November 13, 2014, 4:00pm
Henry Wolkowicz, Department of Mathematics, University of Waterloo.
Taking Advantage of Degeneracy in Cone Optimization: with Applications to Sensor Network Localization
Abstract: The elegant theoretical results for strong duality and strict complementarity for linear programming, LP, lie behind the success of current algorithms. However, the theory and preprocessing techniques that are successful for LP can fail for cone programming over nonpolyhedral cones.
Surprisingly, many important applications of semidefinite programming, SDP, that arise from relaxations of hard combinatorial problems are degenerate. (Slater’s constraint qualification fails.) This includes relaxations for problems such as the: Quadratic Assignment; Graph Partitioning; Set Covering and partitioning; and sensor network localization and molecular conformation. Rather than being a disadvantage, we show that this degeneracy can be exploited. In particular, several huge instances of SDP completion problems can be solved quickly and to extremely high accuracy. In particular, we illustrate this on the sensor network localization problem.
November 18, 2014, 4:00pm
Daniela Witten, Departments of Biostatistics and Statistics, UW.
Flexible Graphical Modeling
Abstract: In recent years, there has been considerable interest in estimating conditional dependence relationships among random variables in the high-dimensional setting, in which the number of variables far exceeds the number of available observations. Most prior work has assumed that the variables are multivariate Gaussian, or that the conditional means of the variables are linear. Unfortunately, if these assumptions are violated, then the resulting estimates can be inaccurate.
I will present two recent lines of work on learning the conditional dependence graph of a set of random variables in the non-Gaussian setting. First, I will present a semi-parametric method that allows the conditional means of the features to take on an arbitrary additive form. Next, I will present an approach for learning a graph in which the distribution of each node, conditioned on the others, may have a different parametric form. Each approach is formulated as the solution to a convex optimization problem corresponding to a penalized log likelihood.
This is joint work with Ali Shojaie, Shizhe Chen, and Arend Voorman.
October 14, 2014, 4:00pm
Annie Raymond, Department of Mathematics, University of Washington.
A New Conjecture for Union-Closed Families
Abstract: The Frankl union-closed sets conjecture states that there exists an element present in at least half of the sets forming a union-closed family. We reformulate the conjecture as an optimization problem and present an integer program to model it. The computations done with this program lead to a new conjecture: we claim that the maximum number of sets in a non-empty union-closed family in which each element is present at most a times is independent of the number n of elements spanned by the sets if n is greater or equal to log_2(a)+1. We prove that this is true when n is greater or equal to a. We also discuss the impact that this new conjecture would have on the Frankl conjecture if it turns out to be true. This is joint work with Jonad Pulaj and Dirk Theis.