@misc{WitzigBerthold2019, author = {Witzig, Jakob and Berthold, Timo}, title = {Conflict-Free Learning for Mixed Integer Programming}, issn = {1438-0064}, doi = {10.1007/978-3-030-58942-4_34}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-75338}, year = {2019}, abstract = {Conflict learning plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. A major step for MIP conflict learning is to aggregate the LP relaxation of an infeasible subproblem to a single globally valid constraint, the dual proof, that proves infeasibility within the local bounds. Among others, one way of learning is to add these constraints to the problem formulation for the remainder of the search. We suggest to not restrict this procedure to infeasible subproblems, but to also use global proof constraints from subproblems that are not (yet) infeasible, but can be expected to be pruned soon. As a special case, we also consider learning from integer feasible LP solutions. First experiments of this conflict-free learning strategy show promising results on the MIPLIB2017 benchmark set.}, language = {en} } @article{GamrathGleixnerKochetal.2019, author = {Gamrath, Gerald and Gleixner, Ambros and Koch, Thorsten and Miltenberger, Matthias and Kniasew, Dimitri and Schl{\"o}gel, Dominik and Martin, Alexander and Weninger, Dieter}, title = {Tackling Industrial-Scale Supply Chain Problems by Mixed-Integer Programming}, volume = {37}, journal = {Journal of Computational Mathematics}, doi = {10.4208/jcm.1905-m2019-0055}, pages = {866 -- 888}, year = {2019}, abstract = {The modeling flexibility and the optimality guarantees provided by mixed-integer programming greatly aid the design of robust and future-proof decision support systems. The complexity of industrial-scale supply chain optimization, however, often poses limits to the application of general mixed-integer programming solvers. In this paper we describe algorithmic innovations that help to ensure that MIP solver performance matches the complexity of the large supply chain problems and tight time limits encountered in practice. Our computational evaluation is based on a diverse set, modeling real-world scenarios supplied by our industry partner SAP.}, language = {en} } @article{Pulaj2019, author = {Pulaj, Jonad}, title = {Cutting planes for families implying Frankl's conjecture}, journal = {Mathematics of Computation}, doi = {10.1090/mcom/3461}, year = {2019}, abstract = {We find previously unknown families of sets which ensure Frankl's conjecture holds for all families that contain them using an algorithmic framework. The conjecture states that for any nonempty finite union-closed (UC) family there exists an element of the ground set in at least half the sets of the considered UC family. Poonen's Theorem characterizes the existence of weights which determine whether a given UC family implies the conjecture for all UC families which contain it. We design a cutting-plane method that computes the explicit weights which satisfy the existence conditions of Poonen's Theorem. This method enables us to answer several open questions regarding structural properties of UC families, including the construction of a counterexample to a conjecture of Morris from 2006.}, language = {en} } @article{BreuerBussieckFiandetal.2019, author = {Breuer, Thomas and Bussieck, Michael and Fiand, Frederik and Cao, Karl-Ki{\^e}n and Gils, Hans Christian and Wetzel, Manuel and Gleixner, Ambros and Koch, Thorsten and Rehfeldt, Daniel and Khabi, Dmitry}, title = {BEAM-ME: Ein interdisziplin{\"a}rer Beitrag zur Erreichung der Klimaziele}, journal = {OR-News : das Magazin der GOR}, number = {66}, pages = {6 -- 8}, year = {2019}, language = {de} } @misc{Serrano2019, author = {Serrano, Felipe}, title = {Visible points, the separation problem, and applications to MINLP}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74016}, year = {2019}, abstract = {In this paper we introduce a technique to produce tighter cutting planes for mixed-integer non-linear programs. Usually, a cutting plane is generated to cut off a specific infeasible point. The underlying idea is to use the infeasible point to restrict the feasible region in order to obtain a tighter domain. To ensure validity, we require that every valid cut separating the infeasible point from the restricted feasible region is still valid for the original feasible region. We translate this requirement in terms of the separation problem and the reverse polar. In particular, if the reverse polar of the restricted feasible region is the same as the reverse polar of the feasible region, then any cut valid for the restricted feasible region that \emph{separates} the infeasible point, is valid for the feasible region. We show that the reverse polar of the \emph{visible points} of the feasible region from the infeasible point coincides with the reverse polar of the feasible region. In the special where the feasible region is described by a single non-convex constraint intersected with a convex set we provide a characterization of the visible points. Furthermore, when the non-convex constraint is quadratic the characterization is particularly simple. We also provide an extended formulation for a relaxation of the visible points when the non-convex constraint is a general polynomial. Finally, we give some conditions under which for a given set there is an inclusion-wise smallest set, in some predefined family of sets, whose reverse polars coincide.