[Course description]
Mathematical optimization has pervasive applications in not only almost all natural science and engineering disciplines such as electrical engineering (especially in control, communication, signal processing, and electronic design automation), computer science, and operations research, just to name a few, but also social sciences such as economics. Among various types of optimization problems, convex optimization problems are special in that they can be solved very efficiently, and thus they have gained much attention. Moreover, many optimization problems fall into this broad category. For instances, linear least squares, linear programming, quadratic programming, geometric programming, and semidefinite programming are special cases of convex optimization.
The importance of convex optimization is becoming more and more apparent as there are many more emerging scientific and engineering problems being solved efficiently in this manner. This course primarily focuses on developing students’ capability of recognizing and formulating convex optimization problems arising from their own research fields, and when time permits, it also introduces how such problems are solved and applied. Students are also encouraged to develop heuristic approaches based on convex optimization theory and techniques to tackle general nonlinear, nonconvex optimization problems encountered in their fields of interest.
[Course topics]
1. Introduction to convex optimization
2. Convex sets
3. Convex functions
3. Convex optimization problems
4. Duality theory (optimality conditions; sensitivity analysis)
5. Algorithms* (descent methods; Newton's method; interior-point methods)
6. Applications* (approximation and fitting, statistical estimation, etc.)