ECE 580
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.
SPRING 2019 OFFERING
Instructor : Professor Tamer Başar
Office : 356 CSL (Phone: 3-3607)
Email : basar1@illinois.edu
Text : D. G. Luenberger, Optimization
by Vector Space Methods,
Wiley, 1997.
Meeting times : Tuesdays and Thursdays,
9:30 a.m. - 10:50 a.m. (possibly starting at 9:00 a.m. some weeks) in 4070 ECEB
COURSE OUTLINE
- An introduction to functional analytic approach to
optimization; Finite- versus infinite-dimensional spaces; Application
examples (1 hr)
- Normed linear spaces (3 hrs)
- Optimization of functionals -- General results on existence and
uniqueness of an optimum (1 hr)
- 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)
- Hilbert Spaces -- The Projection Theorem; Minimum distance to a
convex set (2 hrs)
- Examples of complete orthonormal sequences; Wavelets (2 hrs)
- Hilbert Spaces of random variables and stochastic processes;
Least-squares estimation (3 hrs)
- Dual Spaces. The Hahn-Banach Theorem, with applications to
minimum norm problems (5 hrs)
- Linear operators and adjoints (4 hrs)
- Calculus in Banach Spaces; Gateaux and Frechet derivatives.
Local theory of unconstrained optimization; Euler-Lagrange
equations (3 hrs)
- Global theory of unconstrained optimization; Fenchel
duality theory (2 hrs)
- Constrained optimization of functionals; Local and global
theory. Nonlinear programming and the Kuhn-Tucker Theorem in infinite
dimensions (4 hrs)
- Optimal control and Pontryagin's Minimum Principle (3 hrs)
- Differential Games (2 hrs)
- Numerical Methods (1 hr)
- Other related topics of interest, such as artificial neural networks,
infinite dimensional linear systems, H-infinity control for distributed parameter systems
(as time permits)
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