Knapsack Polytopes – A Survey

The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy a given single linear inequality with non-negative coefficients. This paper provides a comprehensive overview of knapsack polytopes. We discuss basic polyhedral properties, (lifted) cover and other valid inequalities, cases for which complete linear descriptions are known, geometric properties for small dimensions, … Read more

Resilient layout, design and operation of energy-efficient water distribution networks for high-rise buildings using MINLP

Water supply of high-rise buildings requires pump systems to ensure pressure requirements. The design goal of these systems are energy and cost efficiency, both in terms of fixed cost as well as during operation. In this paper, cost optimal decentralized and tree-shaped water distribution networks are computed, where placements of pumps at different locations in … Read more

The SCIP Optimization Suite 6.0

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

Maximizing the storage capacity of gas networks: a global MINLP approach

In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler … Read more

The SCIP Optimization Suite 5.0

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

Global optimization of mixed-integer ODE constrained network problems using the example of stationary gas transport

In this paper we propose a new approach for finding global solutions of mixed-integer nonlinear optimization problems with ordinary differential equation constraints on networks. Instead of using a first discretize then optimize approach, we combine spatial and variable branching with appropriate discretizations of the differential equations to derive relaxations of the original problem. To construct … Read more

Algorithmic Results for Potential-Based Flows: Easy and Hard Cases

Potential-based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize … Read more

Extended Formulations for Column Constrained Orbitopes

In the literature, packing and partitioning orbitopes were discussed to handle symmetries that act on variable matrices in certain binary programs. In this paper, we extend this concept by restrictions on the number of 1-entries in each column. We develop extended formulations of the resulting polytopes and present numerical results that show their effect on … Read more

On the Size of Integer Programs with Bounded Coefficients or Sparse Constraints

Integer programming formulations describe optimization problems over a set of integer points. A fundamental problem is to determine the minimal size of such formulations, in particular, if the size of the coefficients or sparsity of the constraints is bounded. This article considers lower and upper bounds on these sizes both in the original and in … Read more

The SCIP Optimization Suite 4.0

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 … Read more