@misc{AchterbergKochMartin, author = {Achterberg, Tobias and Koch, Thorsten and Martin, Alexander}, title = {Branching on History Information}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6990}, number = {02-32}, abstract = {Mixed integer programs (\$MIPs\$) are commonly solved with branch and bound algorithms based on linear programming. The success and the speed of the algorithm strongly depends on the strategy used to select the branching variables. Today's state-of-the-art strategy is called \emph{pseudocost branching} and uses information of previous branchings to determine the current branching. We propose a modification of \emph{pseudocost branching} which we call \emph{history branching}. This strategy has been implemented in \$SIP\$, a state-of-the-art \$MIP\$ solver. We give computational results that show the superiority of the new strategy.}, language = {en} } @misc{AchterbergBerthold, author = {Achterberg, Tobias and Berthold, Timo}, title = {Improving the Feasibility Pump}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8754}, number = {05-42}, abstract = {The Feasibility Pump of Fischetti, Glover, Lodi, and Bertacco has proved to be a very successful heuristic for finding feasible solutions of mixed integer programs. The quality of the solutions in terms of the objective value, however, tends to be poor. This paper proposes a slight modification of the algorithm in order to find better solutions. Extensive computational results show the success of this variant: in 89 out of 121 MIP instances the modified version produces improved solutions in comparison to the original Feasibility Pump.}, language = {en} } @misc{Helmberg, author = {Helmberg, Christoph}, title = {Numerical Evaluation of SBmethod}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6637}, number = {01-37}, abstract = {We report numerical results for SBmethod --- a publically available implementation of the spectral bundle method --- applied to the 7\$^{th}\$ DIMACS challenge test sets that are semidefinite relaxations of combinatorial optimization problems. The performance of the code is heavily influenced by parameters that control bundle update and eigenvalue computation. Unfortunately, no mathematically sound guidelines for setting them are known. Based on our experience with SBmethod, we propose heuristics for dynamically updating the parameters as well as a heuristc for improving the starting point. These are now the default settings of SBmethod Version 1.1. We compare their performance on the DIMACS instances to our previous best choices for Version 1.0. SBmethod Version 1.1 is also part of the independent DIMACS benchmark by H.~Mittelmann. Based on these results we try to analyze strengths and weaknesses of our approach in comparison to other codes for large scale semidefinite programming.}, language = {en} } @misc{Achterberg, author = {Achterberg, Tobias}, title = {SCIP - a framework to integrate Constraint and Mixed Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7947}, number = {04-19}, abstract = {Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today's state-of-the-art algorithms of both fields have several strategies in common, in particular the branch-and-bound process to recursively divide the problem into smaller sub problems. On the other hand, the main techniques to process each sub problem are different, and it was observed that they have complementary strenghts. We propose a programming framework {\sffamily SCIP} that integrates techniques from both fields in order to exploit the strenghts of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, {\sffamily SCIP} does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, in-depth control and high performance.}, language = {en} } @misc{AchterbergKochMartin, author = {Achterberg, Tobias and Koch, Thorsten and Martin, Alexander}, title = {Branching rules revisited}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7886}, number = {04-13}, abstract = {Mixed integer programs are commonly solved with linear programming based branch-and-bound algorithms. The success of the algorithm strongly depends on the strategy used to select the variable to branch on. We present a new generalization called {\sl reliability branching} of today's state-of-the-art {\sl strong branching} and {\sl pseudocost branching} strategies for linear programming based branch-and-bound algorithms. After reviewing commonly used branching strategies and performing extensive computational studies we compare different parameter settings and show the superiority of our proposed newstrategy.}, language = {en} } @misc{KosterZymolka, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Minimum Converter Wavelength Assignment in All-Optical Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7673}, number = {03-45}, abstract = {Finding conflict-free wavelength assignments with a minimum number of required conversions for a routing of the lightpaths is one of the important tasks within the design of all-optical networks. We consider this problem in multi-fiber networks with different types of WDM systems. We give a detailed description of the problem and derive its theoretical complexity. For practical application, we propose several sequential algorithms to compute appropriate wavelength assignments. We also perform computational experiments to evaluate their performance. For the iterative algorithms, we identify characteristic patterns of progression. Two of these algorithms qualify for application in practice.