MTH 630 Functional Analysis - Optimization

Prof. Orlando Merino   merino@math.uri.edu         874 4442

MTH630 Functional Analysis   is designed for students in mathematics, natural sciences, computer science, engineering, and other fields.  It is a good course for graduate students of engineering and basic sciences who would like to strengthen mathematical skills and knowledge.
The main topic of MTH629 is continuous optimization. Some specific methods we will discuss include Steepest Descent, Newton and Quasi-Newton methods, Trust-Region methods.  There is some flexibility in the choice of topics, and I plan to include certain topics depending on interest of the registered students.

The requirements of the course are familiarity with linear algebra, advanced calculus, mathematical proofs, and programming in a computer language (matlab, mathematica, maple, C, fortran, etc.)  Evaluation is based on homework (which will include programming problems), and a project which is to be presented at the end of the semester.

The main text is Numerical Optimization, by J. Nocedal and S. Wright, (Springer Verlag 1999). In addition to the main text, I will add supplementary material.  The presentation of material is elementary. It begins with vector spaces, matrices, and norms, and the theory is gradually developed, and then it is followed by applications to optimization.

TOPICS of MTH 630

Functional Analysis Concepts
Review of basic concepts. Bases, subspaces, norms. Inner Product Spaces. Linear Functionals. Convexity.  Duality.  Hahn-BanachTheorems.  Distance problems. Nonlinear Operators. Frechet derivatives.
Optimization
Line search methods.  Trust-Region methods.  Conjugate Gradient methods.Quasi Newton Methods.  Nonlinear Least Squares Problems.  Theory of Constrained Optimization. Linear Programming. Interior Point Methods.  Semidefinite Programming.

The topics listed here may be more than we can cover in one semester; some of the topics may be left out.

References

Numerical Optimization, by J. Nocedal and S. Wright.  Springer Verlag, 1999.  This is the main text.

Other useful references are:

Practical Optimization, by P. Gill, W. Murray, and M. Wright.  Academic Press, 1981.
Numerical Methods for Unconstrained Optimization and Nonlinear Equations, by J. Dennis and R. Schnabel. SIAM 1983.
Iterative Methods for Optimization, by C. T. Kelley. SIAM, 1999.
Practical Methods of Optimization, by  R. Fletcher.  Wiley, 1997.
Numerical Linear Algebra, by L.N. Threfethen and D. Bau.  SIAM, 1997.