A computational status update for exact rational mixed integer programming

The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework th...

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Published in:	Mathematical programming Vol. 197; no. 2; pp. 793 - 812
Main Authors:	Eifler, Leon, Gleixner, Ambros
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2023 Springer
Subjects:	Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical Mixed integer programming Symbolic computations Exact computation Rational arithmetic Certificate of correctness
ISSN:	0025-5610, 1436-4646
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Abstract	The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.
AbstractList	The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours. The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.
Audience	Academic
Author	Gleixner, Ambros Eifler, Leon
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Cites_doi	10.1007/978-3-030-73879-2_12 10.1007/s12532-020-00194-3 10.1287/ijoc.2018.0857 10.1007/978-3-540-79719-7_8 10.1007/s00500-018-3365-9 10.1007/s12532-016-0104-z 10.1145/358438.349318 10.1007/978-3-319-09284-3_31 10.1287/ijoc.1120.0501 10.1007/978-3-540-78800-3_24 10.1287/ijoc.2016.0692 10.1145/2382585.2382589 10.1016/j.orl.2013.08.007 10.1090/mcom/3461 10.1007/s10107-003-0433-3 10.1016/j.orl.2004.04.002 10.1007/s12532-013-0055-6 10.1007/978-3-030-04651-4_46 10.1007/978-3-030-49988-4 10.1007/978-3-319-59250-3_13 10.1016/j.orl.2006.12.010 10.1007/s10107-019-01444-6 10.1287/ijoc.1090.0324 10.1007/978-3-642-38189-8_18 10.1137/1.9780898718027 10.1016/j.scico.2007.08.001
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Keywords	Mixed integer programming Symbolic computations Exact computation Rational arithmetic Certificate of correctness
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Comput.201353305344310237310.1007/s12532-013-0055-61305.90310 Gleixner, A., Gottwald, L., Hoen, A.: PaPILO: Parallel Presolve for Integer and Linear Optimization. https://github.com/scipopt/papilo (accessed May 28, 2021) NeumaierAShcherbinaOSafe bounds in linear and mixed-integer programmingMath. Program.200299283296203904110.1007/s10107-003-0433-31098.90043 Gleixner, A., Hendel, G., Gamrath, G., Achterberg, T., Bastubbe, M., Berthold, T., Christophel, P.M., Jarck, K., Koch, T., Linderoth, J., Lübbecke, M., Mittelmann, H.D., Ozyurt, D., Ralphs, T.K., Salvagnin, D., Shinano, Y.: MIPLIB 2017: Data-Driven Compilation of the 6th Mixed-Integer Programming Library. 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References_xml	– reference: Gleixner, A., Gottwald, L., Hoen, A.: PaPILO: Parallel Presolve for Integer and Linear Optimization. https://github.com/scipopt/papilo (accessed May 28, 2021) – reference: NeumaierAShcherbinaOSafe bounds in linear and mixed-integer programmingMath. Program.200299283296203904110.1007/s10107-003-0433-31098.90043 – reference: BertholdTMeasuring the impact of primal heuristicsOp. Res. Lett.2013416611614313183210.1016/j.orl.2013.08.0071287.90037 – reference: GleixnerASteffyDELinear programming using limited-precision oraclesMath. Program.2020183525554413783010.1007/s10107-019-01444-61450.90006 – reference: Applegate, D., Bixby, R., Chvatal, V., Cook, W.: Concorde TSP Solver (2006) – reference: BurtonBAOzlenMComputing the crosscap number of a knot using integer programming and normal surfacesACM Trans. Math. Softw.2012300277310.1145/2382585.23825891295.57006 – reference: ApplegateDCookWDashSEspinozaDGExact solutions to linear programming problemsOp. Res. Lett.2007356693699236103610.1016/j.orl.2006.12.0101177.90282 – reference: Kenter, F., Skipper, D.: Integer-programming bounds on pebbling numbers of cartesian-product graphs. In: Kim, D., Uma, R.N., Zelikovsky, A. (eds.) Combinatorial Optimization and Applications. pp. 681–695 (2018). https://doi.org/10.1007/978-3-030-04651-4_46 – reference: Lancia, G., Pippia, E., Rinaldi, F.: Using integer programming to search for counterexamples: A case study. In: Kononov, A., Khachay, M., Kalyagin, V.A., Pardalos, P. (eds.) Mathematical Optimization Theory and Operations Research. pp. 69–84 (2020). https://doi.org/10.1007/978-3-030-49988-4 – reference: Wetzler, N., Heule, M.J.H., Hunt, W.A.: DRAT-trim: Efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) Theory and Applications of Satisfiability Testing – SAT 2014. pp. 422–429 (2014). https://doi.org/10.1007/978-3-319-09284-3_31 – reference: AssarfBGawrilowEHerrKJoswigMLorenzBPaffenholzARehnTComputing convex hulls and counting integer points with polymakeMath. Program. Comput.201791138361301210.1007/s12532-016-0104-z1370.90009 – reference: GleixnerASteffyDEWolterKIterative refinement for linear programmingInforms. J. Comput.2016283449464350558110.1287/ijoc.2016.06921348.90460 – reference: FaureGNieuwenhuisROliverasARodríguez-CarbonellEKleine BüningHZhaoXSAT Modulo the Theory of Linear Arithmetic: Exact, Inexact and Commercial SolversTheory and Applications of Satisfiability Testing - SAT 20082008Berlin Heidelberg, Berlin, HeidelbergSpringer779010.1007/978-3-540-79719-7_81138.68537 – reference: Eifler, L., Gleixner, A., Pulaj, J.: A safe computational framework for integer programming applied to Chvátal’s conjecture (2020) – reference: Gleixner, A., Hendel, G., Gamrath, G., Achterberg, T., Bastubbe, M., Berthold, T., Christophel, P.M., Jarck, K., Koch, T., Linderoth, J., Lübbecke, M., Mittelmann, H.D., Ozyurt, D., Ralphs, T.K., Salvagnin, D., Shinano, Y.: MIPLIB 2017: Data-Driven Compilation of the 6th Mixed-Integer Programming Library. 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Snippet	The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision...
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Title	A computational status update for exact rational mixed integer programming
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