@misc{BertholdWitzig2020, author = {Berthold, Timo and Witzig, Jakob}, title = {Conflict Analysis for MINLP}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78964}, year = {2020}, abstract = {The generalization of MIP techniques to deal with nonlinear, potentially non-convex, constraints have been a fruitful direction of research for computational MINLP in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers towards the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared towards two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality.}, language = {en} } @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} } @inproceedings{WitzigBerthold2020, author = {Witzig, Jakob and Berthold, Timo}, title = {Conflict-Free Learning for Mixed Integer Programming}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2020}, number = {12296}, publisher = {Springer, Cham.}, doi = {10.1007/978-3-030-58942-4_34}, pages = {521 -- 530}, year = {2020}, 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} } @misc{GamrathAndersonBestuzhevaetal.2020, author = {Gamrath, Gerald and Anderson, Daniel and Bestuzheva, Ksenia and Chen, Wei-Kun and Eifler, Leon and Gasse, Maxime and Gemander, Patrick and Gleixner, Ambros and Gottwald, Leona and Halbig, Katrin and Hendel, Gregor and Hojny, Christopher and Koch, Thorsten and Le Bodic, Pierre and Maher, Stephen J. and Matter, Frederic and Miltenberger, Matthias and M{\"u}hmer, Erik and M{\"u}ller, Benjamin and Pfetsch, Marc and Schl{\"o}sser, Franziska and Serrano, Felipe and Shinano, Yuji and Tawfik, Christine and Vigerske, Stefan and Wegscheider, Fabian and Weninger, Dieter and Witzig, Jakob}, title = {The SCIP Optimization Suite 7.0}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78023}, year = {2020}, abstract = {The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming frame- work SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op- timization problems and can be used stand-alone or integrated into SCIP via a presolver plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im- provements in the Benders' decomposition solver of SCIP, user-defined decomposition structures can be read, which are used by the automated Benders' decomposition solver and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation that is used to predict the completion of the overall solving process and potentially trigger restarts. Moreover, substantial performance improvements of the MIP core were achieved by new developments in presolving, primal heuristics, branching rules, conflict analysis, and symmetry handling. Last, not least, the report presents updates to other components and extensions of the SCIP Optimization Suite, in particular, the LP solver SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.}, language = {en} } @misc{HendelAndersonLeBodicetal.2020, author = {Hendel, Gregor and Anderson, Daniel and Le Bodic, Pierre and Pfetsch, Marc}, title = {Estimating the Size of Branch-And-Bound Trees}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78144}, year = {2020}, abstract = {This paper investigates the estimation of the size of Branch-and-Bound (B\&B) trees for solving mixed-integer programs. We first prove that the size of the B\&B tree cannot be approximated within a factor of~2 for general binary programs, unless P equals NP. Second, we review measures of the progress of the B\&B search, such as the gap, and propose a new measure, which we call leaf frequency. We study two simple ways to transform these progress measures into B\&B tree size estimates, either as a direct projection, or via double-exponential smoothing, a standard time-series forecasting technique. We then combine different progress measures and their trends into nontrivial estimates using Machine Learning techniques, which yields more precise estimates than any individual measure. The best method we have identified uses all individual measures as features of a random forest model. In a large computational study, we train and validate all methods on the publicly available MIPLIB and Coral general purpose benchmark sets. On average, the best method estimates B\&B tree sizes within a factor of 3 on the set of unseen test instances even during the early stage of the search, and improves in accuracy as the search progresses. It also achieves a factor 2 over the entire search on each out of six additional sets of homogeneous instances we have tested. All techniques are available in version 7 of the branch-and-cut framework SCIP.}, language = {en} } @inproceedings{BertholdGleixner2013, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover Branching}, volume = {7933}, booktitle = {Experimental Algorithms, 12th International Symposium, SEA 2013, Rome, Italy, June 5-7, 2013, Proceedings}, editor = {Bonifaci, Vincenzo and Demetrescu, Camil and Marchetti-Spaccamela, Alberto}, doi = {10.1007/978-3-642-38527-8_20}, pages = {212 -- 223}, year = {2013}, abstract = {In this paper, we present a new branching strategy for nonconvex MINLP that aims at driving the created subproblems towards linearity. It exploits the structure of a minimum cover of an MINLP, a smallest set of variables that, when fixed, render the remaining system linear: whenever possible, branching candidates in the cover are preferred. Unlike most branching strategies for MINLP, Undercover branching is not an extension of an existing MIP branching rule. It explicitly regards the nonlinearity of the problem while branching on integer variables with a fractional relaxation solution. Undercover branching can be naturally combined with any variable-based branching rule. We present computational results on a test set of general MINLPs from MINLPLib, using the new strategy in combination with reliability branching and pseudocost branching. The computational cost of Undercover branching itself proves negligible. While it turns out that it can influence the variable selection only on a smaller set of instances, for those that are affected, significant improvements in performance are achieved.