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34
Noncommercial Software for MixedInteger Linear Programming
, 2004
"... We present an overview of noncommercial software tools for the solution of mixedinteger linear programs (MILPs). We first review solution methodologies for MILPs and then present an overview of the available software, including detailed descriptions of eight software packages available under open s ..."
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Cited by 24 (1 self)
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We present an overview of noncommercial software tools for the solution of mixedinteger linear programs (MILPs). We first review solution methodologies for MILPs and then present an overview of the available software, including detailed descriptions of eight software packages available under open source or other noncommercial licenses. Each package is categorized as a black box solver, a callable library, a solver framework, or some combination of these. The distinguishing features of all eight packages are described. The paper concludes with case studies that illustrate the use of two of the solver frameworks to develop custom solvers for specific problem classes and with benchmarking of the six black box solvers.
Conflict analysis in mixed integer programming
, 2006
"... Conflict analysis for infeasible subproblems is one of the key ingredients in modern SAT solvers. In contrast, it is common practice for today’s mixed integer programming solvers to discard infeasible subproblems and the information they reveal. In this paper, we try to remedy this situation by gene ..."
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Cited by 22 (6 self)
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Conflict analysis for infeasible subproblems is one of the key ingredients in modern SAT solvers. In contrast, it is common practice for today’s mixed integer programming solvers to discard infeasible subproblems and the information they reveal. In this paper, we try to remedy this situation by generalizing SAT infeasibility analysis to mixed integer programming. We present heuristics for branchandcut solvers to generate valid inequalities from the current infeasible subproblem and the associated branching information. SAT techniques can then be used to strengthen the resulting constraints. Extensive computational experiments show the potential of our method. Conflict analysis greatly improves the performance on particular models, while it does not interfere with the solving process on the other instances. In total, the number of required branching nodes on general MIP instances was reduced by 18 % in the geometric mean, and the solving time was reduced by 11%. On infeasible MIPs arising in the context of chip verification and on a model for a particular combinatorial game, the number of nodes was reduced by 80%, thereby reducing the solving time by 50%.
COMPUTING OPTIMAL MORSE MATCHINGS
, 2004
"... Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computation ..."
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Cited by 20 (0 self)
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Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.
A Feasibility Pump Heuristic for General MixedInteger Problems
 UNIVERSITÀ DI BOLOGNA – D.E.I.S. – OPERATIONS RESEARCH
, 2007
"... Finding a feasible solution of a given MixedInteger Programming (MIP) model is a very important (NPcomplete) problem that can be extremely hard in practice. Very recently, Fischetti, Glover and Lodi proposed a heuristic scheme for finding a feasible solution to general MIPs, called Feasibility Pum ..."
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Cited by 19 (1 self)
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Finding a feasible solution of a given MixedInteger Programming (MIP) model is a very important (NPcomplete) problem that can be extremely hard in practice. Very recently, Fischetti, Glover and Lodi proposed a heuristic scheme for finding a feasible solution to general MIPs, called Feasibility Pump (FP). According to the computational analysis reported by these authors, FP is indeed quite effective in finding feasible solutions of hard 01 MIPs. However, MIPs with generalinteger variables seem much more difficult to solve by using the FP approach. In this paper we elaborate on the FischettiGloverLodi approach and extend it in two main directions, namely (i) handling as effectively as possible MIP problems with both binary and generalinteger variables, and (ii) exploiting the FP information to drive a subsequent enumeration phase. Extensive computational results on large sets of test instances from the literature are reported, showing the effectiveness of our improved FP scheme for finding feasible solutions to hard MIPs with generalinteger variables.
EXACT AND APPROXIMATE SPARSE SOLUTIONS OF UNDERDETERMINED LINEAR EQUATIONS
, 2007
"... In this paper, we empirically investigate the NPhard problem of finding sparsest solutions to linear equation systems, i.e., solutions with as few nonzeros as possible. This problem has received considerable interest in the sparse approximation and signal processing literature, recently. We use a ..."
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Cited by 13 (2 self)
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In this paper, we empirically investigate the NPhard problem of finding sparsest solutions to linear equation systems, i.e., solutions with as few nonzeros as possible. This problem has received considerable interest in the sparse approximation and signal processing literature, recently. We use a branchandcut approach via the maximum feasible subsystem problem to compute optimal solutions for small instances and investigate the uniqueness of the optimal solutions. We furthermore discuss five (modifications of) heuristics for this problem that appear in different parts of the literature. For small instances, the exact optimal solutions allow us to evaluate the quality of the heuristics, while for larger instances we compare their relative performance. One outcome is that the socalled basis pursuit heuristic performs worse, compared to the other methods. Among the best heuristics are a method due to Mangasarian and a bilinear approach.
