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18-20
Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone
generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where
$1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$.
Several combinatorial problems reduce to a linear optimization problem over $K$.
This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a
completely positive representation of $K$.
We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.
18-35
The amazing success of computational mathematical optimization over the last decades has been driven more by insights into mathematical structures than by the advance of computing technology. In this vein, we address applications, where nonconvexity in the model poses principal difficulties.
This paper summarizes the dissertation of Jonas Schweiger for the occasion of the GOR dissertation award 2018. We focus on the work on non-convex quadratic programs and show how problem specific structure can be used to obtain tight relaxations and speed up Branch&Bound methods. Both a classic general QP and the Pooling Problem as an important practical application serve as showcases.
18-13
We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which products are mixed in intermediate pools in order to meet quality targets at their destinations. In this technical report, we characterize the extreme points of the convex hull of our non-convex set, and show that they are not finite, i.e., the convex hull is not polyhedral. This analysis was used to derive valid nonlinear convex inequalities and show that, for a specific case, they characterize the convex hull of our set. The new valid inequalities and computational results are presented in ZIB Report 18-12.
18-12
We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by considering a single product, a single attibute, and a single pool. The convex hull of the resulting nonconvex set is not polyhedral. We derive valid linear and convex nonlinear inequalities for the convex hull, and demonstrate that different subsets of these inequalities define the convex hull of the nonconvex set in three cases determined by the parameters of the set. Computational results on literature instances and newly created larger test instances demonstrate that the inequalities can significantly strengthen the convex relaxation of the pq-formulation of the pooling problem, which is the relaxation known to have the strongest bound.
18-14
During the last decades, X-ray (micro-)computed tomography has gained increasing attention for the description of porous skeletal and shell structures of various organism groups. However, their quantitative analysis is often hampered by the difficulty to discriminate cavities and pores within the object from the surrounding region. Herein, we test the ambient occlusion (AO) algorithm and newly implemented optimisations for the segmentation of cavities (implemented in the software Amira). The segmentation accuracy is evaluated as a function of (i) changes in the ray length input variable, and (ii) the usage of AO (scalar) field and other AO-derived (scalar) fields. The results clearly indicate that the AO field itself outperforms all other AO-derived fields in terms of segmentation accuracy and robustness against variations in the ray length input variable. The newly implemented optimisations improved the AO field-based segmentation only slightly, while the segmentations based on the AO-derived fields improved considerably. Additionally, we evaluated the potential of the AO field and AO-derived fields for the separation and classification of cavities as well as skeletal structures by comparing them with commonly used distance-map-based segmentations. For this, we tested the zooid separation within a bryozoan colony, the stereom classification of an ophiuroid tooth, the separation of bioerosion traces within a marble block and the calice (central cavity)-pore separation within a dendrophyllid coral. The obtained results clearly indicate that the ideal input field depends on the three-dimensional morphology of the object of interest. The segmentations based on the AO-derived fields often provided cavity separations and skeleton classifications that were superior to or impossible to obtain with commonly used distance- map-based segmentations. The combined usage of various AO-derived fields by supervised or unsupervised segmentation algorithms might provide a promising target for future research to further improve the results for this kind of high-end data segmentation and classification. Furthermore, the application of the developed segmentation algorithm is not restricted to X-ray (micro-)computed tomographic data but may potentially be useful for the segmentation of 3D volume data from other sources.
18-52
Gene Regulatory Networks are powerful models for describing the mechanisms and dynamics inside a cell. These networks are generally large in dimension and seldom yield analytical formulations. It was shown that studying the conditional expectations between dimensions (vertices or species) of a network could lead to drastic dimension reduction. These conditional expectations were classically given by solving equations of motions derived from the Chemical Master Equation. In this paper we deviate from this convention and take an Algebraic approach instead. That is, we explore the consequences of conditional expectations being described by a polynomial function. There are two main results in this work. Firstly: if the conditional expectation can be described by a polynomial function, then coefficients of this polynomial function can be reconstructed using the classical moments. And secondly: there are dimensions in Gene Regulatory Networks which inherently have conditional expectations with algebraic forms. We demonstrate through examples, that the theory derived in this work can be used to develop new and effective numerical schemes for forward simulation and parameter inference. The algebraic line of investigation of conditional expectations has considerable scope to be applied to many different aspects of Gene Regulatory Networks; this paper serves as a preliminary commentary in this direction.
18-18
In molecular structure analysis and visualization, the molecule’s atoms are often modeled as hard spheres parametrized by their positions and radii. While the atom positions result from experiments or molecular simulations, for the radii typically values are taken from literature. Most often, van der Waals (vdW) radii are used, for which diverse values exist. As a consequence, different visualization and analysis tools use different atomic radii, and the analyses are less objective than often believed. Furthermore, for the geometric accessibility analysis of molecular structures, vdW radii are not well suited. The reason is that during the molecular dynamics simulation, depending on the force field and the kinetic energy in the system, non-bonded atoms can come so close to each other that their vdW spheres intersect. In this paper, we introduce a new kind of atomic radius, called atomic accessibility radius’, that better characterizes the accessibility of an atom in a given molecular trajectory. The new radii reflect the movement possibilities of atoms in the simulated physical system. They are computed by solving a linear program that maximizes the radii of the atoms under the constraint that non-bonded spheres do not intersect in the considered molecular trajectory. Using this data-driven approach, the actual accessibility of atoms can be visualized more precisely.
18-25
The void space of granular materials is generally divided into larger local volumes denoted as pores and throats connecting pores. The smallest section in a throat is usually denoted as constriction. A correct description of pores and constrictions may help to understand the processes related to the transport of fluid or fine particles through granular materials, or to build models of imbibition for unsaturated granular media. In the case of numerical granular materials involving packings of spheres, different methods can be used to compute the pore space properties. However, these methods generally induce an over-segmentation of the pore network and a merging step is usually applied to mitigate such undesirable artifacts even if a precise delineation of a pore is somewhat subjective. This study provides a comparison between different merging criteria for pores in packing of spheres and a discussion about their implication on both the pore size distribution and the constriction size distribution of the material. A correspondence between these merging techniques is eventually proposed as a guide for the user.
18-27
We study the problem of finding subpaths with high demand in a given network that is traversed by several users. The demand of a subpath is the number of users who completely cover this subpath during their trip.
Especially with large instances, an efficient algorithm for computing all subpaths' demands is necessary. We introduce a path-graph to prevent multiple generations of the same subpath and give a recursive approach to compute the demands of all subpaths.
Our runtime analysis shows, that the presented approach compares very well against the
theoretical minimum runtime.