Students are expected to have a solid background in - Linear Algebra Prior knowledge on optimization is not required.

Optimization theory is nowadays a well-developed area, both in the theoretical and practical aspects. This graduate course introduces the basic concepts for solving optimization problems and illustrates this theory with many recent applications in signal processing and machine learning. Students attending this course will: - Develop a solid theoretical basis for solving convex optimization problems arising in industry and research. - Learn manipulations to unveil the hidden convexity of optimization problems and relaxation techniques to treat non-convex optimization problems. - Be able to characterize the solution of convex and non-convex optimization problems either analytically or algorithmically. - Learn the usage of some of the more popular optimization toolboxes.

Unit 1. Introduction - Optimization problems and constraints - On closed-form optimization: analytical versus algorithmic solutions - Types of optimization problems - Modelling and applied linear algebra Unit 2. Convex Optimization - Convex sets and convex functions - Convex optimization problems - Disciplined convex programming, CVX - Quadratic optimization - Lagrange duality and KKT conditions Unit 3. Optimization Algorithms - Local optimization algorithms - Stochastic optimization - Global optimization - Integer programming and metaheuristics Unit 4. Applications - Optimization for machine learning

- Theoretical sessions: theoretical basis of optimization theory, illustrated with different applications and examples. Material for out-of-class work. - Problem sessions: formulation and solution of exercises motivated by different problems from communications, signal processing and machine learning. - Practical sessions: popular toolboxes for convex and non-convex optimization. The proposed projects will be solved in Matlab and/or Python programming environments.