The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP . The focus of this paper is on the role of the SCIP Optimization Suite in supporting research. SCIP ’s main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP ’s application as a research tool and as a platform for further developments. Further, the paper gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP .
Abstract. The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 8.0 of the SCIP Optimization Suite. Major updates in SCIP include improvements in symmetry handling and decomposition algorithms, new cutting planes, a new plugin type for cut selection, and a complete rework of the way nonlinear constraints are handled. Additionally, SCIP 8.0 now supports interfaces for Julia as well as Matlab. Further, UG now includes a unified framework to parallelize all solvers, a utility to analyze computational experiments has been added to GCG, dual solutions can be postsolved by PaPILO, new heuristics and presolving methods were added to SCIP-SDP, and additional problem classes and major performance improvements are available in SCIP-Jack. Keywords Constraint integer programming • linear programming • mixed-integer linear programming • mixed-integer nonlinear programming • optimization solver • branch-andcut • branch-and-price • column generation • parallelization • mixed-integer semidefinite programming Mathematics Subject Classification 90C05 • 90C10 • 90C11 • 90C30 • 90C90 • 65Y05 * Extended author information is available at the end of the paper.
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Round robin tournaments are omnipresent in sport competitions and beyond. We propose two new integer programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations with the linear relaxation of a well-known traditional formulation. We find that the matching formulation is stronger than the other formulations, while its LP relaxation is still being solvable in polynomial time. In addition, we provide an exponentially sized class of valid inequalities for the matching formulation. Complementing our theoretical assessment of the strength of the different formulations, we also experimentally show that the matching formulation is superior on a broad set of instances. Finally, we describe a branch-and-price algorithm for finding round robin tournaments that is based on the matching formulation.
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