  Claude Le Pape,et al.  Exploring relaxation induced neighborhoods to improve MIP solutions , 2005, Math. Program..

[15]  Martin W. P. Savelsbergh,et al.  Restrict-and-relax search for 0-1 mixed-integer programs , 2013, EURO J. Comput. Optim..

[16]  Fred W. Glover,et al.  General Purpose Heuristics for Integer Programming—Part II , 1997, J. Heuristics.

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

[18]  Fred W. Glover,et al.  Solving zero-one mixed integer programming problems using tabu search , 1998, European Journal of Operational Research.

[19]  Jonathan Eckstein,et al.  Pivot, Cut, and Dive: a heuristic for 0-1 mixed integer programming , 2007, J. Heuristics.

[20]  Maria Grazia Speranza,et al.  A heuristic framework for the bi-objective enhanced index tracking problem , 2016 .

[21]  Pierre Hansen,et al.  Variable neighborhood search and local branching , 2004, Comput. Oper. Res..

[22]  Fred W. Glover.  Parametric tabu-search for mixed integer programs , 2006, Comput. Oper. Res..

[23]  Maria Grazia Speranza,et al.  Kernel Search: An application to the index tracking problem , 2012, Eur. J. Oper. Res..

[24]  Stefano Lucidi,et al.  A New Class of Functions for Measuring Solution Integrality in the Feasibility Pump Approach , 2013, SIAM J. Optim..

[25]  Jordi Castro,et al.  Using the analytic center in the feasibility pump , 2011, Oper. Res. Lett..

[26]  Maria Grazia Speranza,et al.  Kernel search: A general heuristic for the multi-dimensional knapsack problem , 2010, Comput. Oper. Res..

[27]  Matteo Fischetti,et al.  Repairing MIP infeasibility through local branching , 2008, Comput. Oper. Res..

[28]  A. Lodi,et al.  Heuristics in Mixed Integer Programming , 2011 .

[29]  Vinícius Amaral Armentano,et al.  A computational study of parametric tabu search for 0-1 mixed integer programs , 2011, Comput. Oper. Res..

[30]  Maria Grazia Speranza,et al.  Kernel Search: a new heuristic framework for portfolio selection , 2012, Comput. Optim. Appl..

[31]  Matteo Fischetti,et al.  A feasibility pump heuristic for general mixed-integer problems , 2007, Discret. Optim..