Dept of Electrical Engineering & Computer Sciences

Convex optimization and applications

**Volume:** 3 units

**Lectures:** 60 Evans Hall, M/W/F 1:00-2:00.

**Instructor:**Laurent
El Ghaoui

**Office Hours: **Wed 2:00-4:00.

**Grading: **Homework 25%, Midterm 30%, Project 45%.

Convex optimization relates to a class of nonlinear optimization problems where the objective to be minimized, and the constraints, are both convex. Contrarily to the more classical linear programming framework, convex programs often go unrecognized, and this is a pity since a large class of convex optimization problems can now be efficiently solved. In addition, it is possible to address hard, non convex problems (such as "combinatorial optimization" problems) using convex approximations that are more efficient than classical linear ones. Convex optimization is especially relevant when the data of the problem at hand is uncertain, and "robust" solutions are sought.

The 3-unit course covers some convex optimization theory and algorithms, and describes various applications, with a special emphasis on problems with incomplete/unknown data. A large number of examples arising in a variety of fields will be given, covering analysis, design and control of complex systems, and in various identification, data fitting and estimation problems.

**Intended audience:** Students interested in engineering problems
where decision-making (system analysis and design, engineering trade-offs,
risk analysis, etc) has to do with scientific computations. Related departements/fields
include: Bioengineering, Civil and Environmental Engineering (structure
optimization and control) Electrical Engineering (signal processing, control
and communications, robotics, CAD) Computer Science (computer graphics,
artificial intelligence and decision theory, data mining, algorithms &
complexity, computational geometry), Industrial Engineering and Operations
Research, Mechanical Engineering, Scientific Computing and Computational
Mathematics, Statistics, Finance, Economics.

**Required background:** Basic linear algebra such as matrices, eigenvectors,
symmetric matrices, positive-definite matrices. A prior exposure to optimization,
such as an introductory course on linear programming, helps, but is not
necessary.

**Tentative syllabus: (**Each item is approximately a week, i.e.
3 lectures)

- introduction
- convex sets and functions
- convex optimization problems
- duality
- unconstrained minimization
- interior-point methods
- localization methods
- some applications in engineering
- problems with uncertainty
- uncertainty models
- robust optimization
- applications to data fitting and estimation
- applications to control
- conclusions