90C11 Mixed integer programming
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This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for i) general integer programs, ii) problems with local structure such as knapsack constraints, and iii) problems with 0-1 coefficient matrices, such as set packing, are examined in turn. Finally the use of valid inequalities for classes of problems with structure, such as network design, is explored.
In this article, strategical infrastructure planning problems in the design of large-scale telecommunication networks are discussed based on experiences from three projects with industrial partners: The access network planning of the German Gigabit-Wissenschaftsnetz (G-WiN) for DFN (Verein zur Förderung eines Deutschen Forschungsnetzes e.V.), the mobile network switching center location planning project for E-Plus Mobilfunk, and the fixed network switching center location planning project for TELEKOM AUSTRIA. We introduce a mathematical model for a hierarchical multi-commodity capacitated facility location problem, present adaptions of this basic model to the specific requirements within the different projects and discuss the individual peculiarities and model decisions made. Eventually, we present and discuss computational results of three associated case studies, illustrating '"how we did the job`` with mathematical methods.
Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today's state-of-the-art algorithms of both fields have several strategies in common, in particular the branch-and-bound process to recursively divide the problem into smaller sub problems. On the other hand, the main techniques to process each sub problem are different, and it was observed that they have complementary strenghts. We propose a programming framework {\sffamily SCIP} that integrates techniques from both fields in order to exploit the strenghts of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, {\sffamily SCIP} does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, in-depth control and high performance.
MIPLIB 2003
(2005)
This paper reports on the fourth version of the Mixed Integer Programming Library. Since ({\sc miplib}) is to provide a concise set of challenging problems, it became necessary to purge instances that became too easy. We present an overview of the 27 new problems and statistical data for all 60 instances.
MIPLIB 2010
(2010)
This paper reports on the fifth version of the Mixed 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.
Contemporary supercomputers can easily provide years of
CPU time per wall-clock hour. One challenge of today's software
development is how to harness this wast computing power in order to solve
really hard mixed integer programming instances. In 2010, two out of
six open MIPLIB2003 instances could be solved by ParaSCIP in more than
ten consecutive runs, restarting from checkpointing files.
The contribution of this paper is threefold:
For the first time, we present computational results of single runs for
those two instances. Secondly, we provide new improved upper and lower
bounds for all of the remaining four open MIPLIB2003 instances.
Finally, we explain which new developments led to these results and
discuss the current progress of ParaSCIP. Experiments were conducted on
HLRNII, on HLRN III, and on the Titan supercomputer, using up to 35,200 cores.
In this paper, we introduce the Maximum Diversity Assortment Selection Problem (MADASS), which is a generalization of the 2-dimensional Cutting Stock Problem (2CSP). Given a set of rectangles and a rectangular container, the goal of 2CSP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is covered. In MADASS, we need to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as minimum or average normalized Hamming-Distance of all assortment pairs. The MADASS Problem was used in the 11th AIMMS-MOPTA Competition in 2019. The methods we describe in this article and the computational results won the contest.
In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2CSP literature.