Convex Analysis and Optimization 0365-4409
Time and Place: Winter Semester 2019/2020 - Sunday 16-19 pm, Bldg Schreiber-- Room: 209
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 .
- About 8 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.
Material and Some Useful References
- There is no required textbook for the course. The lectures will be comprehensive
and cover all the necessary material.
- All the lecture notes will be distributed to the students (more infos on this will be given in class).
- However, the students are strongly encouraged to consult
the following references for further reading and study.
- R. T. Rockafellar, Convex Analysis, Princeton University Press,
- O. L. Mangasarian, Nonlinear programming, McGraw-Hill Publishing
- 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.
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.
Optimization: Constrained optimization, The KKT Theorem,
Second Order Optimality Conditions in Nonlinear Programming. Existence theorems.
Duality Theory: Lagrangian duality, Conjugate Functions,
Fenchel duality, Min-Max Theorems, Applications.