ECE 580 / Math 587

Optimization by Vector Space Methods

This is an introductory course in functional analysis and infinite dimensional optimization, with applications in least-squares estimation, nonlinear programming in Banach spaces, optimal and robust control of lumped and distributed parameter systems, and differential games.

The course was last offered in Fall 2006, and will next be offered Spring 2008. The following is the course outline based on the Fall 2006 offering.

For further information contact Professor Tamer Başar

FALL 2006 OFFERING

Instructor : Professor Tamer Başar

Office : 356 CSL (Phone: 3-3607)

Email : tbasar@control.csl.uiuc.edu

Text : D. G. Luenberger, Optimization by Vector Space Methods, Wiley, 1997.

Meeting times : Mondays and Wednesdays, 11:00 a.m. - 12:25 p.m. in 260 MEB

COURSE OUTLINE

  1. An introduction to functional analytic approach to optimization; Finite- versus infinite-dimensional spaces; Application examples (1 hr)
  2. Normed linear spaces (3 hrs)
  3. Optimization of functionals -- General results on existence and uniqueness of an optimum (1 hr)
  4. Fixed points of transformations on Banach Spaces -- Applications to solutions of differential (ordinary and partial) and integral equations; Minimax and Nash equilibrium theorems of game theory (5 hrs)
  5. Hilbert Spaces -- The Projection Theorem; Minimum distance to a convex set (2 hrs)
  6. Examples of complete orthonormal sequences; Wavelets (2 hrs)
  7. Hilbert Spaces of random variables and stochastic processes; Least-squares estimation (3 hrs)
  8. Dual Spaces. The Hahn-Banach Theorem, with applications to minimum norm problems (5 hrs)
  9. Linear operators and adjoints (4 hrs)
  10. Calculus in Banach Spaces; Gateaux and Frechet derivatives. Local theory of unconstrained optimization; Euler-Lagrange equations (3 hrs)
  11. Global theory of unconstrained optimization; Fenchel duality theory (2 hrs)
  12. Constrained optimization of functionals; Local and global theory. Nonlinear programming and the Kuhn-Tucker Theorem in infinite dimensions (4 hrs)
  13. Optimal control and Pontryagin's Minimum Principle (3 hrs)
  14. Differential Games (2 hrs)
  15. Numerical Methods (1 hr)
  16. Other related topics of interest, such as artificial neural networks, infinite dimensional linear systems, H-infinity control for distributed parameter systems (as time permits)