Welcome to Combinatorics and Optimization

Title:The omega-Condition Number: Applications to Preconditioning and Low Rank Generalized Jacobian Updating

Abstract: Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g.,~in quasi-Newton methods. We study a nonclassic matrix condition number, the omega-condition number}, omega for short. omega is the ratio of: the arithmetic and geometric means of the singular values, rather than the largest and smallest for the classical kappa-condition number. The simple functions in omega allow one to exploit  first order optimality conditions. We use this fact to derive explicit formulae for (i) omega-optimal low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) omega-optimal preconditioners of special structure for  iterative methods for linear systems. In the latter context, we analyze the benefits of omega for (a) improving the clustering of eigenvalues; (b) reducing the number of iterations; and (c) estimating the actual condition of a linear system. Moreover we show strong theoretical connections between the omega-optimal preconditioners and incomplete Cholesky factorizations, and highlight the misleading effects arising from the inverse invariance of kappa. Our results confirm the efficacy of using the omega-condition number compared to the kappa-condition number.

(Joint work with: Woosuk L. Jung, David Torregrosa-Belen.)

Title:Sizes of witnesses in covtree

Abstract: Here is a purely combinatorial problem that arose in causal set theory.  Let {P_1, ... , P_k} be distinct unlabelled posets all with n elements.  Suppose there is a poset Q such that {P_1, ... , P_k} is exactly the set of downsets of Q of size n up to isomorphism. Given n and k can we give a tight upper bound on the minimum size of such a Q? As with newspaper headlines, the answer to the question is no, at least for the moment, but I'll explain what we do know.  Joint work with Jette Gutzeit, Kimia Shaban, and Stav Zalel.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,

Title:Automated Sequence Asymptotics

Abstract:Computing with any sort of object requires a way of encoding it on a computer, which poses a problem in enumerative combinatorics where the objects of interest are (infinite) combinatorial sequences. Thankfully, the generating function of a combinatorial sequence often satisfies natural algebraic/differential/functional equations, which can then be viewed as data structures for the sequence. In this talk we survey methods to take a sequence encoded by such data structures and automatically determine asymptotic behaviour using techniques from the field of analytic combinatorics. We also discuss methods to automatically characterize the asymptotic behaviour of multivariate sequences using analytic combinatorics in several variables (ACSV). The focus of each topic will be rigorous algorithms that have already been implemented in computer algebra systems and can be easily used by anyone.