TY - CHAP A1 - Gamrath, Gerald A1 - Melchiori, Anna A1 - Berthold, Timo A1 - Gleixner, Ambros A1 - Salvagnin, Domenico T1 - Branching on Multi-aggregated Variables T2 - Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2015 N2 - In mixed-integer programming, the branching rule is a key component to a fast convergence of the branch-and-bound algorithm. The most common strategy is to branch on simple disjunctions that split the domain of a single integer variable into two disjoint intervals. Multi-aggregation is a presolving step that replaces variables by an affine linear sum of other variables, thereby reducing the problem size. While this simplification typically improves the performance of MIP solvers, it also restricts the degree of freedom in variable-based branching rules. We present a novel branching scheme that tries to overcome the above drawback by considering general disjunctions defined by multi-aggregated variables in addition to the standard disjunctions based on single variables. This natural idea results in a hybrid between variable- and constraint-based branching rules. Our implementation within the constraint integer programming framework SCIP incorporates this into a full strong branching rule and reduces the number of branch-and-bound nodes on a general test set of publicly available benchmark instances. For a specific class of problems, we show that the solving time decreases significantly. Y1 - 2015 U6 - https://doi.org/10.1007/978-3-319-18008-3_10 VL - 9075 SP - 141 EP - 156 ER - TY - JOUR A1 - Berthold, Timo A1 - Hendel, Gregor T1 - Shift-and-Propagate JF - Journal of Heuristics N2 - In recent years, there has been a growing interest in the design of general purpose primal heuristics for use inside complete mixed integer programming solvers. Many of these heuristics rely on an optimal LP solution, which may take a significant amount of time to find. In this paper, we address this issue by introducing a pre-root primal heuristic that does not require a previously found LP solution. This heuristic, named Shift-and-Propagate , applies domain propagation techniques to quickly drive a variable assignment towards feasibility. Computational experiments indicate that this heuristic is a powerful supplement to existing rounding and propagation heuristics. Y1 - 2014 U6 - https://doi.org/10.1007/s10732-014-9271-0 VL - 21 IS - 1 SP - 73 EP - 106 ER - TY - JOUR A1 - Vigerske, Stefan A1 - Gleixner, Ambros T1 - SCIP: Global Optimization of Mixed-Integer Nonlinear Programs in a Branch-and-Cut Framework JF - Optimization Methods & Software N2 - This paper describes the extensions that were added to the constraint integer programming framework SCIP in order to enable it to solve convex and nonconvex mixed-integer nonlinear programs (MINLPs) to global optimality. SCIP implements a spatial branch-and-bound algorithm based on a linear outer-approximation, which is computed by convex over- and underestimation of nonconvex functions. An expression graph representation of nonlinear constraints allows for bound tightening, structure analysis, and reformulation. Primal heuristics are employed throughout the solving process to find feasible solutions early. We provide insights into the performance impact of individual MINLP solver components via a detailed computational study over a large and heterogeneous test set. Y1 - 2018 U6 - https://doi.org/10.1080/10556788.2017.1335312 VL - 33 IS - 3 SP - 563 EP - 593 ER - TY - GEN A1 - Gleixner, Ambros A1 - Steffy, Daniel A1 - Wolter, Kati T1 - Iterative Refinement for Linear Programming 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. T3 - ZIB-Report - 15-15 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55118 SN - 1438-0064 ER - TY - GEN A1 - Gamrath, Gerald A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Winkler, Michael T1 - Structure-based primal heuristics for mixed integer programming N2 - Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They help to reach optimality faster and provide good feasible solutions early in the solving process. In this paper, we present two new primal heuristics which take into account global structures available within MIP solvers to construct feasible solutions at the beginning of the solving process. These heuristics follow a large neighborhood search (LNS) approach and use global structures to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. The definition of the neighborhood is done by iteratively fixing variables and propagating these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. The neighborhood is solved as a sub-MIP and solutions are transferred back to the original problem. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about every third instance and therewith help to improve the average solving time. T3 - ZIB-Report - 15-26 KW - mixed-integer programming, large neighborhood search, primal heuristics, domain propagation Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55518 SN - 1438-0064 ER - TY - CHAP A1 - Gamrath, Gerald A1 - Hiller, Benjamin A1 - Witzig, Jakob T1 - Reoptimization Techniques in MIP Solvers N2 - Recently, there have been many successful applications of optimization algorithms that solve a sequence of quite similar mixed-integer programs (MIPs) as subproblems. Traditionally, each problem in the sequence is solved from scratch. In this paper we consider reoptimization techniques that try to benefit from information obtained by solving previous problems of the sequence. We focus on the case that subsequent MIPs differ only in the objective function or that the feasible region is reduced. We propose extensions of the very complex branch-and-bound algorithms employed by general MIP solvers based on the idea to ``warmstart'' using the final search frontier of the preceding solver run. We extend the academic MIP solver SCIP by these techniques to obtain a reoptimizing branch-and-bound solver and report computational results which show the effectiveness of the approach. Y1 - 2015 SN - 978-3-319-20086-6 U6 - https://doi.org/10.1007/978-3-319-20086-6_14 VL - 9125 SP - 181 EP - 192 PB - Springer ET - Experimental Algorithms, Lecture Notes in Computer Science ER - TY - GEN A1 - Gamrath, Gerald A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Winkler, Michael T1 - Structure-driven fix-and-propagate heuristics for mixed integer programming N2 - Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They often provide good feasible solutions early in the solving process and help to solve instances to optimality faster. In this paper, we present a scheme for primal start heuristics that can be executed without previous knowledge of an LP solution or a previously found integer feasible solution. It uses global structures available within MIP solvers to iteratively fix integer variables and propagate these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. If sufficiently many variables can be fixed that way, the resulting problem is solved as an LP and the solution is rounded. If the rounded solution did not provide a feasible solution already, a sub-MIP is solved for the neighborhood defined by the variable fixings performed in the first phase. The global structures help to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. We present three primal heuristics that use this scheme based on different global structures. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about three out of five instances and therewith help to improve several performance measures for MIP solvers, including the primal integral and the average solving time. T3 - ZIB-Report - 17-56 KW - mixed-integer programming KW - primal heuristics KW - fix-and-propagate KW - large neighborhood search KW - domain propagation Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-65387 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 - TY - JOUR A1 - Gamrath, Gerald A1 - Koch, Thorsten A1 - Martin, Alexander A1 - Miltenberger, Matthias A1 - Weninger, Dieter T1 - Progress in presolving for mixed integer programming JF - Mathematical Programming Computation N2 - This paper describes three presolving techniques for solving mixed integer programming problems (MIPs) that were implemented in the academic MIP solver SCIP. The task of presolving is to reduce the problem size and strengthen the formulation, mainly by eliminating redundant information and exploiting problem structures. The first method fixes continuous singleton columns and extends results known from duality fixing. The second analyzes and exploits pairwise dominance relations between variables, whereas the third detects isolated subproblems and solves them independently. The performance of the presented techniques is demonstrated on two MIP test sets. One contains all benchmark instances from the last three MIPLIB versions, while the other consists of real-world supply chain management problems. The computational results show that the combination of all three presolving techniques almost halves the solving time for the considered supply chain management problems. For the MIPLIB instances we obtain a speedup of 20 % on affected instances while not degrading the performance on the remaining problems. Y1 - 2015 U6 - https://doi.org/10.1007/s12532-015-0083-5 VL - 7 IS - 4 SP - 367 EP - 398 ER -