TY - GEN A1 - Witzig, Jakob A1 - Berthold, Timo A1 - Heinz, Stefan T1 - Experiments with Conflict Analysis in Mixed Integer Programming N2 - The analysis of infeasible subproblems plays an import 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 the analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept 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. The dual LP solution provides a set of multipliers that can be used to generate a single new globally valid linear constraint. The main contribution of this short paper is an empirical evaluation of two ways to combine both approaches. Experiments are carried out on general MIP instances from standard public test sets such as Miplib2010; the presented algorithms have been implemented within the non-commercial MIP solver SCIP. Moreover, we present a pool-based approach to manage conflicts which addresses the way a MIP solver traverses the search tree better than aging strategies known from SAT solving. T3 - ZIB-Report - 16-63 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61087 SN - 1438-0064 ER - TY - GEN A1 - Cheung, Kevin K. H. A1 - Gleixner, Ambros A1 - Steffy, Daniel T1 - Verifying Integer Programming Results N2 - Software for mixed-integer linear programming can return incorrect results for a number of reasons, one being the use of inexact floating-point arithmetic. Even solvers that employ exact arithmetic may suffer from programming or algorithmic errors, motivating the desire for a way to produce independently verifiable certificates of claimed results. Due to the complex nature of state-of-the-art MILP solution algorithms, the ideal form of such a certificate is not entirely clear. This paper proposes such a certificate format, illustrating its capabilities and structure through examples. The certificate format is designed with simplicity in mind and is composed of a list of statements that can be sequentially verified using a limited number of simple yet powerful inference rules. We present a supplementary verification tool for compressing and checking these certificates independently of how they were created. We report computational results on a selection of mixed-integer linear programming instances from the literature. To this end, we have extended the exact rational version of the MIP solver SCIP to produce such certificates. T3 - ZIB-Report - 16-58 KW - correctness, verification, proof, certificate, optimality, infeasibility, mixed-integer linear programming Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61044 SN - 1438-0064 ER - TY - GEN A1 - Gamrath, Gerald A1 - Fischer, Tobias A1 - Gally, Tristan A1 - Gleixner, Ambros A1 - Hendel, Gregor A1 - Koch, Thorsten A1 - Maher, Stephen J. A1 - Miltenberger, Matthias A1 - Müller, Benjamin A1 - Pfetsch, Marc A1 - Puchert, Christian A1 - Rehfeldt, Daniel A1 - Schenker, Sebastian A1 - Schwarz, Robert A1 - Serrano, Felipe A1 - Shinano, Yuji A1 - Vigerske, Stefan A1 - Weninger, Dieter A1 - Winkler, Michael A1 - Witt, Jonas T. A1 - Witzig, Jakob T1 - The SCIP Optimization Suite 3.2 N2 - The SCIP Optimization Suite is a software toolbox for generating and solving various classes of mathematical optimization problems. Its major components are the modeling language ZIMPL, the linear programming solver SoPlex, the constraint integer programming framework and mixed-integer linear and nonlinear programming solver SCIP, the UG framework for parallelization of branch-and-bound-based solvers, and the generic branch-cut-and-price solver GCG. It has been used in many applications from both academia and industry and is one of the leading non-commercial solvers. This paper highlights the new features of version 3.2 of the SCIP Optimization Suite. Version 3.2 was released in July 2015. This release comes with new presolving steps, primal heuristics, and branching rules within SCIP. In addition, version 3.2 includes a reoptimization feature and improved handling of quadratic constraints and special ordered sets. SoPlex can now solve LPs exactly over the rational number and performance improvements have been achieved by exploiting sparsity in more situations. UG has been tested successfully on 80,000 cores. A major new feature of UG is the functionality to parallelize a customized SCIP solver. GCG has been enhanced with a new separator, new primal heuristics, and improved column management. Finally, new and improved extensions of SCIP are presented, namely solvers for multi-criteria optimization, Steiner tree problems, and mixed-integer semidefinite programs. T3 - ZIB-Report - 15-60 KW - mixed-integer linear and nonlinear programming KW - MIP solver KW - MINLP solver KW - linear programming KW - LP solver KW - simplex method KW - modeling KW - parallel branch-and-bound KW - branch-cut-and-price framework KW - generic column generation KW - Steiner tree solver KW - multi-criteria optimization KW - mixed-integer semidefinite programming Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-57675 SN - 1438-0064 ER - TY - GEN A1 - Gamrath, Gerald A1 - Gleixner, Ambros A1 - Koch, Thorsten A1 - Miltenberger, Matthias A1 - Kniasew, Dimitri A1 - Schlögel, Dominik A1 - Martin, Alexander A1 - Weninger, Dieter T1 - Tackling Industrial-Scale Supply Chain Problems by Mixed-Integer Programming N2 - SAP's decision support systems for optimized supply network planning rely on mixed-integer programming as the core engine to compute optimal or near-optimal solutions. The modeling flexibility and the optimality guarantees provided by mixed-integer programming greatly aid the design of a robust and future-proof decision support system for a large and diverse customer base. In this paper we describe our coordinated efforts to ensure that the performance of the underlying solution algorithms matches the complexity of the large supply chain problems and tight time limits encountered in practice. T3 - ZIB-Report - 16-45 KW - supply chain management, supply network optimization, mixed-integer linear programming, primal