The course aims to provide students with the necessary tools to deal, from a quantitative point of view, with the main problems that arise in the economic and financial framework. After an introduction on some advanced notions of Linear Algebra and Calculus for functions of several variables, unconstrained and constrained optimization problems are introduced and their applicability in the economic-financial field is presented. The solution of optimization problems will be treated with the classical results deriving from the first and second order optimality conditions and from the properties of the Lagrangian function.
1. Linear algebra: eigenvalues and eigenvectors of a matrix, quadratic forms, sign of a quadratic form, definite and semidefinite matrices.
2. Differential calculus: functions of several variables, level sets, partial derivatives for functions of several variables, convex functions.
3. Unconstrained optimization: first order optimality conditions, second order optimality conditions.
4. Constrained optimization: the Weierstrass theorem. Optimality problems with equality constraints, Lagrange's theorem. Lagrangian function and optimality conditions. Optimal problems with inequality constraints, the Kuhn-Tucker theorem. Convex problems.
The exam is written. An intermediate test will be offered during the course; at the end of the course a final test will be scheduled. The final grade is the mean of the two partial grades.
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