Heinz Bauschke
Department of Mathematics
Okanagan University College
The method of cyclic projections --- the inconsistent case
Abstract:
The method of cyclic projections is an algorithm for solving
convex feasibility problems in Hilbert space: a starting
point is projected onto the first set, the resulting point onto the
second, and so on, until the last set is visited and the cycle starts
anew.
It is well known that this method converges weakly to a point in the
intersection of the constraints --- assuming there is one! The
inconsistent case, however, is much less clear and will be the focus of
my talk. I will present classical and recent results, and tantalizing
conjectures.
James V. Burke
Department of Mathematics
University of Washington
Approximating of subdifferentials by random sampling of gradients
Abstract:
Many interesting non-smooth functions on Euclidean space are in fact
differentiable almost everywhere. All Lipschitz function have this
property, but so do non-Lipschitzian functions such as the spectral
abscissa of a matrix. In practise, the gradient, when it exists, is
often easy to compute. In this talk, we investigate to what extent one
can approximate the Clarke subdifferential of such functions by the
convex hull of randomly sampled gradients in the neighborhood of a point
of interest. We also show how these ideas can be used to construct
successful numerical methods for minimizing non-smooth and potentially
non-Lipschitzian objective functions.
Lisa Korf
Department of Mathematics
University of Washington
Duality Theorems in Stochastic Programming
Abstract:
This talk will give an overview of some of the existing duality theorems
in stochastic programming in the setting of an infinite probability space.
These theorems provide dual problems whose solution space (in the second
variable) is L1. They are, however restricted to primal constraint
functions which lie in Lºº, and problems satisfying rather
stringent constraint qualifications such as strict feasibility,
boundedness of the constraint set, and relatively complete recourse.
These restrictions are not satisfied by a lot of emerging applications,
notably in mathematical finance. A new duality theory is presented
which allows the constraint functions to lie in L1, does not require
the constraint sets to be bounded, and relies on much weaker constraint
qualifications to achieve dual problems whose solution space (in the
second variable) is Lºº.
A motivating example from mathematical finance will be briefly noted.
Note: Please read Lºº as "L-infinity".
Mason Macklem
Department of Mathematics and Statistics
Simon Fraser University
Current Models in Image Quality Evaluation
Abstract:
Traditionally, objective image quality assessment focussed on the Mean-Squared
Error (MSE) model; image quality can be represented by a
single number, calculated using only local, ie. pixel-wise, comparison of image
matrices. With the evident failure of this model, focus has
shifted towards observed trends in human vision; results from psychophysical
experiments have replaced mathematical analysis independent of
the visual system. This talk will briefly summarize the Human Visual System
(HVS) image quality model, and will present results from an open
implementation of one particular evaluation model.
R. T. Rockafellar
Department of Mathematics
University of Washington
Variational Geometry and Equilibrium
Abstract:
Variational inequalities and even quasi-variational inequalities, as a
means of expressing constrained equilibrium, have utilized geometric
propteries of convex sets, but the general theory of tangent cones and
normal cones has yet to be fully exploited. Much progress has been
made in that theory in recent years in understanding the variational
geometry of nonconvex sets as well as convex sets and in applying it to
optimization problems. Parallel applications to equilibrium problems
could be pursued now as well.
In this talk, it will be explained how normal cone mappings and their
calculus offer an attractive framework for many purposes, and how the
properties of the graphs of such mappings furnish powerful tools for use
in ascertaining how an equilibrium is affected by perturbations. Many
challenges remain, however, in developing conditions that guarantee
the existence of an equilibrium in a nonconvex setting.
Stephen M. Simons
Department of Mathematics
University of California, Santa Barbara
Hahn-Banach and minimax theorems
Abstract:
We introduce a generalized form of the Hahn--Banach theorem, which we
will use to prove various classical results on the existence of linear
functionals, and also to prove a minimax theorem.
We will also explain why one cannot generalize the minimax theorem too
much, in the sense that any reasonable attempt to generalize the minimax
theorem to a pair of noncompact sets is probably doomed to failure.
We will, however, mention an unreasonable generalization that is true,
hard and related to R. C. James's "sup theorem''.
Herre Wiersma
Department of Mathematics and Statistics
Simon Fraser University
A C1
function that is even on a sphere and has no critical points in
the ball
Abstract:
I will construct a real-valued C1-function on the closed
ball in R2 that is even on the boundary of the ball, and has no
critical points inside the ball. This provides a counterexample to a
nonsmooth Rolle-type theorem sought in Borwein and Fitzpatrick's paper,
"Duality Inequalities and Sandwiched Functions".
Jim Zhu
Department of Mathematics and Statistics
Western Michigan University
Necessary conditions for constrained optimization problems
in smooth Banach spaces and applications
Abstract:
We derive necessary optimality conditions for constrained
minimization problems with lower semicontinuous inequality
constraints, continuous equality constraints and a set constraint in
smooth Banach spaces. Then we apply these necessary optimality
conditions to derive subdifferential representations of singular normal
vectors to the epigrah and the graph of functions, subdifferential
calculus and necessary optimality conditions for mathematical
programs with equilibrium constraints.