For sparse graphs, the odd cycle polytope can be used to compute useful bounds for the maximum stable set problem quickly. Yannakakis introduced an extended formulation for the odd cycle inequalities of the stable set polytope in 1991, which provides a direct way to optimize over the odd cycle polytope in polynomial time, although there … Read more
In state-of-the-art mixed-integer programming solvers, a large array of reduction techniques are applied to simplify the problem and strengthen the model formulation before starting the actual branch-and-cut phase. Despite their mathematical simplicity, these methods can have significant impact on the solvability of a given problem. However, a crucial property for employing presolving techniques successfully is … Read more
Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, such that … Read more
Interior point methods have attracted most of the attention in the recent decades for solving large scale convex quadratic programming problems. In this paper we take a different route as we present an augmented Lagrangian method for convex quadratic programming based on recent developments for nonlinear programming. In our approach, box constraints are penalized while … Read more
We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all, provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is … Read more
An attractive feature of the face method \cite{pan14} for solving LP problems is that it uses the orthogonal projection of the negative objective gradient on the related null space as the search direction. However, the method would not be amenable for solving large sparse problems, since it handles the involved normal system by orthogonal transformations. … Read more
The implementation of a linear programming interior point solver is described that is based on iterative linear algebra. The linear systems are preconditioned by a basis matrix, which is updated from one interior point iteration to the next to bound the entries in a certain tableau matrix. The update scheme is based on simplex-type pivot … Read more
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 … Read more
In this paper, we propose a new framework to implement interior point method (IPM) in order to solve some very large scale linear programs (LP). Traditional IPMs typically use Newton’s method to approximately solve a subproblem that aims to minimize a log-barrier penalty function at each iteration. Due its connection to Newton’s method, IPM is … Read more
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 … Read more