Library

Description: Mixed-integer linear programming (MILP) plays a crucial role in the field of mathematical optimization and is especially relevant for practical applications due to the broad range of problems that can be modeled in that fashion. The vast majority of MILP solvers employ the LP-based branch-and-cut approach. As the name suggests, the linear programming (LP) subproblems that need to be solved therein influence their behavior and performance significantly. This thesis explores the impact of various LP solvers as well as LP solving techniques on the constraint integer programming framework SCIP Optimization Suite. SCIP allows for comparisons between academic and open-source LP solvers like Clp and SoPlex, as well as commercially developed, high-end codes like CPLEX, Gurobi, and Xpress. We investigate how the overall performance and stability of an MILP solver can be improved by new algorithmic enhancements like LP solution polishing and persistent scaling that we have implemented in the LP solver SoPlex. The former decreases the fractionality of LP solutions by selecting another vertex on the optimal hyperplane of the LP relaxation, exploiting degeneracy. The latter provides better numerical properties for the LP solver throughout the MILP solving process by preserving and extending the initial scaling factors, effectively also improving the overall performance of SCIP. Both enhancement techniques are activated by default in the SCIP Optimization Suite. Additionally, we provide an analysis of numerical conditions in SCIP through the lens of the LP solver by comparing different measures and how these evolve during the different stages of the solving process. A side effect of our work on this topic was the development of TreeD: a new and convenient way of presenting the search tree interactively and animated in the three-dimensional space. This visualization technique facilitates a better understanding of the MILP solving process of SCIP. Furthermore, this thesis presents the various algorithmic techniques like the row representation and iterative refinement that are implemented in SoPlex and that distinguish the solver from other simplex-based codes. Although it is often not as performant as its competitors, SoPlex demonstrates the ongoing research efforts in the field of linear programming with the simplex method. Aside from that, we demonstrate the rapid prototyping of algorithmic ideas and modeling approaches via PySCIPOpt, the Python interface to the SCIP Optimization Suite. This tool allows for convenient access to SCIP's internal data structures from the user-friendly Python programming language to implement custom algorithms and extensions without any prior knowledge of SCIP's programming language C. TreeD is one such example, demonstrating the use of several Python libraries on top of SCIP. PySCIPOpt also provides an intuitive modeling layer to formulate problems directly in the code without having to utilize another modeling language or framework. All contributions presented in this thesis are readily accessible in source code in SCIP Optimization Suite or as separate projects on the public code-sharing platform GitHub.

Description: The SCIP Optimization Suite is a software toolbox for generating and solving various classes of mathematical optimization problems. Its major components are the modeling language ZIMPL, the linear programming solver SoPlex, the constraint integer programming framework and mixed-integer linear and nonlinear programming solver SCIP, the UG framework for parallelization of branch-and-bound-based solvers, and the generic branch-cut-and-price solver GCG. It has been used in many applications from both academia and industry and is one of the leading non-commercial solvers. This paper highlights the new features of version 3.2 of the SCIP Optimization Suite. Version 3.2 was released in July 2015. This release comes with new presolving steps, primal heuristics, and branching rules within SCIP. In addition, version 3.2 includes a reoptimization feature and improved handling of quadratic constraints and special ordered sets. SoPlex can now solve LPs exactly over the rational number and performance improvements have been achieved by exploiting sparsity in more situations. UG has been tested successfully on 80,000 cores. A major new feature of UG is the functionality to parallelize a customized SCIP solver. GCG has been enhanced with a new separator, new primal heuristics, and improved column management. Finally, new and improved extensions of SCIP are presented, namely solvers for multi-criteria optimization, Steiner tree problems, and mixed-integer semidefinite programs.

Description: State-of-the-art solvers for mixed integer programs (MIP) govern a variety of algorithmic components. Ideally, the solver adaptively learns to concentrate its computational budget on those components that perform well on a particular problem, especially if they are time consuming. We focus on three such algorithms, namely the classes of large neighborhood search and diving heuristics as well as Simplex pricing strategies. For each class we propose a selection strategy that is updated based on the observed runtime behavior, aiming to ultimately select only the best algorithms for a given instance. We review several common strategies for such a selection scenario under uncertainty, also known as Multi Armed Bandit Problem. In order to apply those bandit strategies, we carefully design reward functions to rank and compare each individual heuristic or pricing algorithm within its respective class. Finally, we discuss the computational benefits of using the proposed adaptive selection within the SCIP Optimization Suite on publicly available MIP instances.

