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Random sampling and machine learning to understand good decompositions

  • S.I.: Decomposition Methods for Hard Optimization Problems
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Abstract

Motivated by its implications in the development of general purpose solvers for decomposable Mixed Integer Programs (MIPs), we address a fundamental research question, that is how to exploit data-driven techniques to obtain automatic decomposition methods. We preliminary investigate the link between static properties of MIP input instances and good decomposition patterns. We devise a random sampling algorithm, considering a set of generic MIP base instances, and generate a large, balanced and well diversified set of decomposition patterns, that we analyze with machine learning tools. We also propose and test a minimal proof of concept framework performing data-driven automatic decomposition. The use of supervised techniques highlights interesting structures of random decompositions, as well as proving (under certain conditions) that data-driven methods are fruitful in our context, triggering at the same time perspectives for future research.

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References

  • Abdi, H., & Williams, L. J. (2010). Principal component analysis. Wiley Interdisciplinary Reviews: Computational Statistics, 2(4), 433–459.

    Article  Google Scholar 

  • Achterberg, T. (2009). SCIP: solving constraint integer programs. Mathematical Programming Computation, 1(1), 1–41.

    Article  Google Scholar 

  • Achterberg, T., Koch, T., & Martin, A. (2006). MIPLIB 2003. Operations Research Letters, 34(4), 361–372.

    Article  Google Scholar 

  • Basso, S., & Ceselli, A. (2017). Asynchronous column generation. In Proceedings of the ninteenth workshop on algorithm engineering and experiments (ALENEX) (pp. 197–206).

  • Basso, S., Ceselli, S., & Tettamanzi, A. (2018). Understanding good decompositions: An exploratory data analysis. Technical report, Università degli Studi di Milano. http://hdl.handle.net/2434/487931.

  • Bergner, M., Caprara, A., Ceselli, A., Furini, F., Lübbecke, M., Malaguti, E., et al. (2015). Automatic Dantzig–Wolfe reformulation of mixed integer programs. Mathematical Programming A, 149(1–2), 391–424.

    Article  Google Scholar 

  • Bettinelli, A., Ceselli, A., & Righini, G. (2010). A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint. Annals of Operations Research, 179, 221–241.

    Article  Google Scholar 

  • Brooks, J. P., & Lee, E. K. (2010). Analysis of the consistency of a mixed integer programming-based multi-category constrained discriminant model. Annals of Operations Research, 174(1), 147–168.

    Article  Google Scholar 

  • Burges, C. (1998). A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2), 121–167.

    Article  Google Scholar 

  • Ceselli, A., Liberatore, F., & Righini, G. (2009). A computational evaluation of a general branch-and-price framework for capacitated network location problems. Annals of Operations Research, 167, 209–251.

    Article  Google Scholar 

  • Delorme, M., Iori, M., & Martello, S. (2016). Bin packing and cutting stock problems: Mathematical models and exact algorithms. European Journal of Operational Research, 255(1), 1–20.

    Article  Google Scholar 

  • Desaulniers, G., Desrosiers, J., & Solomon, M. M. (Eds.). (2005). Column generation. Berlin: Springer.

    Google Scholar 

  • FICO xpress webpage. (2017). http://www.fico.com/en/products/fico-xpress-optimization-suite. Last accessed March, 2017

  • Fisher, R. A. (1992). Statistical methods for research workers. In S. Kotz & N. L. Johnson (Eds.), Breakthroughs in statistics. Springer series in statistics (perspectives in statistics). New York, NY: Springer.

    Google Scholar 

  • Gamrath, G., & Lübbecke, M. E. (2010). Experiments with a generic Dantzig–Wolfe decomposition for integer programs. LNCS 6049 (pp. 239–252).

  • GUROBI webpage. (2017). http://www.gurobi.com. Last accessed March, 2017

  • He, H., & Garcia, E. A. (2009). Learning from imbalanced data. IEEE Transactions on Knowledge and Data Engineering, 21(9), 1263–1284.

    Article  Google Scholar 

  • Hutter, F., Xu, L., Hoos, H. H., & Leyton-Brown, K. (2014). Algorithm runtime prediction: Methods & evaluation. Artificial Intelligence, 206(1), 79–111.

    Article  Google Scholar 

  • IBM Cplex webpage. (2016). http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/index.html. Last accessed August, 2016

  • Khalil, E. B. (2016). Machine learning for integer programming. In Proceedings of the twenty-fifth international joint conference on artificial intelligence.

  • Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R. E., et al. (2011). MIPLIB 2010. Mathematical Programming Computation, 3(2), 103–163.

    Article  Google Scholar 

  • Kruber, M., Luebbecke, M. E., & Parmentier, A. (2016). Learning when to use a decomposition. RWTH technical report 2016-037.

  • Larose, D. T., & Larose, C. D. (2015). Data mining and predictive analytics. Hoboken: Wiley.

    Google Scholar 

  • Mitzenmacher, M., & Upfal, E. (2005). Probability and computing: Randomized algorithms and probabilistic analysis. New York, NY: Cambridge University Press.