}, language = {en} } @inproceedings{GleixnerWeltge2013, author = {Gleixner, Ambros and Weltge, Stefan}, title = {Learning and Propagating Lagrangian Variable Bounds for Mixed-Integer Nonlinear Programming}, volume = {7874}, booktitle = {Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 10th International Conference, CPAIOR 2013, Yorktown Heights, NY, USA, May 18-22, 2013}, doi = {10.1007/978-3-642-38171-3_26}, pages = {355 -- 361}, year = {2013}, abstract = {Optimization-based bound tightening (OBBT) is a domain reduction technique commonly used in nonconvex mixed-integer nonlinear programming that solves a sequence of auxiliary linear programs. Each variable is minimized and maximized to obtain the tightest bounds valid for a global linear relaxation. This paper shows how the dual solutions of the auxiliary linear programs can be used to learn what we call Lagrangian variable bound constraints. These are linear inequalities that explain OBBT's domain reductions in terms of the bounds on other variables and the objective value of the incumbent solution. Within a spatial branch-and-bound algorithm, they can be learnt a priori (during OBBT at the root node) and propagated within the search tree at very low computational cost. Experiments with an implementation inside the MINLP solver SCIP show that this reduces the number of branch-and-bound nodes and speeds up solution times.}, language = {en} } @inproceedings{GleixnerSteffyWolter2012, author = {Gleixner, Ambros and Steffy, Daniel and Wolter, Kati}, title = {Improving the Accuracy of Linear Programming Solvers with Iterative Refinement}, booktitle = {ISSAC '12. Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation}, doi = {10.1145/2442829.2442858}, pages = {187 -- 194}, year = {2012}, language = {en} } @article{MuellerSerranoGleixner2020, author = {M{\"u}ller, Benjamin and Serrano, Felipe and Gleixner, Ambros}, title = {Using two-dimensional Projections for Stronger Separation and Propagation of Bilinear Terms}, volume = {30}, journal = {SIAM Journal on Optimization}, number = {2}, doi = {10.1137/19M1249825}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-72767}, pages = {1339 -- 1365}, year = {2020}, 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{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{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} } @misc{AndersonHendelLeBodicetal.2019, author = {Anderson, Daniel and Hendel, Gregor and Le Bodic, Pierre and Viernickel, Jan Merlin}, title = {Clairvoyant Restarts in Branch-and-Bound Search Using Online Tree-Size Estimation}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-72653}, year = {2019}, abstract = {We propose a simple and general online method to measure the search progress within the Branch-and-Bound algorithm, from which we estimate the size of the remaining search tree. We then show how this information can help solvers algorithmically at runtime by designing a restart strategy for Mixed-Integer Programming (MIP) solvers that decides whether to restart the search based on the current estimate of the number of remaining nodes in the tree. We refer to this type of algorithm as clairvoyant. Our clairvoyant restart strategy outperforms a state-of-the-art solver on a large set of publicly available MIP benchmark instances. It is implemented in the MIP solver SCIP and will be available in future releases.}, language = {en} } @misc{GleixnerBertholdMuelleretal.2016, author = {Gleixner, Ambros and Berthold, Timo and M{\"u}ller, Benjamin and Weltge, Stefan}, title = {Three Enhancements for Optimization-Based Bound Tightening}, issn = {1438-0064}, doi = {10.1007/s10898-016-0450-4}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57803}, year = {2016}, abstract = {Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT. Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs. We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17\% to 19\% on average. Most importantly, more instances can be solved when using OBBT.}, language = {en} } @misc{VigerskeGleixner2016, author = {Vigerske, Stefan and Gleixner, Ambros}, title = {SCIP: Global Optimization of Mixed-Integer Nonlinear Programs in a Branch-and-Cut Framework}, issn = {1438-0064}, doi = {10.1080/10556788.2017.1335312}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59377}, year = {2016}, abstract = {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.}, language = {en} } @misc{BertholdGleixner2012, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover: a primal MINLP heuristic exploring a largest sub-MIP}, issn = {1438-0064}, doi = {10.1007/s10107-013-0635-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14631}, number = {12-07}, year = {2012}, abstract = {We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programming (MINLP) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a minimal set of variables that need to be fixed in order to linearize each constraint, a so-called cover. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. We apply domain propagation and conflict analysis to try to avoid infeasibilities and learn from them, respectively. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and general MINLPs from MINLPLib. It turns out that the majority of these instances allow for small covers. Although general in nature, the heuristic appears most promising for MIQCPs, and complements nicely with existing root node heuristics in different state-of-the-art solvers.}, language = {en} } @phdthesis{Gleixner2015, author = {Gleixner, Ambros}, title = {Exact and Fast Algorithms for Mixed-Integer Nonlinear Programming}, publisher = {Logos Verlag Berlin}, isbn = {978-3-8325-4190-3}, pages = {341}, year = {2015}, abstract = {Mixed-integer nonlinear programming (MINLP) comprises the broad class of finite-dimensional mathematical optimization problems from mixed-integer linear programming and global optimization. The combination of the two disciplines allows us to construct more accurate models of real-world systems, while at the same time it increases the algorithmic challenges that come with solving them. This thesis presents new methods that improve the numerical reliability and the computational performance of global MINLP solvers. Since state-of-the-art algorithms for nonconvex MINLP fundamentally rely on solving linear programming (LP) relaxations, we address numerical accuracy directly for LP by means of LP iterative refinement: a new algorithm to solve linear programs to arbitrarily high levels of precision. The thesis is supplemented by an exact extension of the LP solver SoPlex, which proves on average 1.85 to 3 times faster than current state-of-the-art software for solving general linear programs exactly over the rational numbers. These methods can be generalized to quadratic programming. We study their application to numerically difficult multiscale LP models for