TY - CHAP A1 - Witzig, Jakob A1 - Berthold, Timo T1 - Conflict-Free Learning for Mixed Integer Programming T2 - Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2020 N2 - 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. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-58942-4_34 IS - 12296 SP - 521 EP - 530 PB - Springer, Cham. ER - TY - CHAP A1 - Berthold, Timo A1 - Gleixner, Ambros ED - Bonifaci, Vincenzo ED - Demetrescu, Camil ED - Marchetti-Spaccamela, Alberto T1 - Undercover Branching T2 - Experimental Algorithms, 12th International Symposium, SEA 2013, Rome, Italy, June 5-7, 2013, Proceedings N2 - 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. Y1 - 2013 U6 - https://doi.org/10.1007/978-3-642-38527-8_20 VL - 7933 SP - 212 EP - 223 ER - TY - CHAP A1 - Gleixner, Ambros A1 - Weltge, Stefan T1 - Learning and Propagating Lagrangian Variable Bounds for Mixed-Integer Nonlinear Programming T2 - 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 N2 - 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. Y1 - 2013 U6 - https://doi.org/10.1007/978-3-642-38171-3_26 VL - 7874 SP - 355 EP - 361 ER - TY - CHAP A1 - Gleixner, Ambros A1 - Steffy, Daniel A1 - Wolter, Kati T1 - Improving the Accuracy of Linear Programming Solvers with Iterative Refinement T2 - ISSAC '12. Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation Y1 - 2012 U6 - https://doi.org/10.1145/2442829.2442858 SP - 187 EP - 194 ER - TY - CHAP A1 - Witzig, Jakob A1 - Berthold, Timo A1 - Heinz, Stefan T1 - A Status Report on Conflict Analysis in Mixed Integer Nonlinear Programming T2 - Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2019 N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1007/978-3-030-19212-9_6 VL - 11494 SP - 84 EP - 94 PB - Springer ER - TY - CHAP A1 - Berthold, Timo A1 - Stuckey, Peter A1 - Witzig, Jakob T1 - Local Rapid Learning for Integer Programs T2 - Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2019 N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1007/978-3-030-19212-9_5 VL - 11494 SP - 67 EP - 83 PB - Springer ER - TY - CHAP A1 - Anderson, Daniel A1 - Hendel, Gregor A1 - Le Bodic, Pierre A1 - Viernickel, Jan Merlin T1 - Clairvoyant Restarts in Branch-and-Bound Search Using Online Tree-Size Estimation T2 - AAAI-19: Proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence N2 - 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. Y1 - 2018 ER - TY - CHAP A1 - Gleixner, Ambros A1 - Steffy, Daniel T1 - Linear Programming using Limited-Precision Oracles T2 - A. Lodi, V. Nagarajan (eds), Integer Programming and Combinatorial Optimization: 20th International Conference, IPCO 2019 N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1007/978-3-030-17953-3_30 SP - 399 EP - 412 ER - TY - CHAP A1 - D'Andreagiovanni, Fabio A1 - Gleixner, Ambros T1 - Towards an accurate solution of wireless network design problems T2 - Cerulli R., Fujishige S., Mahjoub A. (eds) Combinatorial Optimization. ISCO 2016 N2 - 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. Y1 - 2016 U6 - https://doi.org/10.1007/978-3-319-45587-7_12 SP - 135 EP - 147 ER - TY - CHAP A1 - Berthold, Timo A1 - Gleixner, Ambros ED - Bonami, Pierre ED - Liberti, Leo ED - Miller, Andrew J. ED - Sartenaer, Annick T1 - Undercover – a primal heuristic for MINLP based on sub-MIPs generated by set covering T2 - Proceedings of the European Workshop on Mixed Integer Nonlinear Programming, April 12-16, 2010, Marseilles, France N2 - 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. Y1 - 2010 SP - 103 EP - 112 ER - TY - CHAP A1 - Bley, Andreas A1 - Gleixner, Ambros A1 - Koch, Thorsten A1 - Vigerske, Stefan T1 - Comparing MIQCP Solvers to a Specialised Algorithm for Mine Production Scheduling T2 - Modeling, Simulation and Optimization of Complex Processes. Proceedings of the Fourth International Conference on High Performance Scientific Computing, March 2-6, 2009, Hanoi, Vietnam N2 - 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. Y1 - 2012 U6 - https://doi.org/10.1007/978-3-642-25707-0_3 SP - 25 EP - 39 ER - TY - CHAP A1 - Hendel, Gregor A1 - Miltenberger, Matthias A1 - Witzig, Jakob T1 - Adaptive Algorithmic Behavior for Solving Mixed Integer Programs Using Bandit Algorithms T2 - OR 2018: International Conference on Operations Research N2 - 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. Y1 - 2018 ER -