20603 - OPTIMIZATION
Course taught in English
Go to class group/s: 31
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.
Vector spaces, Banach spaces, Hilbert spaces. Separable spaces: $ell^2$ and $L^2_T$. Operators: norms and fixed points.
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.
- Carry out a formal mathematical proof.
- Recognize the abstract mathematical structures that underline modern theories.
- Master infinite-dimensional vector spaces techniques.
- Model optimization problems from calculus of variations.
- Model optimal control problems.
- Model dynamic optimization problems.
- Apply to data science, to social sciences, and to economics the techniques of mathematical optimization.
- Work out both the quantitative and the qualitative perspectives.
- Solve infinite-dimensional optimization problems.
- Solve optimal control problems.
- Solve dynamic optimization problems.
- Face-to-face lectures
- Exercises (exercises, database, software etc.)
- Group assignments
Every one/two weeks there is a problem session where mathematical problems concerning the topics taught in class are discussed and solved.
|Continuous assessment||Partial exams||General exam|
Written exam. Depending on the covid restrictions, group assignments might also be evaluated.
Lecture notes. A textbook is in preparation but it may not be available at the beginning of the course. In this case, notes will be available.