metabolic networks in systems biology. To improve the computational performance of LP-based MINLP solvers, we show how the expensive, but effective, bound-tightening technique called optimization-based bound tightening can be approximated more efficiently via feasibility-based bound tightening. The resulting implementation increases the number of instances that can be solved and reduces the average running time of the MINLP solver SCIP by 17-19\% on hard mixed-integer nonlinear programs. Last, we present branching rules that exploit the presence of nonlinear integer variables, i.e., variables both contained in nonlinear terms and required to be integral. The new branching rules prefer integer variables when performing spatial branching, and favor variables in nonlinear terms when resolving integer infeasibility. They reduce the average running time of SCIP by 17\% on affected instances. Most importantly, all of the new methods enable us to solve problems which could not be solved before, either due to their numerical complexity or because of limited computing resources.}, language = {en} } @misc{GleixnerEiflerGallyetal.2017, author = {Gleixner, Ambros and Eifler, Leon and Gally, Tristan and Gamrath, Gerald and Gemander, Patrick and Gottwald, Robert Lion and Hendel, Gregor and Hojny, Christopher and Koch, Thorsten and Miltenberger, Matthias and M{\"u}ller, Benjamin and Pfetsch, Marc and Puchert, Christian and Rehfeldt, Daniel and Schl{\"o}sser, Franziska and Serrano, Felipe and Shinano, Yuji and Viernickel, Jan Merlin and Vigerske, Stefan and Weninger, Dieter and Witt, Jonas T. and Witzig, Jakob}, title = {The SCIP Optimization Suite 5.0}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-66297}, year = {2017}, abstract = {This article describes new features and enhanced algorithms made available in version 5.0 of the SCIP Optimization Suite. In its central component, the constraint integer programming solver SCIP, remarkable performance improvements have been achieved for solving mixed-integer linear and nonlinear programs. On MIPs, SCIP 5.0 is about 41 \% faster than SCIP 4.0 and over twice as fast on instances that take at least 100 seconds to solve. For MINLP, SCIP 5.0 is about 17 \% faster overall and 23 \% faster on instances that take at least 100 seconds to solve. This boost is due to algorithmic advances in several parts of the solver such as cutting plane generation and management, a new adaptive coordination of large neighborhood search heuristics, symmetry handling, and strengthened McCormick relaxations for bilinear terms in MINLPs. Besides discussing the theoretical background and the implementational aspects of these developments, the report describes recent additions for the other software packages connected to SCIP, in particular for the LP solver SoPlex, the Steiner tree solver SCIP-Jack, the MISDP solver SCIP-SDP, and the parallelization framework UG.}, language = {en} } @article{Hendel2018, author = {Hendel, Gregor}, title = {Adaptive Large Neighborhood Search for Mixed Integer Programming}, journal = {Mathematical Programming Computation}, year = {2018}, abstract = {Large Neighborhood Search (LNS) heuristics are among the most powerful but also most expensive heuristics for mixed integer programs (MIP). Ideally, a solver learns adaptively which LNS heuristics work best for the MIP problem at hand in order to concentrate its limited computational budget. To this end, this work introduces Adaptive Large Neighborhood Search (ALNS) for MIP, a primal heuristic that acts a framework for eight popular LNS heuristics such as Local Branching and Relaxation Induced Neighborhood Search (RINS). We distinguish the available LNS heuristics by their individual search domains, which we call neighborhoods. The decision which neighborhood should be executed is guided by selection strategies for the multi armed bandit problem, a related optimization problem during which suitable actions have to be chosen to maximize a reward function. In this paper, we propose an LNS-specific reward function to learn to distinguish between the available neighborhoods based on successful calls and failures. A second, algorithmic enhancement is a generic variable fixing priorization, which ALNS employs to adjust the subproblem complexity as needed. This is particularly useful for some neighborhoods which do not fix variables by themselves. The proposed primal heuristic has been implemented within the MIP solver SCIP. An extensive computational study is conducted to compare different LNS strategies within our ALNS framework on a large set of publicly available MIP instances from the MIPLIB and Coral benchmark sets. The results of this simulation are used to calibrate the parameters of the bandit selection strategies. A second computational experiment shows the computational benefits of the proposed ALNS framework within the MIP solver SCIP.}, language = {en} } @inproceedings{WitzigBertholdHeinz2019, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {A Status Report on Conflict Analysis in Mixed Integer Nonlinear Programming}, volume = {11494}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2019}, publisher = {Springer}, doi = {10.1007/978-3-030-19212-9_6}, pages = {84 -- 94}, year = {2019}, abstract = {Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer programming (MIP) solvers in the sense that they are based on a branch-and-cut algorithm, enhanced by various heuristics, domain propagation, and presolving techniques. However, the analysis of infeasible subproblems, which is an important component of most major MIP solvers, has been hardly studied in the context of MINLPs. There are two main approaches for infeasibility analysis in MIP solvers: conflict graph analysis, which originates from artificial intelligence and constraint programming, and dual ray analysis. The main contribution of this short paper is twofold. Firstly, we present the first computational study regarding the impact of dual ray analysis on convex and nonconvex MINLPs. In that context, we introduce a modified generation of infeasibility proofs that incorporates linearization cuts that are only locally valid. Secondly, we describe an extension of conflict analysis that works directly with the nonlinear relaxation of convex MINLPs instead of considering a linear relaxation. This is work-in-progress, and this short paper is meant to present first theoretical considerations without a computational study for that part.