Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences

Questions in computational molecular biology generate various discrete optimization problems, such as DNA sequence alignment and RNA secondary structure prediction. However, the optimal solutions are fundamentally dependent on the parameters used in the objective functions. The goal of a parametric analysis is to elucidate such dependencies, especially as they pertain to the accuracy and robustness of the optimal solutions. Techniques from geometric combinatorics, including polytopes and their normal fans, have been used previously to give parametric analyses of simple models for DNA sequence alignment and RNA branching configurations. Here, we present a new computational framework, and proof-of-principle results, which give the first complete parametric analysis of the branching portion of the nearest neighbor thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure

Volumetric Guidance for Handling Triple Products in Spatial Branch-and-Bound

Spatial branch-and-bound (sBB) is the workhorse algorithmic framework used to globally solve mathematical mixed-integer non-linear optimization (MINLO) problems. Formulating a problem using this paradigm allows both the non-linearities of a system and any discrete design choices to be modeled effectively. Because of the generality of this approach, MINLO is used in a wide variety of applications, from chemical engineering problems and network design, to medical applications and problems in the airline industry. Due in part to their generality (and therefore wide applicability), MINLO problems are very difficult in general, and consequently, the best ways to implement many details of sBB are not wholly understood. In this work, we provide analytic results guiding the implementation of sBB for a simple but frequently occurring function ‘building block’. As opposed to computationally demonstrating that our techniques work only for a particular set of test problems, we analytically establish results that hold for all problems of the given form. In this way, we also demonstrate that analytic results are indeed obtainable for certain sBB implementation decisions. In particular, we use volume as a geometric measure to compare different convex relaxations for functions involving trilinear monomials (or any three quantities multiplied together). We consider different choices for convexifying the graph of a triple product, and obtain formulae for the volume (in terms of the variable upper and lower bounds) for each of these convexifications. We are then able to order the convexifications with regard to their volume. We also provide computational evidence to support our choice of volume as an effective comparison measure, and show that in the context of triple products, volume is an excellent predictor of the objective function gap. Finally, we use the volume measure to provide guidance regarding branching-point selection in the implementation of sBB.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/136973/1/eespeakm_1.pd

Geometric comparison of combinatorial polytopes

We survey some analytic methods of volume calculation and introduce a distance function for pairs of polytopes based on their volumes. We study the distance function in the context of three familiar settings of combinatorial optimization: (i) ChvatalGomory rounding, (ii) fixed charge problems, and (iii) vertex packing on threshold graphs.