Description: State-of-the-art solvers for mixed integer programs (MIP) govern a variety of algorithmic components. Ideally, the solver adaptively learns to concentrate its computational budget on those components that perform well on a particular problem, especially if they are time consuming. We focus on three such algorithms, namely the classes of large neighborhood search and diving heuristics as well as Simplex pricing strategies. For each class we propose a selection strategy that is updated based on the observed runtime behavior, aiming to ultimately select only the best algorithms for a given instance. We review several common strategies for such a selection scenario under uncertainty, also known as Multi Armed Bandit Problem. In order to apply those bandit strategies, we carefully design reward functions to rank and compare each individual heuristic or pricing algorithm within its respective class. Finally, we discuss the computational benefits of using the proposed adaptive selection within the \scip Optimization Suite on publicly available MIP instances.

Description: This article describes new features and enhanced algorithms made available in version 5.0 of the SCIP Optimization Suite. In its central component, the constraint integer programming solver SCIP, remarkable performance improvements have been achieved for solving mixed-integer linear and nonlinear programs. On MIPs, SCIP 5.0 is about 41 % faster than SCIP 4.0 and over twice as fast on instances that take at least 100 seconds to solve. For MINLP, SCIP 5.0 is about 17 % faster overall and 23 % faster on instances that take at least 100 seconds to solve. This boost is due to algorithmic advances in several parts of the solver such as cutting plane generation and management, a new adaptive coordination of large neighborhood search heuristics, symmetry handling, and strengthened McCormick relaxations for bilinear terms in MINLPs. Besides discussing the theoretical background and the implementational aspects of these developments, the report describes recent additions for the other software packages connected to SCIP, in particular for the LP solver SoPlex, the Steiner tree solver SCIP-Jack, the MISDP solver SCIP-SDP, and the parallelization framework UG.

Description: The SCIP Optimization Suite is a powerful collection of optimization software that consists of the branch-cut-and-price framework and mixed-integer programming solver SCIP, the linear programming solver SoPlex, the modeling language Zimpl, the parallelization framework UG, and the generic branch-cut-and-price solver GCG. Additionally, it features the extensions SCIP-Jack for solving Steiner tree problems, PolySCIP for solving multi-objective problems, and SCIP-SDP for solving mixed-integer semidefinite programs. The SCIP Optimization Suite has been continuously developed and has now reached version 4.0. The goal of this report is to present the recent changes to the collection. We not only describe the theoretical basis, but focus on implementation aspects and their computational consequences.

Description: 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 6.0 of the SCIP Optimization Suite. Besides performance improvements of the MIP and MINLP core achieved by new primal heuristics and a new selection criterion for cutting planes, one focus of this release are decomposition algorithms. Both SCIP and the automatic decomposition solver GCG now include advanced functionality for performing Benders’ decomposition in a generic framework. GCG’s detection loop for structured matrices and the coordination of pricing routines for Dantzig-Wolfe decomposition has been significantly revised for greater flexibility. Two SCIP extensions have been added to solve the recursive circle packing problem by a problem-specific column generation scheme and to demonstrate the use of the new Benders’ framework for stochastic capacitated facility location. Last, not least, the report presents updates and additions to the other components and extensions of the SCIP Optimization Suite: the LP solver SoPlex, the modeling language Zimpl, the parallelization framework UG, the Steiner tree solver SCIP-Jack, and the mixed-integer semidefinite programming solver SCIP-SDP.

Description: The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming frame- work SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op- timization problems and can be used stand-alone or integrated into SCIP via a presolver plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im- provements in the Benders’ decomposition solver of SCIP, user-defined decomposition structures can be read, which are used by the automated Benders’ decomposition solver and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation that is used to predict the completion of the overall solving process and potentially trigger restarts. Moreover, substantial performance improvements of the MIP core were achieved by new developments in presolving, primal heuristics, branching rules, conflict analysis, and symmetry handling. Last, not least, the report presents updates to other components and extensions of the SCIP Optimization Suite, in particular, the LP solver SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.