Convex Analysis and Optimization 0365-4409
Time and Place:
Winter Semester 2013/2014 - Sunday 16-19 pm, Schreiber Bldg -- Room: 8
Intended Audience and Prerequisites
This is a graduate core course for the M.Sc. in Operations Research.
Graduate Students in Mathematics, Statistics,
Computer Sciences, and Engineering are strongly
encouraged to register. Previous exposure to optimization courses is not
necessary. The only prerequisite is a good working
knowledge of Linear Algebra and Advanced Calculus. However, these
elementary tools and the much advanced ones which will be covered in the course
will require some mathematical maturity on the part of the student
to a better appreciation of the subject.
The course will provide the mathematical foundations of optimization
theory which relies essentially on Convex Analysis .
Course requirements/Exams
About 10 Homeworks will be assigned during the semester. The homework
assignements are mandatory .
Solutions to the most
interesting/difficult problems will be discussed in class.
Late homework will not be accepted.
The final grade will be mainly determined by the final exam.
However, the submitted homeworks (remember mandatory...)
will also be used to improve grades,-- when and if-- necessary.
Some Useful References
There is no required textbook for the course. The lectures will be comprehensive
and cover the necessary material. Handouts will also be distributed.
However, the students are strongly encouraged to consult
the following references for further reading and study:
- R. T. Rockafellar, Convex Analysis, Princeton University Press,
1970.
- O. L. Mangasarian, Nonlinear programming, McGraw-Hill Publishing
Company, 1969.
- D. P. Bertsekas, Convex analysis and Optimization
, Athenas Scientific, 2003.
- S. Boyd and L. Vandenberghe, Convex Optimization Cambridge University Press, 2004.
- A. W. Roberts and D. E. Varberg, Convex Functions, Academic
Press, New-York and London, 1973.
Approximate Syllabus
Convex sets:
Functional and topologial properties, representation
theorems (Carhateodory, Helly, Radon), Asymptotic cones, Separation
and supporting theorems, Alternative theorems.
Convex functions: Characterization, Functional operations with convex
functions, differentiablity properties, Asymptotic functions,
Existence theorems.
Optimization: Constrained optimization, The KKT Theorem,
Second Order Optimality Conditions in Nonlinear Programming.
Duality Theory: Lagrangian duality, Conjugate Functions,
Fenchel duality, Min-Max Theorems, Applications.