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This paper describes a new instance library for quadratic programming (QP), i.e., the family of continuous and (mixed)-integer optimization problems where the objective function and/or the constraints are quadratic. QP is a very diverse class of problems, comprising sub-classes ranging from trivial to undecidable. This diversity is reflected in the variety of QP solution methods, ranging from entirely combinatorial approaches to completely continuous algorithms, including many methods for which both aspects are fundamental. Selecting a set of instances of QP that is at the same time not overwhelmingly onerous but sufficiently challenging for the different, interested communities is therefore important. We propose a simple taxonomy for QP instances leading to a systematic problem selection mechanism. We then briefly survey the field of QP, giving an overview of theory, methods and solvers. Finally, we describe how the library was put together, and detail its final contents.
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.
This paper addresses the problem of generating synthetic test cases for experimentation in linear programming. We propose a method which maps instance generation and instance space search to an alternative encoded space. This allows us to develop a generator for feasible bounded linear programming instances with controllable properties. We show that this method is capable of generating any feasible bounded linear program, and that parameterised generators and search algorithms using this approach generate only feasible bounded instances. Our results demonstrate that controlled generation and instance space search using this method achieves feature diversity more effectively than using a direct representation.
Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the so-called symmetric mappings that can be used to derive strong cuts for the proposed relaxation model. We show that the bounds based on the new models are comparable to some strong bounds in the literature. Promising experimental results based on the new relaxations are reported.
We address the solution of a very challenging (and previously unsolved) instance of the quadratic 3-dimensional assignment problem, arising in digitalwireless communications.The paper describes the techniques developed to solve this instance to optimality, from the choice of an appropriate mixed-integer programming formulation, to cutting planes and symmetry handling. Using these techniques we were able to solve the target instance with moderate computational effort (2.5 million nodes and 1 week of computations on a standard PC).
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.