Polynomial Linear Programming with Gaussian Belief Propagation

Interior-point methods are state-of-the-art algorithms for solving linear programming (LP) problems with polynomial complexity. Specifically, the Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where $n$ is the number of unknown variables. Karmarkar's celebrated algorithm is known to be an instance of the log-barrier method using the Newton iteration. The main computational overhead of this method is in inverting the Hessian matrix of the Newton iteration. In this contribution, we propose the application of the Gaussian belief propagation (GaBP) algorithm as part of an efficient and distributed LP solver that exploits the sparse and symmetric structure of the Hessian matrix and avoids the need for direct matrix inversion. This approach shifts the computation from realm of linear algebra to that of probabilistic inference on graphical models, thus applying GaBP as an efficient inference engine. Our construction is general and can be used for any interior-point algorithm which uses the Newton method, including non-linear program solvers.Comment: 7 pages, 1 figure, appeared in the 46th Annual Allerton Conference on Communication, Control and Computing, Allerton House, Illinois, Sept. 200

The world crisis: the implications of globalised finance

The term “globalisation” has survived its first significant sell-by date in modern times. Rightly, it continues to attract policy attention and debate at the very highest levels. Together with just a handful of others—economic growth and inequality, financial crisis, climate change—with all of which it remains inextricably intertwined, only globalisation among economic phenomena has both effects and causes observable from outer space. Its impact on the welfare of humanity is therefore singular. This is even before one considers the sweeping changes in culture and politics that ever greater global integration both requires and engenders. This article cannot hope to cover the massive body of modern thinking that surrounds globalisation. Instead, what it seeks to do is two-fold: first, flag, with the benefit of hindsight, some of the key background points that any continuing discussion of globalisation needs to keep in mind; and second, offer conjecture where the most likely contentious issues in the near future might be. To keep within space constraints, careful and exhaustive discussion of empirical evidence is omitted. Instead, just the largest salient facts are provided where needed

Comment on Donohue

A comment on John J. Donohue III\u27s article on the effects of fee-shifting rules on the rate of settlements in lawsuits is presented. The article bears out the idea that something may work fine in practice but it remains to be proven if it will work in theory

Circular groups, planar groups, and the Euler class

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the Euler class of C^infty diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^infty action with arbitrary Euler class. On the other hand, we show that Z oplus Z actions satisfy a homological rigidity property: every orientation-preserving C^1 action of Z oplus Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R^2 in every degree of smoothness.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper15.abs.htm

Cosmology and Hierarchy in Stabilized Randall-Sundrum Models

We consider the cosmology and hierarchy of scales in models with branes immersed in a five-dimensional curved spacetime subject to radion stabilization. The universe naturally find itself in the radiation-dominated epoch when the inter-brane spacing is static and stable, independent of the form of the stabilizing potential. We recover the standard Friedmann equations without assuming a specific form for the bulk energy-momentum tensor. We address the hierarchy problem in the context of a quartic and exponential stabilizing potential, and find that in either case the presence of a negative tension brane is required and that the string scale can be as low as the electroweak scale. In the situation of self-tuning branes (corresponding to an exponential potential) where the bulk cosmological constant is set to zero, the brane tensions have hierarchical values.Comment: 3 pages, 1 figure, 1 table. Talk given at DPF 2000, Columbus, OH, August 12, 200