@misc{ArnoldBertholdHeinzetal., author = {Arnold, Thomas and Berthold, Timo and Heinz, Stefan and Vigerske, Stefan and Henrion, Ren{\´e} and Gr{\"o}tschel, Martin and Koch, Thorsten and Tischendorf, Caren and R{\"o}misch, Werner}, title = {A Jack of all Trades? Solving stochastic mixed-integer nonlinear constraint programs}, series = {MATHEON - Mathematics for Key Technologies}, volume = {1}, journal = {MATHEON - Mathematics for Key Technologies}, editor = {Deuflhard, Peter and Gr{\"o}tschel, Martin and H{\"o}mberg, Dietmar and Horst, Ulrich and Kramer, J{\"u}rg and Mehrmann, Volker and Polthier, Konrad and Schmidt, Frank and Sch{\"u}tte, Christof and Skutella, Martin and Sprekels, J{\"u}rgen}, publisher = {European Mathematical Society}, doi = {10.4171/137}, pages = {135 -- 146}, abstract = {Natural gas is one of the most important energy sources in Germany and Europe. In recent years, political regulations have led to a strict separation of gas trading and gas transport, thereby assigning a central role in energy politics to the transportation and distribution of gas. These newly imposed political requirements influenced the technical processes of gas transport in such a way that the complex task of planning and operating gas networks has become even more intricate. Mathematically, the combination of discrete decisions on the configuration of a gas transport network, the nonlinear equations describing the physics of gas, and the uncertainty in demand and supply yield large-scale and highly complex stochastic mixed-integer nonlinear optimization problems. The Matheon project "Optimization of Gas Transport" takes the key role of making available the necessary core technology to solve the mathematical optimization problems which model the topology planning and the operation of gas networks. An important aspect of the academic impact is the free availability of our framework. As a result of several years of research and development, it is now possible to download a complete state-of-the-art framework for mixed-integer linear and nonlinear programming in source code at http://scip.zib.de}, language = {en} } @article{BertholdHendelKoch2017, author = {Berthold, Timo and Hendel, Gregor and Koch, Thorsten}, title = {From feasibility to improvement to proof: three phases of solving mixed-integer programs}, series = {Optimization Methods and Software}, volume = {33}, journal = {Optimization Methods and Software}, number = {3}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2017.1392519}, pages = {499 -- 517}, year = {2017}, abstract = {Modern mixed-integer programming (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 distinct 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 behaviour of the non-commercial MIP solver Scip at the predicted phase transition points.}, language = {en} } @inproceedings{HendelBertholdAchterberg, author = {Hendel, Gregor and Berthold, Timo and Achterberg, Tobias}, title = {Rounding and Propagation Heuristics for Mixed Integer Programming}, series = {Operations Research Proceedings 2011}, booktitle = {Operations Research Proceedings 2011}, pages = {71 -- 76}, abstract = {Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB~2010 benchmark instances.}, language = {en} } @article{BertholdGleixner2014, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover: a primal MINLP heuristic exploring a largest sub-MIP}, series = {Mathematical Programming}, volume = {144}, journal = {Mathematical Programming}, number = {1-2}, doi = {10.1007/s10107-013-0635-2}, pages = {315 -- 346}, year = {2014}, 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} } @article{BertholdHendel, author = {Berthold, Timo and Hendel, Gregor}, title = {Shift-and-Propagate}, series = {Journal of Heuristics}, volume = {21}, journal = {Journal of Heuristics}, number = {1}, doi = {10.1007/s10732-014-9271-0}, pages = {73 -- 106}, abstract = {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.}, language = {en} } @inproceedings{GamrathMelchioriBertholdetal., author = {Gamrath, Gerald and Melchiori, Anna and Berthold, Timo and Gleixner, Ambros and Salvagnin, Domenico}, title = {Branching on Multi-aggregated Variables}, series = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2015}, volume = {9075}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2015}, doi = {10.1007/978-3-319-18008-3_10}, pages = {141 -- 156}, abstract = {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.}, language = {en} } @incollection{GamrathBertholdHeinzetal., author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-Based Primal Heuristics for Mixed Integer Programming}, series = {Optimization in the Real World}, 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}, 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} } @article{GleixnerBertholdMuelleretal., author = {Gleixner, Ambros and Berthold, Timo and M{\"u}ller, Benjamin and Weltge, Stefan}, title = {Three Enhancements for Optimization-Based Bound Tightening}, series = {Journal of Global Optimization}, volume = {67}, journal = {Journal of Global Optimization}, number = {4}, doi = {10.1007/s10898-016-0450-4}, pages = {731 -- 757}, 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} } @inproceedings{BertholdFarmerHeinzetal., author = {Berthold, Timo and Farmer, James and Heinz, Stefan and Perregaard, Michael}, title = {Parallelization of the FICO Xpress-Optimizer}, series = {Mathematical Software - ICMS 2016, 5th International Conference Berlin, Germany, July 11-14, 2016 Proceedings}, booktitle = {Mathematical Software - ICMS 2016, 5th International Conference Berlin, Germany, July 11-14, 2016 Proceedings}, doi = {10.1007/978-3-319-42432-3_31}, pages = {251 -- 258}, language = {en} } @inproceedings{ShinanoBertholdHeinz, author = {Shinano, Yuji and Berthold, Timo and Heinz, Stefan}, title = {A First Implementation of ParaXpress: Combining Internal and External Parallelization to Solve MIPs on Supercomputers}, series = {Mathematical Software - ICMS 2016, 5th International Conference Berlin, Germany, July 11-14, 2016 Proceedings}, booktitle = {Mathematical Software - ICMS 2016, 5th International Conference Berlin, Germany, July 11-14, 2016 Proceedings}, doi = {10.1007/978-3-319-42432-3_38}, pages = {308 -- 316}, language = {en} }