}, language = {en} } @inproceedings{BertholdStuckeyWitzig2019, author = {Berthold, Timo and Stuckey, Peter and Witzig, Jakob}, title = {Local Rapid Learning for Integer Programs}, volume = {11494}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2019}, publisher = {Springer}, doi = {10.1007/978-3-030-19212-9_5}, pages = {67 -- 83}, year = {2019}, abstract = {Conflict learning algorithms are an important component of modern MIP and CP solvers. But strong conflict information is typically gained by depth-first search. While this is the natural mode for CP solving, it is not for MIP solving. Rapid Learning is a hybrid CP/MIP approach where CP search is applied at the root to learn information to support the remaining MIP solve. This has been demonstrated to be beneficial for binary programs. In this paper, we extend the idea of Rapid Learning to integer programs, where not all variables are restricted to the domain {0, 1}, and rather than just running a rapid CP search at the root, we will apply it repeatedly at local search nodes within the MIP search tree. To do so efficiently, we present six heuristic criteria to predict the chance for local Rapid Learning to be successful. Our computational experiments indicate that our extended Rapid Learning algorithm significantly speeds up MIP search and is particularly beneficial on highly dual degenerate problems.}, language = {en} } @inproceedings{AndersonHendelLeBodicetal.2018, author = {Anderson, Daniel and Hendel, Gregor and Le Bodic, Pierre and Viernickel, Jan Merlin}, title = {Clairvoyant Restarts in Branch-and-Bound Search Using Online Tree-Size Estimation}, booktitle = {AAAI-19: Proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence}, year = {2018}, abstract = {We propose a simple and general online method to measure the search progress within the branch-and-bound algorithm, from which we estimate the size of the remaining search tree. We then show how this information can help solvers algorithmically at runtime by designing a restart strategy for Mixed-Integer Programming (MIP) solvers that decides whether to restart the search based on the current estimate of the number of remaining nodes in the tree. We refer to this type of algorithm as clairvoyant. Our clairvoyant restart strategy outperforms a state-of-the-art solver on a large set of publicly available MIP benchmark instances. It is implemented in the MIP solver SCIP and will be available in future releases.}, language = {de} } @inproceedings{GleixnerSteffy2019, author = {Gleixner, Ambros and Steffy, Daniel}, title = {Linear Programming using Limited-Precision Oracles}, booktitle = {A. Lodi, V. Nagarajan (eds), Integer Programming and Combinatorial Optimization: 20th International Conference, IPCO 2019}, doi = {10.1007/978-3-030-17953-3_30}, pages = {399 -- 412}, year = {2019}, abstract = {Linear programming is a foundational tool for many aspects of integer and combinatorial optimization. This work studies the complexity of solving linear programs exactly over the rational numbers through use of an oracle capable of returning limited-precision LP solutions. It is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. Previous work has often considered oracles that provide solutions of an arbitrary specified precision. While this leads to polynomial-time algorithms, the level of precision required is often unrealistic for practical computation. In contrast, our work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.}, language = {en} } @article{GamrathBertholdHeinzetal.2019, author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-driven fix-and-propagate heuristics for mixed integer programming}, volume = {11}, journal = {Mathematical Programming Computation}, number = {4}, publisher = {Springer}, address = {Berlin Heidelberg}, doi = {10.1007/s12532-019-00159-1}, pages = {675 -- 702}, year = {2019}, abstract = {Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They often provide good feasible solutions early and help to reduce the time needed to prove optimality. In this paper, we present a scheme for 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 first as an LP, and then as an auxiliary MIP if the rounded LP solution does not provide a feasible solution already. 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 60 \% of the instances and by this, help to improve several performance measures for MIP solvers, including the primal integral and the average solving time.}, language = {en} } @misc{BertholdStuckeyWitzig2018, author = {Berthold, Timo and Stuckey, Peter and Witzig, Jakob}, title = {Local Rapid Learning for Integer Programs}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71190}, year = {2018}, abstract = {Conflict learning algorithms are an important component of modern MIP and CP solvers. But strong conflict information is typically gained by depth-first search. While this is the natural mode for CP solving, it is not for MIP solving. Rapid Learning is a hybrid CP/MIP approach where CP search is applied at the root to learn information to support the remaining MIP solve. This has been demonstrated to be beneficial for binary programs. In this paper, we extend the idea of Rapid Learning to integer programs, where not all variables are restricted to the domain {0, 1}, and rather than just running a rapid CP search at the root, we will apply it repeatedly at local search nodes within the MIP search tree. To do so efficiently, we present six heuristic criteria to predict the chance for local Rapid Learning to be successful. Our computational experiments indicate that our extended Rapid Learning algorithm significantly speeds up MIP search and is particularly beneficial on highly dual degenerate problems.