Advanced Topics in Convex Optimization

The course provides an in-depth treatment of classical and modern concepts in convex optimization (with emphasis on the latter) that are relevant in control, decision making and data science problems. The course articulates around the following four topics:

• Basics 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 course is not aimed at teaching the use of specific optimization toolbox or software. Conversely, emphasis will be placed 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.