Invited Session Wed.2.H 2038

Wednesday, 13:15 - 14:45 h, Room: H 2038

Cluster 4: Conic programming [...]

Conic and convex programming in statistics and signal processing III

 

Chair: Venkat Chandrasekaran

 

 

Wednesday, 13:15 - 13:40 h, Room: H 2038, Talk 1

Deanna Needell
Randomized projection algorithms for overdetermined linear systems

 

Abstract:
In this talk we discuss variations to projection onto convex sets (POCS) type methods for overdetermined linear systems. POCS methods have found many applications
ranging from computer tomography to digital signal and image processing. The
Kaczmarz method is one of the most popular solvers for overdetermined systems
of linear equations due to its speed and simplicity. Here we introduce and analyze
extensions of this method which provide exponential convergence to the solution in expectation which in some settings significantly improves upon the convergence rate of the standard method.

 

 

Wednesday, 13:45 - 14:10 h, Room: H 2038, Talk 2

Stephen Joseph Wright
Packing ellipsoids (and chromosomes)

Coauthor: Caroline Uhler

 

Abstract:
Problems of packing shapes with maximal density, possibly into a
container of restricted size, are classical in mathematics. We
describe here the problem of packing ellipsoids of given (and varying)
dimensions into a finite container of given size, allowing overlap
between adjacent ellipsoids but requiring some measure of total
overlap to be minimized. A trust-region bilevel optimization algorithm
is described for finding local solutions of this problem - both the
general case and the more elementary special case in which the
ellipsoids are in fact spheres. Tools from conic optimization,
especially semidefinite programming and duality, are key to the
algorithm. Theoretical and computational results will be
summarized. Our work is motivated by a problem in structural biology -
chromosome arrangement in cell nuclei - for which results are
described.

 

 

Wednesday, 14:15 - 14:40 h, Room: H 2038, Talk 3

James Saunderson
Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

Coauthor: Pablo A. Parrilo

 

Abstract:
The hyperbolicity cones associated with the elementary symmetric polynomials provide an intriguing family of non-polyhedral relaxations of the non-negative orthant that preserve its low-dimensional faces and successively discard higher dimensional structure. A similar construction gives a family of outer approximations for any spectrahedral cone (i.e. slice of the psd cone), and more generally for any hyperbolicity cone. We show, by a simple and explicit construction, that these derivative relaxations of spectrahedral cones have polynomial-sized representations as projections of slices of the psd cone. This, for example, allows us to solve the associated linear cone program using semidefinite programming, and allows us to give corresponding explicit semidefinite representations for the (thus far poorly understood) duals of the derivative relaxations of spectrahedral cones.

 

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