}, language = {en} } @misc{Hendel2018, author = {Hendel, Gregor}, title = {Adaptive Large Neighborhood Search for Mixed Integer Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71167}, year = {2018}, abstract = {Large Neighborhood Search (LNS) heuristics are among the most powerful but also most expensive heuristics for mixed integer programs (MIP). Ideally, a solver learns adaptively which LNS heuristics work best for the MIP problem at hand in order to concentrate its limited computational budget. To this end, this work introduces Adaptive Large Neighborhood Search (ALNS) for MIP, a primal heuristic that acts a framework for eight popular LNS heuristics such as Local Branching and Relaxation Induced Neighborhood Search (RINS). We distinguish the available LNS heuristics by their individual search domains, which we call neighborhoods. The decision which neighborhood should be executed is guided by selection strategies for the multi armed bandit problem, a related optimization problem during which suitable actions have to be chosen to maximize a reward function. In this paper, we propose an LNS-specific reward function to learn to distinguish between the available neighborhoods based on successful calls and failures. A second, algorithmic enhancement is a generic variable fixing priorization, which ALNS employs to adjust the subproblem complexity as needed. This is particularly useful for some neighborhoods which do not fix variables by themselves. The proposed primal heuristic has been implemented within the MIP solver SCIP. An extensive computational study is conducted to compare different LNS strategies within our ALNS framework on a large set of publicly available MIP instances from the MIPLIB and Coral benchmark sets. The results of this simulation are used to calibrate the parameters of the bandit selection strategies. A second computational experiment shows the computational benefits of the proposed ALNS framework within the MIP solver SCIP.}, language = {en} } @misc{WitzigBertholdHeinz2018, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {A Status Report on Conflict Analysis in Mixed Integer Nonlinear Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71170}, year = {2018}, abstract = {Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer programming (MIP) solvers in the sense that they are based on a branch-and-cut algorithm, enhanced by various heuristics, domain propagation, and presolving techniques. However, the analysis of infeasible subproblems, which is an important component of most major MIP solvers, has been hardly studied in the context of MINLPs. There are two main approaches for infeasibility analysis in MIP solvers: conflict graph analysis, which originates from artificial intelligence and constraint programming, and dual ray analysis. The main contribution of this short paper is twofold. Firstly, we present the first computational study regarding the impact of dual ray analysis on convex and nonconvex MINLPs. In that context, we introduce a modified generation of infeasibility proofs that incorporates linearization cuts that are only locally valid. Secondly, we describe an extension of conflict analysis that works directly with the nonlinear relaxation of convex MINLPs instead of considering a linear relaxation. This is work-in-progress, and this short paper is meant to present first theoretical considerations without a computational study for that part.}, language = {en} } @article{WitzigGleixner2020, author = {Witzig, Jakob and Gleixner, Ambros}, title = {Conflict-Driven Heuristics for Mixed Integer Programming}, volume = {33}, journal = {INFORMS Journal on Computing}, number = {2}, doi = {10.1287/ijoc.2020.0973}, pages = {706 -- 720}, year = {2020}, abstract = {Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving heuristic that targets the generation of valid conflict constraints from the Farkas dual. We show that in the primal this is equivalent to the optimistic strategy of diving towards the best bound with respect to the objective function. Secondly, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classical coefficient diving. The computational performance of both methods is evaluated using an implementation in the source-open MIP solver SCIP. Experiments are carried out on publicly available test sets including Miplib 2010 and Cor@l.}, language = {en} } @article{GleixnerHendelGamrathetal.2021, author = {Gleixner, Ambros and Hendel, Gregor and Gamrath, Gerald and Achterberg, Tobias and Bastubbe, Michael and Berthold, Timo and Christophel, Philipp M. and Jarck, Kati and Koch, Thorsten and Linderoth, Jeff and L{\"u}bbecke, Marco and Mittelmann, Hans and Ozyurt, Derya and Ralphs, Ted and Salvagnin, Domenico and Shinano, Yuji}, title = {MIPLIB 2017: Data-Driven Compilation of the 6th Mixed-Integer Programming Library}, volume = {13}, journal = {Mathematical Programming Computation}, number = {3}, doi = {10.1007/s12532-020-00194-3}, pages = {443 -- 490}, year = {2021}, abstract = {We report on the selection process leading to the sixth version of the Mixed Integer Programming Library. Selected from an initial pool of over 5,000 instances, the new MIPLIB 2017 collection consists of 1,065 instances. A subset of 240 instances was specially selected for benchmarking solver performance. For the first time, the compilation of these sets was done using a data-driven selection process supported by the solution of a sequence of mixed integer optimization problems, which encoded requirements on diversity and balancedness with respect to instance features and performance data.}, language = {en} } @misc{MuellerMuñozGasseetal.2019, author = {M{\"u}ller, Benjamin and Muñoz, Gonzalo and Gasse, Maxime and Gleixner, Ambros and Lodi, Andrea and Serrano, Felipe}, title = {On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-75179}, year = {2019}, abstract = {The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm's ability to generate strong dual bounds through extensive computational experiments.}, language = {en} } @misc{GeorgesGleixnerGojicetal.2018, author = {Georges, Alexander and Gleixner, Ambros and Gojic, Gorana and Gottwald, Robert Lion and Haley, David and Hendel, Gregor and Matejczyk, Bartlomiej}, title = {Feature-Based Algorithm Selection for Mixed Integer Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68362}, year = {2018}, abstract = {Mixed integer programming is a versatile and valuable optimization tool. However, solving specific problem instances can be computationally demanding even for cutting-edge solvers. Such long running times are often significantly reduced by an appropriate change of the solver's parameters. In this paper we investigate "algorithm selection", the task of choosing among a set of algorithms the ones that are likely to perform best for a particular instance. In our case, we treat different parameter settings of the MIP solver SCIP as different algorithms to choose from. Two peculiarities of the MIP solving process have our special attention. We address the well-known problem of performance variability by using multiple random seeds. Besides solving time, primal dual integrals are recorded as a second performance measure in order to distinguish solvers that timed out. We collected feature and performance data for a large set of publicly available MIP instances. The algorithm selection problem is addressed by several popular, feature-based methods, which have been partly extended for our purpose. Finally, an analysis of the feature space and performance results of the selected algorithms are presented.