Orbitopal fixing
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, PROCEEDINGS OF THE TWELFTH INTERNATIONAL IPCO CONFERENCE, VOLUME 4513 OF LNCS
, 2007
"... The topic of this paper are integer programming models in which a subset of 0/1variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branchandcut algorithms if the order of the subsets of the partition is irrelevant. This kind ..."
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Cited by 12 (2 self)
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The topic of this paper are integer programming models in which a subset of 0/1variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branchandcut algorithms if the order of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branchandcut tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branchandcut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the branchandcut tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been investigated in [11]. It does, however, not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks.
Recoverable Robust Knapsacks: the Discrete Scenario Case
, 2010
"... Admission control problems have been studied extensively in the past. In a typical setting, resources like bandwidth have to be distributed to the different customers according to their demands maximizing the profit of the company. Yet, in realworld applications those demands are deviating and in o ..."
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Cited by 12 (3 self)
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Admission control problems have been studied extensively in the past. In a typical setting, resources like bandwidth have to be distributed to the different customers according to their demands maximizing the profit of the company. Yet, in realworld applications those demands are deviating and in order to satisfy their service requirements often a robust approach is chosen wasting benefits for the company. Our model overcomes this problem by allowing a limited recovery of a previously fixed assignment as soon as the data are known by violating at most k service promises and serving up to ℓ new customers. Applying this approaches to the call admission problem on a single link of a telecommunication network leads to a recoverable robust version of the knapsack problem. In this paper, we show that for a fixed number of discrete scenarios this recoverable robust knapsack problem is weakly NPcomplete and any such instance can be solved in pseudopolynomial time by a dynamic program. If the number of discrete scenarios is part of the input, the problem is strongly NPcomplete and in special cases not approximable in polynomial time, unless P = NP. Next to its complexity status we were interested in obtaining strong polyhedral descriptions for this problem. We thus generalized the wellknown concept of covers to gain valid inequalities for the recoverable robust knapsack polytope. Besides the canonical extension of covers we introduce a second kind of extension exploiting the scenariobased problem structure and producing stronger valid inequalities. Furthermore, we present two extensive computational studies to (i) investigate the influence of parameters k and ℓ to the objective and (ii) evaluate the effectiveness of our new class of valid inequalities.
Decomposition, Reformulation, and Diving in University Course Timetabling
"... In many reallife optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of ..."
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Cited by 9 (3 self)
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In many reallife optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of soft constraint violation. The goal is then to minimise a linear combination of such measures. This paper studies an approach to such problems, which can be thought of as multiphase exploitation of multiple objective/valuerestricted submodels. In this approach, only one computationally difficult component of a problem and the associated subset of objectives is considered at first. This produces partial solutions, which define interesting neighbourhoods in the search space of the complete problem. Often, it is possible to pick the initial component so that variable aggregation can be performed at the first stage, and the neighbourhoods to be explored next are guaranteed to contain feasible solutions. Using integer programming, it is then easy to implement heuristics producing solutions with bounds on their quality.
An integrated solver for optimization problems
, 2009
"... One of the central trends in the optimization community over the past several years has been the steady improvement of generalpurpose solvers. A logical next step in this evolution is to combine mixed integer linear programming, constraint programming, and global optimization in a single system. Re ..."
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Cited by 9 (2 self)
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One of the central trends in the optimization community over the past several years has been the steady improvement of generalpurpose solvers. A logical next step in this evolution is to combine mixed integer linear programming, constraint programming, and global optimization in a single system. Recent research in the area of integrated problem solving suggests that the right combination of different technologies can simplify modeling and speed up computation substantially. Nevertheless, integration often requires special purpose coding, which is timeconsuming and errorprone. We present a general purpose solver, SIMPL, that allows its user to replicate (and sometimes improve on) the results of custom implementations with concise models written in a highlevel language. We apply SIMPL to production planning, product configuration, machine scheduling, and truss structure design problems on which customized integrated methods have shown significant computational advantage. We obtain results that either match or surpass the original codes at a fraction of the implementation effort. 1