}, language = {en} } @misc{MuellerSerranoGleixner2019, author = {M{\"u}ller, Benjamin and Serrano, Felipe and Gleixner, Ambros}, title = {Using two-dimensional Projections for Stronger Separation and Propagation of Bilinear Terms}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-72759}, year = {2019}, abstract = {One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over a strict, non-rectangular subset of the box. In order to exploit this in practice, we propose to compute valid linear inequalities for the projection of the feasible region onto the x-y-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for x, y, and xy over the available projections. Our computational evaluation using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.}, language = {en} } @misc{BertholdGamrathSalvagnin2019, author = {Berthold, Timo and Gamrath, Gerald and Salvagnin, Domenico}, title = {Exploiting Dual Degeneracy in Branching}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-73028}, year = {2019}, abstract = {Branch-and-bound methods for mixed-integer programming (MIP) are traditionally based on solving a linear programming (LP) relaxation and branching on a variable which takes a fractional value in the (single) computed relaxation optimum. In this paper, we study branching strategies for mixed-integer programs that exploit the knowledge of multiple alternative optimal solutions (a cloud ) of the current LP relaxation. These strategies naturally extend common methods like most infeasible branching, strong branching, pseudocost branching, and their hybrids, but we also propose a novel branching rule called cloud diameter branching. We show that dual degeneracy, a requirement for alternative LP optima, is present for many instances from common MIP test sets. Computational experiments show significant improvements in the quality of branching decisions as well as reduced branching effort when using our modifications of existing branching rules. We discuss different ways to generate a cloud of solutions and present extensive computational results showing that through a careful implementation, cloud modifications can speed up full strong branching by more than 10 \% on standard test sets. Additionally, by exploiting degeneracy, we are also able to improve the state-of-the-art hybrid branching rule and reduce the solving time on affected instances by almost 20 \% on average.}, language = {en} } @misc{SerranoSchwarzGleixner2019, author = {Serrano, Felipe and Schwarz, Robert and Gleixner, Ambros}, title = {On the Relation between the Extended Supporting Hyperplane Algorithm and Kelley's Cutting Plane Algorithm}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-73253}, year = {2019}, abstract = {Recently, Kronqvist et al. (2016) rediscovered the supporting hyperplane algorithm of Veinott (1967) and demonstrated its computational benefits for solving convex mixed-integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley's cutting plane algorithm applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by general, not necessarily convex, differentiable functions that satisfy a mild condition.}, language = {en} } @inproceedings{ShinanoRehfeldtKoch2019, author = {Shinano, Yuji and Rehfeldt, Daniel and Koch, Thorsten}, title = {Building Optimal Steiner Trees on Supercomputers by Using up to 43,000 Cores}, volume = {11494}, booktitle = {Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2019}, publisher = {Springer}, doi = {10.1007/978-3-030-19212-9_35}, pages = {529 -- 539}, year = {2019}, abstract = {SCIP-JACK is a customized, branch-and-cut based solver for Steiner tree and related problems. ug [SCIP-JACK, MPI] extends SCIP-JACK to a massively parallel solver by using the Ubiquity Generator (UG) framework. ug [SCIP-JACK, MPI] was the only solver that could run on a distributed environment at the (latest) 11th DIMACS Challenge in 2014. Furthermore, it could solve three well-known open instances and updated 14 best-known solutions to instances from the benchmark libary STEINLIB. After the DIMACS Challenge, SCIP-JACK has been considerably improved. However, the improvements were not reflected on ug [SCIP- JACK, MPI]. This paper describes an updated version of ug [SCIP-JACK, MPI], especially branching on constrains and a customized racing ramp-up. Furthermore, the different stages of the solution process on a supercomputer are described in detail. We also show the latest results on open instances from the STEINLIB.}, language = {en} } @misc{WitzigBertholdHeinz2019, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {Computational Aspects of Infeasibility Analysis in Mixed Integer Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74962}, year = {2019}, abstract = {The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications, obtained by domain propagation, that led to infeasibility. The result of this analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept is called conflict graph analysis and has its origin in solving satisfiability problems and is similarly used in constraint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. Every ray of the dual LP provides a set of multipliers that can be used to generate a single new globally valid linear constraint. This method is called dual proof analysis. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide an intense computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first integrated approach to use both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set MIPLIB 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.}, language = {en} }