20603 - OPTIMIZATION

To feel comfortable in this course you should have attended at least two calculus classes covering the basic concepts of function, derivative, 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 economics is written. The course aims to provide the basic mathematical tools that students need to complete their Economics studies. Moreover, the course develops the analytical thinking skills that students need later on in their academic career.

• Basics on differential equations, separation of variables, linear equations, linear systems. Partial differentiation, free and constrained optimization, gradient methods.

• Vector spaces, Banach spaces, Hilbert spaces. Separable spaces, Fourier series: $ell^2$ and $L^2_T$.

• 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 vector spaces techniques.
• Solve optimization problems from calculus of variations.
• Set up and solve control theory problems.
• Solve dynamic optimization problems.

• Apply to economics and to the social sciences the techniques of contemporary mathematics.
• Work out both the quantitative and the qualitative perspectives.
• Solve dynamic optimization problems which are key in Economic Theory.
• Master topological arguments which are important in Game Theory and Microeconomics.

• 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.

Lecture notes.