}, language = {en} } @misc{AchterbergBrinkmannWedler, author = {Achterberg, Tobias and Brinkmann, Raik and Wedler, Markus}, title = {Property Checking with Constraint Integer Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10376}, number = {07-37}, abstract = {We address the property checking problem for SoC design verification at the register transfer level (RTL) by integrating techniques from integer programming, constraint programming, and SAT solving. Specialized domain propagation and preprocessing algorithms for individual RTL operations extend a general constraint integer programming framework. Conflict clauses are learned by analyzing infeasible LPs and deductions, and by employing reverse propagation. Experimental results show that our approach outperforms SAT techniques for proving the validity of properties on circuits containing arithmetics.}, language = {en} } @misc{SteffyWolter, author = {Steffy, Daniel and Wolter, Kati}, title = {Valid Linear Programming Bounds for Exact Mixed-Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-12332}, number = {11-08}, abstract = {Fast computation of valid linear programming (LP) bounds serves as an important subroutine for solving mixed-integer programming problems exactly. We introduce a new method for computing valid LP bounds designed for this application. The algorithm corrects approximate LP dual solutions to be exactly feasible, giving a valid bound. Solutions are repaired by performing a projection and a shift to ensure all constraints are satisfied; bound computations are accelerated by reusing structural information through the branch-and-bound tree. We demonstrate this method to be widely applicable and faster than solving a sequence of exact LPs. Several variations of the algorithm are described and computationally evaluated in an exact branch-and-bound algorithm within the mixed-integer programming framework SCIP.}, language = {en} } @misc{GamrathSchubert, author = {Gamrath, Gerald and Schubert, Christoph}, title = {Measuring the impact of branching rules for mixed-integer programming}, issn = {1438-0064}, doi = {10.1007/978-3-319-89920-6_23}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64722}, abstract = {Branching rules are an integral component of the branch-and-bound algorithm typically used to solve mixed-integer programs and subject to intense research. Different approaches for branching are typically compared based on the solving time as well as the size of the branch-and-bound tree needed to prove optimality. The latter, however, has some flaws when it comes to sophisticated branching rules that do not only try to take a good branching decision, but have additional side-effects. We propose a new measure for the quality of a branching rule that distinguishes tree size reductions obtained by better branching decisions from those obtained by such side-effects. It is evaluated for common branching rules providing new insights in the importance of strong branching.}, language = {en} } @misc{PfetschFuegenschuhGeissleretal., author = {Pfetsch, Marc and F{\"u}genschuh, Armin and Geißler, Bj{\"o}rn and Geißler, Nina and Gollmer, Ralf and Hiller, Benjamin and Humpola, Jesco and Koch, Thorsten and Lehmann, Thomas and Martin, Alexander and Morsi, Antonio and R{\"o}vekamp, Jessica and Schewe, Lars and Schmidt, Martin and Schultz, R{\"u}diger and Schwarz, Robert and Schweiger, Jonas and Stangl, Claudia and Steinbach, Marc and Vigerske, Stefan and Willert, Bernhard}, title = {Validation of Nominations in Gas Network Optimization: Models, Methods, and Solutions}, issn = {1438-0064}, doi = {10.1080/10556788.2014.888426}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16531}, abstract = {In this article we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: Given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult mixed-integer non-convex nonlinear feasibility problem. The first phase consists of four distinct algorithms facilitating mixed-integer linear, mixed-integer nonlinear, reduced nonlinear, and complementarity constrained methods to compute possible settings for the discrete decisions. The second phase employs a precise continuous nonlinear programming model of the gas network. Using this setup, we are able to compute high quality solutions to real-world industrial instances whose size is significantly larger than networks that have appeared in the literature previously.}, language = {en} } @misc{BestuzhevaGleixnerVigerske, author = {Bestuzheva, Ksenia and Gleixner, Ambros and Vigerske, Stefan}, title = {A Computational Study of Perspective Cuts}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81821}, abstract = {The benefits of cutting planes based on the perspective function are well known for many specific classes of mixed-integer nonlinear programs with on/off structures. However, we are not aware of any empirical studies that evaluate their applicability and computational impact over large, heterogeneous test sets in general-purpose solvers. This paper provides a detailed computational study of perspective cuts within a linear programming based branch-and-cut solver for general mixed-integer nonlinear programs. Within this study, we extend the applicability of perspective cuts from convex to nonconvex nonlinearities. This generalization is achieved by applying a perspective strengthening to valid linear inequalities which separate solutions of linear relaxations. The resulting method can be applied to any constraint where all variables appearing in nonlinear terms are semi-continuous and depend on at least one common indicator variable. Our computational experiments show that adding perspective cuts for convex constraints yields a consistent improvement of performance, and adding perspective cuts for nonconvex constraints reduces branch-and-bound tree sizes and strengthens the root node relaxation, but has no significant impact on the overall mean time.