OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

"Course on Convex Optimization"
Michel Baes (K.U.Leuven) and Bharath Rangarajan (University of Minnesota, USA)

The difference between an efficiently tractable optimization problem and an extremely difficult one is sometimes not easy to see for a non-expert. However, this issue is of crucial importance for practitioners. If their problem is poorly modeled, the chance that the optimization software they use gives a reliable solution is close to zero.

Convex optimization problems consitute an important tractable class of instances. The purpose of this course is to give practitioners some tools to model some real-life problems as convex optimization problems, and to give them an insight on the algorithms used to solve them efficiently. The first part of the course will provide a review of basic convexity theory and of conic duality theory. Next, we will focus on some important classes of convex problems, including Second-Order Cone Programming, Semidefinite Programming, and Geometric Programming. Several application examples will be discussed and some modeling techniques will be described, such as the S-Lemma and Schur's Lemma.

All these problems can be solved very efficiently both in theory and in practice. We will review two important classes of algorithms, the Black-Box Methods and the Interior-Point Methods, and explain what makes them so efficient.

Finally, we will cover some newer topics, related to decomposition techniques of extremely large problems and using interior-point methods to detect infeasibility.


References
[1] Yurii Nesterov, Introductory lectures on Convex Optimization: A Basic course, Springer.
[2] Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press.
[3] Imre Polik and Tamas Terlaky, Survey of S-Lemma, SIAM Review.
[4] Aaron Ben-Tal and Arkadi Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM.

Schedule
Mon. (slides)
9h00-10h30:

• Convexity: definitions, examples.
  
• Optimization: examples and models for convex optimization.

11h00-12h30:

• Optimality conditions
  
• Easy and intractable problems.

Tue.
9h00-10h30: (slides)

• Duality theory
  
• Conjugate functions

11h00-12h30: (slides)

• Introduction to optimization methods.
  
• Black-box methods

Wed.
9h00-10h30: (slides)

• Interior-point methods for conic programming

11h00-12h30: (slides)

• Interior-point methods (cont.)
  
• Dealing with infeasibility

Thu.
9h00-10h30: (slides)

• Second-order, semidefinite representability
  
• S-Lemma and applications

11h00-12h30 (slides)

• Convex modeling: applications and demonstrations.

afternoon: (exercise)

• Computer session: using Sedumi

Fri. 13 (Special topics)
9h00-10h30: (slides)

• Smoothing techniques

11h00-12h30: (slides)

• Decomposition methods for very large problems

***** REGISTRATION *****
If you want to attend the course, please send an e-mail to Ida.Tassens@esat.kuleuven.be before July 3. Please indicate in your email wether you want to get an evaluation or not.

This course on convex optimization is funded by OPTEC and by ICCoS (Identification and Control of Complex Systems), a Scientific Research Network of the Research Foundation - Flanders (FWO-Vlaanderen).