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The Feasibility Pump of Fischetti, Glover, Lodi, and Bertacco has proved to be a very successful heuristic for finding feasible solutions of mixed integer programs. The quality of the solutions in terms of the objective value, however, tends to be poor. This paper proposes a slight modification of the algorithm in order to find better solutions. Extensive computational results show the success of this variant: in 89 out of 121 MIP instances the modified version produces improved solutions in comparison to the original Feasibility Pump.
This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques from SAT solving. SCIP is available in source code and free for non-commercial use. We demonstrate the usefulness of CIP on two tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we employ the CIP framework to solve chip design verification problems, which involve some highly non-linear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the non-linear constraints by employing constraint programming techniques.
This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques for solving satisfiability problems. SCIP is available in source code and free for noncommercial use. We demonstrate the usefulness of CIP on three tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we demonstrate how to use CIP techniques to compute the number of optimal solutions of integer programs. Third, we employ the CIP framework to solve chip design verification problems, which involve some highly nonlinear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the nonlinear constraints by employing constraint programming techniques.
In this paper we give an overview of the heuristics which are integrated into the open source branch-cut-and-price-framework SCIP. We briefly describe the fundamental ideas of different categories of heuristics and present some computational results which demonstrate the impact of heuristics on the overall solving process of SCIP.
In the recent years, a couple of quite successful large neighborhood search heuristics for mixed integer programs has been published. Up to our knowledge, all of them are improvement heuristics. We present a new start heuristic for general MIPs working in the spirit of large neighborhood search. It constructs a sub-MIP which represents the space of all feasible roundings of some fractional point - normally the optimum of the LP-relaxation of the original MIP. Thereby, one is able to determine whether a point can be rounded to a feasible solution and which is the best possible rounding. Furthermore, a slightly modified version of RENS proves to be a well-performing heuristic inside the branch-cut-and-price-framework SCIP.
In modern MIP solvers, primal heuristics play a major role in finding and improving feasible solutions early in the solution process. However, classical performance measures such as time to optimality or number of branch-and-bound nodes reflect the impact of primal heuristics on the overall solving process badly. This article discusses the question of how to evaluate the effect of primal heuristics.
Therefore, we introduce a new performance measure, the "primal integral" which depends on the quality of solutions found during the solving process as well as on the points in time when they are found. Our computational results reveal that heuristics improve the performance of MIP solvers in terms of the primal bound by around 80%. Further, we compare five state-of-the-art MIP solvers w.r.t. the newly proposed measure.
Mixed integer programming (MIP) has become one of the most important techniques in Operations Research and Discrete Optimization. SCIP (Solving Constraint Integer Programs) is currently one of the fastest non-commercial MIP solvers. It is based on the branch-and-bound procedure in which the problem is recursively split into smaller subproblems, thereby creating a so-called branching tree. We present ParaSCIP, an extension of SCIP, which realizes a parallelization on a distributed memory computing environment. ParaSCIP uses SCIP solvers as independently running processes to solve subproblems (nodes of the branching tree) locally. This makes the parallelization development independent of the SCIP development. Thus, ParaSCIP directly profits from any algorithmic progress in future versions of SCIP. Using a first implementation of ParaSCIP, we were able to solve two previously unsolved instances from MIPLIB2003, a standard test set library for MIP solvers. For these computations, we used up to 2048 cores of the HLRN~II supercomputer.
Cloud branching
(2013)
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 state-of-the-art methods like strong branching, pseudocost branching, and their hybrids.
We show that by exploiting dual degeneracy, and thus multiple alternative optimal solutions, it is possible to enhance traditional methods. We present preliminary computational results, applying the newly proposed strategy to full strong branching, which is known to be the MIP branching rule leading to the fewest number of search nodes. It turns out that cloud branching can reduce the mean running time by up to 30% on standard test sets.
Shift-And-Propagate
(2013)
For mixed integer programming, recent years have seen a growing interest in the design of general purpose primal heuristics for use inside complete solvers. Many of these heuristics rely on an optimal LP solution. Finding this may itself take a significant amount of time.
The presented paper addresses this issue by the introduction of the Shift-And-Propagate heuristic. Shift-And-Propagate is a pre-root primal heuristic that does not require a previously found LP solution. It applies domain propagation techniques to quickly drive a variable assignment towards feasibility. Computational experiments indicate that this heuristic is a powerful supplement of existing rounding and propagation heuristics.
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.
"Interior point algorithms are a good choice for solving pure LPs or QPs, but when you solve MIPs, all you need is a dual simplex." This is the common conception which disregards that an interior point solution provides some unique structural insight into the problem at hand. In this paper, we will discuss some of the benefits that an interior point solver brings to the solution of difficult MIPs within FICO Xpress. This includes many different components of the MIP solver such as branching variable selection, primal heuristics, preprocessing, and of course the solution of the LP relaxation.
In the literature for mixed integer programming, heuristic algorithms (particularly primal heuristics) are often considered as stand-alone procedures; in that context, heuristics are treated as an alternative to solving a problem to proven optimality. This conceals the fact that heuristic algorithms are a fundamental component of state-of-the-art global solvers for mixed integer linear programming (MIP) and mixed integer nonlinear programming (MINLP).
In the present thesis, we focus on this latter aspect; we study heuristic algorithms that are tightly integrated within global MINLP solvers and analyze their impact on the overall solution process. Our contributions comprise generalizations of primal heuristics for MIP towards MINLP as well as novel ideas for MINLP primal heuristics and for heuristic algorithms to take branching decisions and to collect global information in MIP. These are:
- Shift-and-Propagate, a novel propagation heuristic for MIP that does not require the solution to an LP relaxation,
- a generic way to generalize large neighborhood search (LNS) heuristics from MIP to MINLP,
- an Objective Feasibility Pump heuristic for nonconvex MINLP that uses second-order information and a dynamic selection of rounding procedures,
- RENS, an LNS start heuristic for MINLP that optimizes over the set of feasible roundings of an LP solution,
- Undercover, an LNS start heuristic for MINLP that solves a largest sub-MIP of a given MINLP,
- Rapid Learning, a heuristic algorithm to generate globally valid conflict constraints for MIPs,
- Cloud Branching, a heuristic algorithm that exploits dual degeneracy to reduce the number of candidates for branching variable selection.
Additionally, we propose a new performance measure, the primal integral, that captures the benefits of primal heuristics better than traditional methods. In our computational study, we compare the performance of the MIP and MINLP solver SCIP with and without primal heuristics on six test sets with altogether 983 instances from academic and industrial sources, including our project partners ForNe, SAP, and Siemens. We observe that heuristics improve the solver performance regarding all measures that we used - by different orders of magnitude. We further see that the harder a problem is to solve to global optimality, the more important the deployment of primal heuristics becomes.
The algorithms presented in this thesis are available in source code as part of the solver SCIP, of which the author has been a main developer for the last years. Methods described in this thesis have also been re-implemented within several commercial and noncommercial MIP and MINLP software packages, including Bonmin, CBC, Cplex, Gams, Sulum, and Xpress.
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.
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.