Title : Optimization by Vector Space Methods
Abstract:
Optimization theory , which is a branch of applied mathematics, looks to
various areas of pure mathematics for its unification, clarification and
general foundation. One such area of particular relevance is functional
analysis. Functional analysis is the study of vector spaces of resulting
from a merging of geometry, linear algebra and analysis. It serves as a
basis for Fourier series, integral and differential equations, numerical
analysis, and any field where linearity plays a key role. Its appeal as a
unifying discipline stems primarily from its geometric character.
This talk will focus on the some aspects of optimization theory which are
derived from a few simple, intuitive, geometric insights. The conceptual
utility of functional analysis will be motivated by showing how it enables
us to extend our three-dimensional geometric insights to complex
infinite-dimensional problems. The projection theorem will be used as the
motivating example. In ordinary three-dimensional space, it states that the
shortest line from a point to a plane is furnished by the perpendicular from
the point to the plane. This result has direct extensions in spaces of
higher dimension and in infinite-dimensional Hilbert space. If time permits,
I will also show how the Hahn-Banach theorem serves as an appropriate
generalization of the projection theorem from Hilbert spaces to arbitrary
normed spaces.