Abstracts

Abstract: A feasibility problem requests solution to the problem

\mbox{Find~~} x\in \bigcap_{i=1}^N C_i

where C_1,C_2,\ldots C_N are closed sets finitely many closed sets lying in a Hilbert space \mathcal{H}. We consider iterative methods based on the non-expansive properties of the metric projection operator

P_C(x):={\rm argmin}_{c \in C} \|x-c\|

or reflection operator R_C:=2P_C-I on a closed convex set C in Hilbert space. These methods work best when the projection on each set C_i is easy to describe or approximate. These methods are especially useful when the number of sets involved is large as the methods are fairly easy to parallelize. The theory is pretty well understood when all sets are convex. The theory is much less clear in the non-convex case. But as we shall see application of this case has had may successes. So this is a fertile area for both pure and applied study. The five hours of lectures will cover the following topics.

  1. Alternating projection methods: background theory, convergence and basic algorithms
  2. The Douglas Rachford reflection method and generalizations
  3. Applications to convex problems and to non-convex combinatorial problems and to matrix completion problems
  4. Protein conformation determination: a detailed case study
  5. Relaxed reflection methods and norm convergence for realistic problems
This is based on joint work with Matt Tam, Brailey Sims and Fran Aragon.

Abstract: We consider important properties of Fréchet subdifferentials, in particular: the non-emptiness of subdifferentials, the non-zeroness of normal cones, the fuzzy calculus, and the extremal principle; all statements being considered in Fréchet sense. Given a nonseparable Banach space X, we show how the validity of these statements is implied by the validity of them in every separable subspace of X. Such a reasoning is called “separable reduction”. We show that, behind this approach, there is a modern and powerful concept of rich subfamily of the family of all separable subspaces of X.

Abstract: Developments in variational analysis have had a really revolutionizing effect on many aspects of optimization theory. On the one hand, new methods developed within variational analysis allow to extend many classical results to extremely broad classes of objects. On the other hand, applied in situations typical for the classical theories, they often allow to single out and illuminate the real cause of phenomena and substantially simplify many arguments. In the lectures we intend to touch upon a wide spectrum of topics from maximum principle in optimal control to the von Neuman method of alternating projections.

Abstract: The goal of these lectures is twofold: first, to introduce students with some background in variational and convex analysis to variational numerical methods and secondly, to introduce students with some experience in more traditional numerical methods to the variational perspective. We apply the concepts of variational analysis to four problems: computed x-ray tomography, optical design, sparsity optimization, and phase retrieval. Students will analyze and implement proximal, and more specifically projection, algorithms for solving the above problems. At the end of the lecture students should be able to implement and analyze the results from classical techniques as well as state-of-the-art algorithms.