20603 - OPTIMIZATION

To feel comfortable in this course you should have attended at least two calculus classes covering the basic concepts of sequences, functions, derivatives, multivariable functions, partial derivatives, finite dimensional optimization, Lagrange multipliers, integrals as well as having a working knowledge of linear algebra (vectors/matrices/eigenvalues/systems).

Mathematics is the language in which most of modern sciences and economics is written. The course aims to provide basic and sophisticated mathematical tools that students need in order to tackle data science challenges and advanced economics studies. The course develops the mathematical point of view of optimization, aiming to form the modeling and thinking skills that students will need later on, during both their academic and professional careers.

• Basics on differential equations, separation of variables, linear equations, linear systems. Quick overview of some nonlinear equations.

• Continuity, convexity, compactness. Fréchet-derivatives. Fixed points, contractions.

• Classical problems in calculus of variations, critical points. Maxima and minima, necessary/sufficient conditions. Convexity.

• Control theory, bang-bang principle. Hamiltonians, the Pontryagin maximum principle.

• Dynamic programming. The Hamilton-Jacobi-Bellman equation.

• Deterministic and stochastic variational approximations in machine learning.

• Carry out a formal mathematical proof
• Master infinite-dimensional vector spaces techniques.
• Model optimization problems from calculus of variations and implementation in the machine learning context.
• Model optimal control problems.
• Model dynamic optimization problems.

• Solve infinite-dimensional optimization problems.
• Apply to data science and to machine learning the techniques of mathematical optimization.
• Work out both the quantitative and the qualitative perspectives.
• Solve optimal control problems.
• Solve dynamic optimization problems.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

Every one/two weeks there is a problem session where mathematical problems concerning the topics taught in class are discussed and solved.

Written exam.

Textbook (with exercises): P. Cannarsa, F. Gazzola, Dynamic optimization for beginners - with prerequisites and applications, EMS, 2021

Textbook (mainly Chapter 10): C.M. Bishop, Pattern Recognition and Machine Learning. Springer, 2006.