# Convex Optimization and Applications

# Description

This course is about convex optimization, a class of mathematical optimization with numerous applications in science and engineering problems. The main advantage of formulating a problem as a convex optimization is that it can be solved numerically efficient and tractable. This course has been designed for graduate students of electrical engineering. After introducing different forms of convex optimization such as linear programming and least-squares, the focus will be on Linear Matrix Inequalities (LMI) and its applications in electrical engineering.

# Course content

- Introduction
- Mathematical optimization
- Least-squares and linear programming
- Convex optimization
- Nonlinear optimization

- Convex sets
- Affine and convex sets
- Operations that preserve convexity

- Convex functions
- Basic properties and examples
- Operations that preserve convexity
- The conjugate function
- Quasiconvex functions

- Convex optimization problems
- Linear optimization problems
- Quadratic optimization problems
- Geometric programming
- Vector optimization

- Linear Matrix Inequalities (LMI)
- What are LMI’s and what are they good for?
- Stability: linear time-invariant, time-varying or non-linear systems

- Performance
- Dissipativity
- Quadratic performance and specializations (H_{1}, passivity)
- H_{2} performance and generalizations

- Synthesis
- State-feedback and estimation problems
- Output feedback synthesis

- Multi-objective Control
- Youla parametrization and genuine multi-objective controller synthesis
- Robust controller design

- Parameter Robust Stability
- Robust stability against time-invariant and time-varying uncertainties
- Parameter dependent Lyapunov functions
- Semi-infinite LMI problems and relaxations

- Robust Optimization and Lagrange Duality
- Introduction to robust optimization and robust LMI problems
- Lagrange duality
- How to construct tractable relaxations

- Dynamic Robustness
- Linear fractional representations
- Robust stability tests with multipliers
- Relations to the structured singular value

- LPV synthesis
- Linear parametrically-varying controller synthesis
- Direct approach
- Multiplier approach

- Polynomial optimization
- Sum of Square (SOS) optimization

# References

- S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, Cambridge, 2004.
- C. Scherer and S. Weiland, Linear Matrix Inequalities in Control, Lecture Notes, 2005.
- S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Studies in Applied Mathematics), SIAM, Philadelphia, 1994.

# Evaluation

- Project 30%
- Midterm 30%
- Final 40%

# Projects

- Convex Optimization in Model Predictive Control, Yousof Koohmaskan
- Hybrid Control Methods for Boost Converters, Hamidreza Toodeji
- Power/Ground Network Optimization, Mehdi Hajizadeh
- Optimization of the Height of Power Transmission Towers, Mohammad Hassan Zamani
- Ellipsoid Approximation for Polyhedral Sets, Applications in Power Systems Optimization, Naghi Moaddabi
- LMI Approach to Stability of 2D positive Systems, Mehdi Zeinali
- Static Output Feedback Control for Linear Discriptor Systems: LMI Approach, Mehdi Zeinali