}, language = {en} } @misc{CookKochSteffyetal., author = {Cook, William and Koch, Thorsten and Steffy, Daniel and Wolter, Kati}, title = {An Exact Rational Mixed-Integer Programming Solver}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-12329}, number = {11-07}, abstract = {We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the MIPLIB and Mittelmann collections.}, language = {en} } @misc{GamrathKochMartinetal., author = {Gamrath, Gerald and Koch, Thorsten and Martin, Alexander and Miltenberger, Matthias and Weninger, Dieter}, title = {Progress in Presolving for Mixed Integer Programming}, issn = {1438-0064}, doi = {10.1007/s12532-015-0083-5}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42530}, abstract = {Presolving attempts to eliminate redundant information from the problem formulation and simultaneously tries to strengthen the formulation. It can be very effective and is often essential for solving instances. Especially for mixed integer programming problems, fast and effective presolving algorithms are very important. In this paper, we report on three new presolving techniques. The first method searches for singleton continuous columns and tries to fix the corresponding variables. Then we present a presolving technique which exploits a partial order of the variables to induce fixings. Finally, we show an approach based on connected components in graphs. Our computational results confirm the profitable use of the algorithms in practice.}, language = {en} } @misc{Gamrath, author = {Gamrath, Gerald}, title = {Improving strong branching by domain propagation}, issn = {1438-0064}, doi = {10.1007/s13675-014-0021-8}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42546}, abstract = {One of the essential components of a branch-and-bound based mixed-integer linear programming (MIP) solver is the branching rule. Strong branching is a method used by many state-of-the-art branching rules to select the variable to branch on. It precomputes the dual bounds of potential child nodes by solving auxiliary linear programs (LPs) and thereby helps to take good branching decisions that lead to a small search tree. In this paper, we describe how these dual bound predictions can be improved by including domain propagation into strong branching. Domain propagation is a technique usually used at every node of the branch-and-bound tree to tighten the local domains of variables. Computational experiments on standard MIP instances indicate that our improved strong branching method significantly improves the quality of the predictions and causes almost no additional effort. For a full strong branching rule, we are able to obtain substantial reductions of the branch-and-bound tree size as well as the solving time. Moreover, also the state-of-the-art hybrid branching rule can be improved this way. This paper extends previous work by the author published in the proceedings of the CPAIOR 2013.}, language = {en} } @misc{Achterberg, author = {Achterberg, Tobias}, title = {Conflict Analysis in Mixed Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8537}, number = {05-19}, abstract = {Conflict analysis for infeasible subproblems is one of the key ingredients in modern SAT solvers to cope with large real-world instances. In contrast, it is common practice for today's mixed integer programming solvers to just discard infeasible subproblems and the information they reveal. In this paper we try to remedy this situation by generalizing the SAT infeasibility analysis to mixed integer programming. We present heuristics for branch-and-cut solvers to generate valid inequalities from the current infeasible subproblem and the associated branching information. SAT techniques can then be used to strengthen the resulting cuts. We performed computational experiments which show the potential of our method: On feasible MIP instances, the number of required branching nodes was reduced by 50\\% in the geometric mean. However, the total solving time increased by 15\\%. on infeasible MIPs arising in the context of chip verification, the number of nodes was reduced by 90\\%, thereby reducing the solving time by 60\\%.}, language = {en} } @misc{BestuzhevaGleixnerVoelker, author = {Bestuzheva, Ksenia and Gleixner, Ambros and V{\"o}lker, Helena}, title = {Strengthening SONC Relaxations with Constraints Derived from Variable Bounds}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88306}, abstract = {Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since the computational cost depends only on the number of terms in the polynomials and does not depend on the degrees of the polynomials. This work is a first step to integrating SONC-based relaxations of polynomial problems into a branch-and-bound algorithm. To this end, the SONC relaxation for constrained optimization problems is extended in order to better utilize variable bounds, since this property is key for the success of a relaxation in the context of branch-and-bound. Computational experiments show that the proposed extension is crucial for making the SONC relaxations applicable to most constrained polynomial optimization problems and for integrating the two approaches.}, language = {en} } @misc{TjusilaBesanconTurneretal., author = {Tjusila, Gennesaret and Besancon, Mathieu and Turner, Mark and Koch, Thorsten}, title = {How Many Clues To Give? A Bilevel Formulation For The Minimum Sudoku Clue Problem}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-90902}, abstract = {It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95\\% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.}, language = {en} }