}, language = {en} } @inproceedings{D'AndreagiovanniGleixner2016, author = {D'Andreagiovanni, Fabio and Gleixner, Ambros}, title = {Towards an accurate solution of wireless network design problems}, booktitle = {Cerulli R., Fujishige S., Mahjoub A. (eds) Combinatorial Optimization. ISCO 2016}, doi = {10.1007/978-3-319-45587-7_12}, pages = {135 -- 147}, year = {2016}, abstract = {The optimal design of wireless networks has been widely studied in the literature and many optimization models have been proposed over the years. However, most models directly include the signal-to-interference ratios representing service coverage conditions. This leads to mixed-integer linear programs with constraint matrices containing tiny coefficients that vary widely in their order of magnitude. These formulations are known to be challenging even for state-of-the-art solvers: the standard numerical precision supported by these solvers is usually not sufficient to reliably guarantee feasible solutions. Service coverage errors are thus commonly present. Though these numerical issues are known and become evident even for small-sized instances, just a very limited number of papers has tried to tackle them, by mainly investigating alternative non-compact formulations in which the sources of numerical instabilities are eliminated. In this work, we explore a new approach by investigating how recent advances in exact solution algorithms for linear and mixed-integer programs over the rational numbers can be applied to analyze and tackle the numerical difficulties arising in wireless network design models.}, language = {en} } @inproceedings{BertholdGleixner2010, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover - a primal heuristic for MINLP based on sub-MIPs generated by set covering}, booktitle = {Proceedings of the European Workshop on Mixed Integer Nonlinear Programming, April 12-16, 2010, Marseilles, France}, editor = {Bonami, Pierre and Liberti, Leo and Miller, Andrew J. and Sartenaer, Annick}, pages = {103 -- 112}, year = {2010}, abstract = {We present Undercover, a primal heuristic for mixed-integer nonlinear programming (MINLP). The heuristic constructs a mixed-integer linear subproblem (sub-MIP) of a given MINLP by fixing a subset of the variables. We solve a set covering problem to identify a minimal set of variables which need to be fixed in order to linearise each constraint. Subsequently, these variables are fixed to approximate values, e.g. obtained from a linear outer approximation. The resulting sub-MIP is solved by a mixed-integer linear programming solver. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. Although general in nature, the heuristic seems most promising for mixed-integer quadratically constrained programmes (MIQCPs). We present computational results on a general test set of MIQCPs selected from the MINLPLib.}, language = {en} } @inproceedings{BleyGleixnerKochetal.2012, author = {Bley, Andreas and Gleixner, Ambros and Koch, Thorsten and Vigerske, Stefan}, title = {Comparing MIQCP Solvers to a Specialised Algorithm for Mine Production Scheduling}, booktitle = {Modeling, Simulation and Optimization of Complex Processes. Proceedings of the Fourth International Conference on High Performance Scientific Computing, March 2-6, 2009, Hanoi, Vietnam}, doi = {10.1007/978-3-642-25707-0_3}, pages = {25 -- 39}, year = {2012}, abstract = {In this paper we investigate the performance of several out-of-the box solvers for mixed-integer quadratically constrained programmes (MIQCPs) on an open pit mine production scheduling problem with mixing constraints. We compare the solvers BARON, Couenne, SBB, and SCIP to a problem-specific algorithm on two different MIQCP formulations. The computational results presented show that general-purpose solvers with no particular knowledge of problem structure are able to nearly match the performance of a hand-crafted algorithm.}, language = {en} } @misc{MuellerKuhlmannVigerske2017, author = {M{\"u}ller, Benjamin and Kuhlmann, Renke and Vigerske, Stefan}, title = {On the performance of NLP solvers within global MINLP solvers}, issn = {1438-0064}, doi = {10.1007/978-3-319-89920-6_84}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64504}, year = {2017}, abstract = {Solving mixed-integer nonlinear programs (MINLPs) to global optimality efficiently requires fast solvers for continuous sub-problems. These appear in, e.g., primal heuristics, convex relaxations, and bound tightening methods. Two of the best performing algorithms for these sub-problems are Sequential Quadratic Programming (SQP) and Interior Point Methods. In this paper we study the impact of different SQP and Interior Point implementations on important MINLP solver components that solve a sequence of similar NLPs. We use the constraint integer programming framework SCIP for our computational studies.}, language = {en} } @inproceedings{HendelMiltenbergerWitzig2018, author = {Hendel, Gregor and Miltenberger, Matthias and Witzig, Jakob}, title = {Adaptive Algorithmic Behavior for Solving Mixed Integer Programs Using Bandit Algorithms}, booktitle = {OR 2018: International Conference on Operations Research}, year = {2018}, abstract = {State-of-the-art solvers for mixed integer programs (MIP) govern a variety of algorithmic components. Ideally, the solver adaptively learns to concentrate its computational budget on those components that perform well on a particular problem, especially if they are time consuming. We focus on three such algorithms, namely the classes of large neighborhood search and diving heuristics as well as Simplex pricing strategies. For each class we propose a selection strategy that is updated based on the observed runtime behavior, aiming to ultimately select only the best algorithms for a given instance. We review several common strategies for such a selection scenario under uncertainty, also known as Multi Armed Bandit Problem. In order to apply those bandit strategies, we carefully design reward functions to rank and compare each individual heuristic or pricing algorithm within its respective class. Finally, we discuss the computational benefits of using the proposed adaptive selection within the SCIP Optimization Suite on publicly available MIP instances.