    Book  Google Scholar 

  • Puchinger, J., Stuckey, P. J., Wallace, M. G., & Brand, S. (2011). Dantzig–Wolfe decomposition and branch-and-price solving in G12. Constraints, 16(1), 77–99.

    Article  Google Scholar 

  • R Core Team. (2016). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/.

  • Ralphs, T. K., & Galati, M. V. (2017). DIP—decomposition for integer programming. https://projects.coin-or.org/Dip. Last accessed March, 2017.

  • Schrijver, A. (1998). Theory of linear and integer programming. Hoboken: Wiley.

    Google Scholar 

  • Smola, A. J., & Scholkopf, B. (2004). A tutorial on support vector regression. Statistics and Computing, 14, 199–222.

    Article  Google Scholar 

  • Vanderbeck, F. (2017). BaPCod—A generic branch-and-price code. https://wiki.bordeaux.inria.fr/realopt/pmwiki.php/Project/BaPCod. Last accessed March, 2017.

  • Vanderbeck, F., & Wolsey, L. (2010). Reformulation and decomposition of integer programs. In M. Jünger, Th M Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, & L. A. Wolsey (Eds.), 50 years of integer programming 1958–2008. Berlin: Springer.

    Google Scholar 

  • Wang, J., & Ralphs, T. (2013). Computational experience with hypergraph-based methods for automatic decomposition in discrete optimization. In C. Gomes & M. Sellmann (Eds.), Integration of AI and OR techniques in constraint programming for combinatorial optimization problems. LNCS 7874 (pp. 394–402).

  • Wolsey, L. (1998). Integer programming. Hoboken: Wiley.

    Google Scholar 

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Acknowledgements

The authors wish to thank the guest editors and three anonymous reviewers: their insightful comments allowed to substantially improve the manuscript.

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Via Bramante 65, 26013, Crema, CR, Italy

    S. Basso & A. Ceselli

  2. CNRS, I3S, Université Côte d’Azur - INRIA, 930 route des Colles, 06903, Sophia Antipolis CEDEX, France

    A. Tettamanzi

Authors
  1. S. Basso
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  2. A. Ceselli
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  3. A. Tettamanzi
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Corresponding author

Correspondence to A. Ceselli.

Additional information

A. Ceselli: The work has been partially funded by Regione Lombardia—Fondazione Cariplo, Grant No. 2015-0717, project REDNEAT, and partially undertook when the second author was visiting INRIA Sophia Antipolis—I3S CNRS Université Côte d’Azur.

A Appendix

A Appendix

1.1 A.1 Base instances of the dataset

For each instance of the dataset, we present in Table 4 its name, its source (MIPLIB), the overall number of variables (Var.), the number of integer (Int.) and binary (Bin.) variables, the overall number of constraints (Constr.) and the percentage of non-zeroes in the constraint matrix (Nzs). We also report the average time required to solve a decomposition (Time) and the index of dispersion (variance over mean) for both the average time (D(Time)) and the average bound (D(Bound)). Statistics of few decompositions that could not be solved in a reasonable time (weeks), or that could not be solved at all, are excluded.

Initially, further base optimization instances (macrophage, harp2, opt1217) were considered but they were discarded during preprocessing operations.

1.2 A.2 Features of the dataset

For each base optimization problem instance the following features were measured:

  • number of variables

  • number of generic integer variables

  • number of binary variables

  • number of continuous variables

  • total number of constraints

  • number of equality constraints

  • number of inequality constraints

For each decomposition we instead measured:

  • number of blocks

  • average, min, max, standard deviation on the number of variables in blocks

  • average, min, max, standard deviation on the number of generic integer variables in blocks

  • average, min, max, standard deviation on the number of binary variables in blocks

  • average, min, max, standard deviation on the number of continuous variables in blocks

  • average, min, max, standard deviation on the number of constraints in blocks

  • average, min, max, standard deviation on the density of blocks (fraction of nonzero coefficients)

  • average, min, max number of equality constraints in blocks

  • average, min, max number of inequality constraints in blocks

  • average, min, max standard deviation of mean constraints right hand side coefficients (rhs) in blocks

  • average, min, max standard deviation of rhs ranges (max rhs − min rhs) in blocks

  • average, min, max, standard deviation of blocks “shape” (number of variables divided by the number of constraints in each block)

  • average, min, max, standard deviation of “Total Unimodularity Coefficient” of blocks

  • average, min, max, standard deviation of mean objective function coefficients in each block

  • average, min, max, standard deviation of the objective function coefficients range (maximum coefficient − minimum coefficient) in each block

  • number of blocks with both positive and negative coefficients in the objective function

  • number of variables in the border

  • number of generic integer variables in the border

  • number of binary variables in border

  • number of continuous variables in border

  • number of constraints in border

  • density of border (fraction of nonzero coefficients)

  • number of equality constraints in border

  • number of inequality constraints in border

  • average, stdandard deviation and range of rhs in the border

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Basso, S., Ceselli, A. & Tettamanzi, A. Random sampling and machine learning to understand good decompositions. Ann Oper Res 284, 501–526 (2020). https://doi.org/10.1007/s10479-018-3067-9

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  • DOI: https://doi.org/10.1007/s10479-018-3067-9

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