University of Wisconsin

We investigate the use of interior-point and semismooth methods for solving

quadratic programming problems with a small number of linear constraints,

where the quadratic term consists of a low-rank update to a positive

semi-definite matrix. Several formulations of the support vector machine

fit into this category. An interesting feature of these particular problems

is the volume of data, which can lead to quadratic programs with between 10

and 100 million variables and, if written explicitly, a dense $Q$ matrix.

Our codes are based on OOQP, an object-oriented interior-point code, with the

linear algebra specialized for the support vector machine application.

For the targeted massive problems, all of the data is stored out of core and

we overlap computation and I/O to reduce overhead. Results are reported for

several linear support vector machine formulations demonstrating that the

methods are reliable and scalable and comparing the two approaches.

The problem of isometric embedding of a Riemannian Manifold into

Euclidean space is a classical issue in differential geometry and

nonlinear PDE. In this talk, I will outline recent work my

co-workers and I have done, using ideas from continuum mechanics as a guide,

formulating the problem, and giving (we hope) some new insight

into the role of " entropy".