heuristics, numerical stability, large-scale optimization Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61107 SN - 1438-0064 ER - TY - GEN A1 - Gottwald, Robert Lion A1 - Maher, Stephen J. A1 - Shinano, Yuji T1 - Distributed domain propagation N2 - Portfolio parallelization is an approach that runs several solver instances in parallel and terminates when one of them succeeds in solving the problem. Despite it's simplicity portfolio parallelization has been shown to perform well for modern mixed-integer programming (MIP) and boolean satisfiability problem (SAT) solvers. Domain propagation has also been shown to be a simple technique in modern MIP and SAT solvers that effectively finds additional domain reductions after a variables domain has been reduced. This paper investigates the impact of distributed domain propagation in modern MIP solvers that employ portfolio parallelization. Computational experiments were conducted for two implementations of this parallelization approach. While both share global variable bounds and solutions they communicate differently. In one implementation the communication is performed only at designated points in the solving process and in the other it is performed completely asynchronously. Computational experiments show a positive performance impact of communicating global variable bounds and provide valuable insights in communication strategies for parallel solvers. T3 - ZIB-Report - 16-71 KW - mixed integer programming KW - parallelization KW - domain propagation KW - portfolio solvers Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61380 SN - 1438-0064 ER - TY - JOUR A1 - Gleixner, Ambros A1 - Steffy, Daniel A1 - Wolter, Kati T1 - Iterative Refinement for Linear Programming JF - INFORMS Journal on Computing N2 - We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification of the objective function, right-hand side, and variable bounds. Exact computation is used to compute and store the exact representation of the transformed problems, while numeric computation is used for solving LPs. At all steps of the algorithm the LP bases encountered in the transformed problems correspond directly to LP bases in the original problem description. We show that this algorithm is effective in practice for computing extended precision solutions and that it leads to a direct improvement of the best known methods for solving LPs exactly over the rational numbers. Our implementation is publically available as an extension of the academic LP solver SoPlex. Y1 - 2016 U6 - https://doi.org/10.1287/ijoc.2016.0692 VL - 28 IS - 3 SP - 449 EP - 464 ER - TY - CHAP A1 - Hendel, Gregor T1 - Exploiting Solving Phases for Mixed-Integer Programs T2 - Operations Research Proceedings 2015 N2 - Modern MIP solving software incorporates dozens of auxiliary algorithmic components for supporting the branch-and-bound search in finding and improving solutions and in strengthening the relaxation. Intuitively, a dynamic solving strategy with an appropriate emphasis on different solving components and strategies is desirable during the search process. We propose an adaptive solver behavior that dynamically reacts on transitions between the three typical phases of a MIP solving process: The first phase objective is to find a feasible solution. During the second phase, a sequence of incumbent solutions gets constructed until the incumbent is eventually optimal. Proving optimality is the central objective of the remaining third phase. Based on the MIP-solver SCIP, we demonstrate the usefulness of the phase concept both with an exact recognition of the optimality of a solution, and provide heuristic alternatives to make use of the concept in practice. Y1 - 2016 U6 - https://doi.org/10.1007/978-3-319-42902-1_1 SP - 3 EP - 9 ER - TY - GEN A1 - Maher, Stephen J. A1 - Miltenberger, Matthias A1 - Pedroso, João Pedro A1 - Rehfeldt, Daniel A1 - Schwarz, Robert A1 - Serrano, Felipe T1 - PySCIPOpt: Mathematical Programming in Python with the SCIP Optimization Suite N2 - SCIP is a solver for a wide variety of mathematical optimization problems. It is written in C and extendable due to its plug-in based design. However, dealing with all C specifics when extending SCIP can be detrimental to development and testing of new ideas. This paper attempts to provide a remedy by introducing PySCIPOpt, a Python interface to SCIP that enables users to write new SCIP code entirely in Python. We demonstrate how to intuitively model mixed-integer linear and quadratic optimization problems and moreover provide examples on how new Python plug-ins can be added to SCIP. T3 - ZIB-Report - 16-64 KW - SCIP, Mathematical optimization, Python, Modeling Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61348 SN - 1438-0064 ER - TY - GEN A1 - Berthold, Timo A1 - Hendel, Gregor A1 - Koch, Thorsten T1 - The Three Phases of MIP Solving N2 - Modern MIP solvers employ dozens of auxiliary algorithmic components to support the branch-and-bound search in finding and improving primal solutions and in strengthening the dual bound. Typically, all components are tuned to minimize the average running time to prove optimality. In this article, we take a different look at the run of a MIP solver. We argue that the solution process consists of three different phases, namely achieving feasibility, improving the incumbent solution, and proving optimality. We first show that the entire solving process can be improved by adapting the search strategy with respect to the phase-specific aims using different control tunings. Afterwards, we provide criteria to predict the transition between the individual phases and evaluate the performance impact of altering the algorithmic behavior of the MIP solver SCIP at the predicted phase transition points. T3 - ZIB-Report - 16-78 KW - optimization software KW - mixed-integer programming KW - branch-and-bound KW - adaptive search behavior KW - optimality prediction Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61607 SN - 1438-0064 ER -