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 .

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The relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) of the set of integer points X contained in a polyhedron is the smallest number of facets of any polyhedron P such that the integer points in P coincide with X. It is a useful tool to investigate the existence of compact linear descriptions of X. In this article, we derive tight and computable upper bounds on $${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$ rc Q ( X ) , a variant of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) in which the polyhedra P are required to be rational, and we show that $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) with the particular focus on the existence of a finite set $$Y \subseteq \mathbb {Z}^d$$ Y ⊆ Z d such that separating X and $$Y \setminus X$$ Y \ X allows us to deduce $${{\,\mathrm{rc}\,}}(X) \ge k$$ rc ( X ) ≥ k . In particular, we show for some choices of X that no such finite set Y exists to certify the value of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) , providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for specific classes of sets X and present the first practically applicable approach to compute $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for sets X that admit a finite certificate.

We consider an extended version of the classical Max-$$k$$ k -Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved.

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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