}, language = {en} } @misc{BertholdGleixner2009, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover - a primal heuristic for MINLP based on sub-MIPs generated by set covering}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11632}, number = {09-40}, year = {2009}, abstract = {We present Undercover, a primal heuristic for mixed-integer nonlinear programming (MINLP). The heuristic constructs a mixed-integer linear subproblem (sub-MIP) of a given MINLP by fixing a subset of the variables. We solve a set covering problem to identify a minimal set of variables which need to be fixed in order to linearise each constraint. Subsequently, these variables are fixed to approximate values, e.g. obtained from a linear outer approximation. The resulting sub-MIP is solved by a mixed-integer linear programming solver. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. Although general in nature, the heuristic seems most promising for mixed-integer quadratically constrained programmes (MIQCPs). We present computational results on a general test set of MIQCPs selected from the MINLPLib.}, language = {en} } @misc{GleixnerWeltge2013, author = {Gleixner, Ambros and Weltge, Stefan}, title = {Learning and Propagating Lagrangian Variable Bounds for Mixed-Integer Nonlinear Programming}, issn = {1438-0064}, doi = {10.1007/978-3-642-38171-3_26}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-17631}, year = {2013}, abstract = {Optimization-based bound tightening (OBBT) is a domain reduction technique commonly used in nonconvex mixed-integer nonlinear programming that solves a sequence of auxiliary linear programs. Each variable is minimized and maximized to obtain the tightest bounds valid for a global linear relaxation. This paper shows how the dual solutions of the auxiliary linear programs can be used to learn what we call Lagrangian variable bound constraints. These are linear inequalities that explain OBBT's domain reductions in terms of the bounds on other variables and the objective value of the incumbent solution. Within a spatial branch-and-bound algorithm, they can be learnt a priori (during OBBT at the root node) and propagated within the search tree at very low computational cost. Experiments with an implementation inside the MINLP solver SCIP show that this reduces the number of branch-and-bound nodes and speeds up solution times.}, language = {en} } @misc{BertholdGleixnerHeinzetal.2013, author = {Berthold, Timo and Gleixner, Ambros and Heinz, Stefan and Vigerske, Stefan}, title = {Analyzing the computational impact of MIQCP solver components}, issn = {1438-0064}, doi = {10.3934/naco.2012.2.739}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-17754}, year = {2013}, abstract = {We provide a computational study of the performance of a state-of-the-art solver for nonconvex mixed-integer quadratically constrained programs (MIQCPs). Since successful general-purpose solvers for large problem classes necessarily comprise a variety of algorithmic techniques, we focus especially on the impact of the individual solver components. The solver SCIP used for the experiments implements a branch-and-cut algorithm based on a linear relaxation to solve MIQCPs to global optimality. Our analysis is based on a set of 86 publicly available test instances.}, language = {en} } @inproceedings{MuellerKuhlmannVigerske2018, author = {M{\"u}ller, Benjamin and Kuhlmann, Renke and Vigerske, Stefan}, title = {On the performance of NLP solvers within global MINLP solvers}, booktitle = {Operations Research Proceedings 2017}, publisher = {Springer International Publishing}, doi = {10.1007/978-3-319-89920-6_84}, pages = {633 -- 639}, year = {2018}, abstract = {Solving mixed-integer nonlinear programs (MINLPs) to global optimality efficiently requires fast solvers for continuous sub-problems. These appear in, e.g., primal heuristics, convex relaxations, and bound tightening methods. Two of the best performing algorithms for these sub-problems are Sequential Quadratic Programming (SQP) and Interior Point Methods. In this paper we study the impact of different SQP and Interior Point implementations on important MINLP solver components that solve a sequence of similar NLPs. We use the constraint integer programming framework SCIP for our computational studies.}, language = {en} } @misc{GleixnerSteffyWolter2012, author = {Gleixner, Ambros and Steffy, Daniel and Wolter, Kati}, title = {Improving the Accuracy of Linear Programming Solvers with Iterative Refinement}, issn = {1438-0064}, doi = {10.1145/2442829.2442858}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-15451}, year = {2012}, abstract = {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 demonstrate that this algorithm is effective in practice for computing extended precision solutions and that this leads to direct improvement of the best known methods for solving LPs exactly over the rational numbers.}, language = {en} } @misc{BleyGleixnerKochetal.2009, author = {Bley, Andreas and Gleixner, Ambros and Koch, Thorsten and Vigerske, Stefan}, title = {Comparing MIQCP solvers to a specialised algorithm for mine production scheduling}, organization = {ZIB}, issn = {1438-0064}, doi = {10.1007/978-3-642-25707-0}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11507}, number = {09-32}, year = {2009}, abstract = {In this paper we investigate the performance of several out-of-the box solvers for mixed-integer quadratically constrained programmes (MIQCPs) on an open pit mine production scheduling problem with mixing constraints. We compare the solvers BARON, Couenne, SBB, and SCIP to a problem-specific algorithm on two different MIQCP formulations. The computational results presented show that general-purpose solvers with no particular knowledge of problem structure are able to nearly match the performance of a hand-crafted algorithm.