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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.
We report on the selection process leading to the sixth version of the Mixed Integer Programming Library, MIPLIB 2017. Selected from an initial pool of 5721 instances, the new MIPLIB 2017 collection consists of 1065 instances. A subset of 240 instances was specially selected for benchmarking solver performance. For the first time, these sets were compiled using a data-driven selection process supported by the solution of a sequence of mixed integer optimization problems, which encode requirements on diversity and balancedness with respect to instance features and performance data.
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: conflict graph analysis and dual proof analysis. While conflict graph analysis detects sets of contradicting variable bounds in an implication graph, dual proof analysis derives valid linear constraints from the proof of the dual LP’s unboundedness. 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 a comprehensive computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first combined approach that uses 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 and the commercial MIP solver .
RENS
(2013)
This article introduces rens, the relaxation enforced neighborhood search, a large neighborhood search algorithm for mixed integer nonlinear programs (MINLPs). It uses a sub-MINLP to explore the set of feasible roundings of an optimal solution ¯ x of a linear or nonlinear relaxation. The sub-MINLP is constructed by fixing integer variables x j with ¯ x j ? Z and bounding the remaining integer variables to x j ? {¯x j , ¯x j }. We describe two different applications of rens: as a standalone algorithm to compute an optimal rounding of the given starting solution and as a primal heuristic inside a complete MINLP solver. We use the former to compare different kinds of relaxations and the impact of cutting planes on the so-called roundability of the corresponding optimal solutions. We further utilize rens to analyze the performance of three rounding heuristics implemented in the branch-cut-and-price framework scip. Finally, we study the impact of rens when it is applied as a primal heuristic inside scip. All experiments were performed on three publicly available test sets of mixed integer linear programs (MIPs), mixed integer quadratically constrained programs (MIQCPs), and MINLPs, using solely software which is available in source code. It turns out that for these problem classes 60 to 70% of the instances have roundable relaxation optima and that the success rate of rens does not depend on the percentage of fractional variables. Last but not least, rens applied as primal heuristic complements nicely with existing primal heuristics in scip.
MIPLIB 2010
(2011)
This paper reports on the fifth version of theMixed Integer Programming Library. The miplib 2010 is the first miplib release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today’s codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.
This paper describes the computational challenge developed for a computational com-
petition held in 2023 for the 20th anniversary of the Mixed Integer Programming
Workshop. The topic of this competition was reoptimization, also known as warm
starting, of mixed integer linear optimization problems after slight changes to the
input data for a common formulation. The challenge was to accelerate the proof of
optimality of the modified instances by leveraging the information from the solving
processes of previously solved instances, all while creating high-quality primal solu-
tions. Specifically, we discuss the competition’s format, the creation of public and
hidden datasets, and the evaluation criteria. Our goal is to establish a methodology
for the generation of benchmark instances and an evaluation framework, along with
benchmark datasets, to foster future research on reoptimization of mixed integer linear
optimization problems.
Learning to use local cuts
(2025)
An essential component in modern solvers for mixed-integer (linear) programs (MIPs)
is the separation of additional inequalities (cutting planes) to tighten the linear pro-
gramming relaxation. Various algorithmic decisions are necessary when integrating
cutting plane methods into a branch-and-bound (B&B) solver as there is always the
trade-off between the efficiency of the cuts and their overhead, given that they tend
to slow down the solution time of the relaxation. One of the most crucial questions
is: Should cuts only be generated globally at the root or also locally at nodes of the
tree? We address this question by a machine learning approach for which we train a
regression forest to predict the speed-up (or slow-down) provided by using local cuts.
We demonstrate with an open implementation that this helps to improve the perfor-
mance of the FICO Xpress MIP solver on a public test set of general MIP instances.
We further report on the impact of a practical implementation inside Xpress on a large,
diverse set of real-world industry MIPs.