Ph D course in nonlinear optimization

Key words:

Nonlinear optimization with or without constraints, optimality conditions, iterative algorithms, convergence analysis, duality, numerical methods

Who should take this course?

The course is directed towards mathematially inclined fourth-year students and Ph D students with a sufficient knowledge in analysis, linear algebra and continuous optimization, and who are interested in a deeper understanding of the theory of nonlinear optimization and classical as well as more modern methodologies.

Time:

Fall term 1999

Credits:

Six

Course start:

Wednesday 8 September at 10.00 in MD9

Schedule:

Wednesdays at 10.00-11.45 in ?.

The first two weeks the schedule is however as follows:
15 September at 15.15-17.00 in S1
22 September at 15.15-17.00 in S2

Teachers and examiners:

Axel Ruhe (numerical analysis), tel: 772 10 96, e-mail: ruhe@math.chalmers.se

Michael Patriksson (optimization), tel: 772 35 29, e-mail: mipat@math.chalmers.se

Literature:

Dimitri P. Bertsekas: Nonlinear Programming
Athena Scientific, Belmont, MA, 1995
ISBN 1-886529-14-0

The book is available for $79 at Amazon online (www.amazon.com).

The book is pedagogical and is excellent for self study.

Some articles will be handed out for use especially in the projects.

Course information:

Teaching is mainly used as a guide for the student's own reading, but some technical parts of the material will also be discussed.

At certain gatherings the exercises and projects will be discussed.

The exercises, which are found in the book, are of a varying character, containing repetition type questions (to be done by all), numerical calculations with Matlab (which can be split between the participants), and theoretical questions often specializing or extending results in the course material (and which also are suitable to split between participants).

In addition to this material there is also a project to be done, the purpose of which is to illustrate and analyze an algorithm for a practically motivated nonlinear optimization problem. The task should mainly be performed using available software, such as Matlab. A project can either be selected by the participants based on own research, or among some suggested topics. Each project is normally done by two persons.

Examination:

Passed on all exercises and projects.

Rough content list:

  1. Chapter 1 (Unconstrained Optimization) (Axel Ruhe): Optimality conditions, gradient methods, least squares problems, convergence analysis (September).

    Reading list, exercises and projects are here.

  2. Chapter 2 (Optimization Over a Convex Set) (Michael Patriksson): Optimality conditions, gradient projection methods, simplicial decomposition, decomposition of optimization problems defined over Cartesian product feasible sets, convergence analysis (October).

    Reading list, exercises and projects are here.

  3. Chapter 3 (Lagrange Multiplier Theory) (Michael Patriksson): Necessary and sufficient optimality conditions, Farkas' Lemma, sensitivity analysis (October)

    Reading list, exercises and projects are here.

  4. Chapter 4 (Lagrange Multiplier Algorithms) (Axel Ruhe): Barrier and other penalty methods, sequential quadratic programming (SQP), interior point methods, convergence analysis (November)

    Reading list, exercises and projects are here.

  5. Chapter 5 (Duality and Convex Programming) (Michael Patriksson): Duality theory, weak and strong duality, duality gaps (November)

    Reading list, exercises and projects are here.

  6. Kapitel 6 (Dual Methods) (Ann-Brith Strömberg): Non-differentiable optimization, subgradient methods, dual ascent, ergodic methods, decomposition/coordination methods (December)

    Reading list, exercises and projects are here.