The aim of the course is to teach students the main techniques to approach multivariable optimization problems, both constrained and unconstrained, and to make students able to solve systems of differential equations and optimal control problems. The theoretical part of each module of the course will be enriched by a numerical part, where the goal is to get students acquainted with the main ideas and methodologies of numerical solutions with Matlab and Julia.
Risultati apprendimento attesi
At the end of the course, students will be expected to possess and be able to use the main techniques for solving multivariable optimization problems, both constrained and unconstrained. Moreover, they will have the necessary backgrounds to categorize and, whenever possible, to solve analytically systems of differential equations and optimal control problems. Everywhere the analytical solution is out of reach, students will be equipped with the necessary numerical tools available in Matlab and Julia.
Static Optimization Topics in linear algebra. Review of basic linear algebra. Linear independence. The rank of a matrix. Main results on linear systems. Eigenvalues. Diagonalization. Quadratic forms. Quadratic forms with linear constraints. Linear programming. A simple maximization problem. Graphical solution procedure. Extreme points and the optimal solution. Special cases. General linear programming notation. Sensitivity analysis and interpretation of solution. Multivariable calculus. Gradients and directional derivatives. Convex sets. Concave and convex functions. Quasiconcave and quasiconvex functions. Taylor's formula. Implicit and inverse function theorems. Degrees of freedom and functional dependence. Differentiability. Existence and uniqueness of solutions of systems of equations. Static optimization. Extreme points. Local extreme points. Equality constraints: the Lagrange problem. Local second-order conditions. Inequality constraints: nonlinear programming. Sufficient conditions. Comparative statics. Nonnegativity constraints. Concave programming. Precise comparative statics results. Existence of Lagrange multipliers. Dynamical Systems Differential equations I: First-order equations in one variable. Introduction. The direction is given, find the path. Separable equations. First-order linear equations. Exact equations and integrating factors. Transformation of variables. Qualitative theory and stability. Existence and uniqueness. Differential equations II: Second-order equations and systems in the plane. Introduction. Linear differential equations. Constant coefficients. Stability for linear equations. Simultaneous equations in the plane. Equilibrium points for linear systems. Phase plane analysis. Stability for nonlinear systems. Saddle points. Control theory: basic techniques. The basic problem. A simple case. Regularity conditions. The standard problem. The maximum principle. Sufficient conditions.
Lecture, tutorial, and lab.
Materiale di riferimento
Static Optimization Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom, Further Mathematics for Economic Analysis, Financial Times Prentice Hall, 2008 (chapters 1,2,3). Dynamical Systems Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom, Further Mathematics for Economic Analysis, Financial Times Prentice Hall, 2008 (chapters 5,6,9).
Modalità di verifica dell’apprendimento e criteri di valutazione