Course Title: Optimization over Vector Spaces

This course module will cover the basic theory of optimization over vector spaces and application of this theory to optimal control problems. This module will be taught in about 5 weeks (about 15 lectures of 1 hour duration). The following topics will be covered:

1.    Normed Linear Spaces-Banach Spaces, Hilbert Spaces (Review)

2.    Dual Spaces-Linear Functionals, Hahn-Banach Theorem: (extension and geometric form)

3.    Optimization of Functionals: Gateaux and Frechet Differentials, Euler Lagrange equations, Min-max theorem

4.    Constrained Optimization:

a.    Lagrange Multiplier theorems

b.    Optimal Control theory (Pontryagin’s maximum principle)

Text: David G Luenberger, “Optimization by Vector Space Methods,” 1969 John Wiley and Sons, New York.

Lecture Notes (Summer 2011)

1.     Hahn Banach Theorem: Extension Form

2.     Hahn Banach Theorem: Geometric Form

3.     Gateaux and Frechet Differential

4.     Constrained Optimization

5.     Constrained Optimization: Inequality