}, language = {en} } @misc{BertholdGleixner2013, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover Branching}, issn = {1438-0064}, doi = {10.1007/978-3-642-38527-8_20}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18030}, year = {2013}, abstract = {In this paper, we present a new branching strategy for nonconvex MINLP that aims at driving the created subproblems towards linearity. It exploits the structure of a minimum cover of an MINLP, a smallest set of variables that, when fixed, render the remaining system linear: whenever possible, branching candidates in the cover are preferred. Unlike most branching strategies for MINLP, Undercover branching is not an extension of an existing MIP branching rule. It explicitly regards the nonlinearity of the problem while branching on integer variables with a fractional relaxation solution. Undercover branching can be naturally combined with any variable-based branching rule. We present computational results on a test set of general MINLPs from MINLPLib, using the new strategy in combination with reliability branching and pseudocost branching. The computational cost of Undercover branching itself proves negligible. While it turns out that it can influence the variable selection only on a smaller set of instances, for those that are affected, significant improvements in performance are achieved.}, language = {en} } @misc{Gleixner2012, author = {Gleixner, Ambros}, title = {Factorization and update of a reduced basis matrix for the revised simplex method}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16349}, year = {2012}, abstract = {In this paper, we describe a method to enhance the FTRAN and BTRAN operations in the revised simplex algorithm by using a reduced basis matrix defined by basic columns and nonbasic rows. This submatrix of the standard basis matrix is potentially much smaller, but may change its dimension dynamically from iteration to iteration. For the classical product form update ("eta update"), the idea has been noted already by Zoutendijk, but only preliminarily tested by Powell in the early 1970s. We extend these ideas to Forrest-Tomlin type update formulas for an LU factorization of the reduced basis matrix, which are suited for efficient implementation within a state-of-the-art simplex solver. The computational advantages of the proposed method apply to pure LP solving as well as to LP-based branch-and-cut algorithms. It can easily be integrated into existing simplex codes.}, language = {en} } @misc{BertholdGleixnerHeinzetal.2011, author = {Berthold, Timo and Gleixner, Ambros and Heinz, Stefan and Vigerske, Stefan}, title = {On the computational impact of MIQCP solver components}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11998}, number = {11-01}, year = {2011}, abstract = {We provide a computational study of the performance of a state-of-the-art solver for nonconvex mixed-integer quadratically constrained programs (MIQCPs). Since successful general-purpose solvers for large problem classes necessarily comprise a variety of algorithmic techniques, we focus especially on the impact of the individual solver components. The solver SCIP used for the experiments implements a branch-and-cut algorithm based on linear outer approximation to solve MIQCPs to global optimality. Our analysis is based on a set of 86 publicly available test instances.}, language = {en} } @inproceedings{MuellerMuñozGasseetal.2020, author = {M{\"u}ller, Benjamin and Muñoz, Gonzalo and Gasse, Maxime and Gleixner, Ambros and Lodi, Andrea and Serrano, Felipe}, title = {On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming}, booktitle = {Integer Programming and Combinatorial Optimization: 21th International Conference, IPCO 2020}, doi = {10.1007/978-3-030-45771-6_25}, pages = {322 -- 337}, year = {2020}, abstract = {The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm's ability to generate strong dual bounds through extensive computational experiments.}, language = {en} } @misc{WitzigGleixner2019, author = {Witzig, Jakob and Gleixner, Ambros}, title = {Conflict-Driven Heuristics for Mixed Integer Programming}, issn = {1438-0064}, doi = {10.1287/ijoc.2020.0973}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-72204}, year = {2019}, abstract = {Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving heuristic that targets the generation of valid conflict constraints from the Farkas dual. We show that in the primal this is equivalent to the optimistic strategy of diving towards the best bound with respect to the objective function. Secondly, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classical coefficient diving. The computational performance of both methods is evaluated using an implementation in the source-open MIP solver SCIP. Experiments are carried out on publicly available test sets including Miplib 2010 and Cor@l.}, language = {en} } @misc{HendelMiltenbergerWitzig2018, author = {Hendel, Gregor and Miltenberger, Matthias and Witzig, Jakob}, title = {Adaptive Algorithmic Behavior for Solving Mixed Integer Programs Using Bandit Algorithms}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69563}, year = {2018}, abstract = {State-of-the-art solvers for mixed integer programs (MIP) govern a variety of algorithmic components. Ideally, the solver adaptively learns to concentrate its computational budget on those components that perform well on a particular problem, especially if they are time consuming. We focus on three such algorithms, namely the classes of large neighborhood search and diving heuristics as well as Simplex pricing strategies. For each class we propose a selection strategy that is updated based on the observed runtime behavior, aiming to ultimately select only the best algorithms for a given instance. We review several common strategies for such a selection scenario under uncertainty, also known as Multi Armed Bandit Problem. In order to apply those bandit strategies, we carefully design reward functions to rank and compare each individual heuristic or pricing algorithm within its respective class. Finally, we discuss the computational benefits of using the proposed adaptive selection within the \scip Optimization Suite on publicly available MIP instances.}, language = {en} } @misc{GamrathKochMartinetal.2013, 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}, year = {2013}, 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} } @incollection{GamrathBertholdHeinzetal.2015, author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-Based Primal Heuristics for Mixed Integer Programming}, volume = {13}, booktitle = {Optimization in the Real World}, publisher = {Springer Japan}, isbn = {978-4-431-55419-6}, doi = {10.1007/978-4-431-55420-2_3}, pages = {37 -- 53}, year = {2015}, abstract = {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.}, language = {en} }