Advanced Topics in Convex Optimization

The course provides an in-depth treatment of classical and modern concepts in convex optimization that are relevant in control, decision making and machine learning problems. The course articulates around the following four topics:

• Fundamentals of convex analysis
• Operator-splitting methods
• Distributed optimization
• Online convex optimization

After an introductory part covering classic and foundational concepts in convex optimization (convex sets and functions; Lagrangian and Fenchel duality; gradient and coordinate descent methods), we will focus on three state-of-the-art topics in convex optimization.

Operator-splitting methods are first-order methods based on monotone operator theory that are particularly suitable to handle non-smooth problems (often arising in control and learning applications). Distributed optimization allows large-scale problems (appearing e.g. in learning-from-big-data and distributed control settings) to be solved by means of local computations and is a central paradigm for the development of network infrastructures (e.g. smart cities, swarm robotics). Online convex optimization is a paradigm for sequential decision making problems where an agent needs to take decisions by solving a series of optimization problems online, thus requiring real-time capable computations and means to take action in the face of uncertainty.

The emphasis of the course is on methodological aspects such as: design principles behind the algorithms; properties of the methods and mathematical tools required to prove them; understanding of the most important features of state-of-the-art algorithms used in applications.