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Optimization Optimization Per i Corsi di laurea: - EPS n.o.; totale cfu 6 Periodo di erogazione 1° trimestre Prof. LA TORRE DAVIDE , DIPARTIMENTO DI ECONOMIA, MANAGEMENT E METODI QUANTITATIVI Indirizzo: 02503 21462 - v. Conservatorio, 7 Mail: davide.latorre@unimi.it L'Insegnamento sviluppa crediti sui seguenti settori scientifico disciplinari 6 cfu SECS-S/06 (6 cfu) Obiettivi Students will learn advanced optimization methods and how to use them to model and solve problems in economics and social sciences. Short Course Description This course introduces the principal methods and algorithms for linear, nonlinear, and multi-objective optimization. Emphasis is on methodology and the underlying mathematical structures. Topics include linear programming, convex optimization, optimality conditions for nonlinear optimization, interior point methods, Newton's method, duality theory, Lagrange multiplier theory, multiobjective decision making, goal programming. Programma 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. Partitioned matrices and their inverses. 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. Multicriteria decisions. Goal programming: formulation and graphical solution. Goal programming: solving more complex problems. Scoring models. Analytic hierarchy process AHP. Establishing priorities using AHP. Using AHP to develop an overall priority ranking. Numerical examples and application to marketing, finance, and operations management with LINGO. Materiale di riferimento Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom, Further Mathematics for Economic Analysis, Financial Times Prentice Hall, 2008 (chapters 1,2,3). Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, Kipp Martin, An Introduction to Management Science: Quantitative Approaches to Decision Making, Cengage Learning, 2010 (chapters 2,3,4,5,8,14). Prerequisiti e modalità d'esame , con riferimento ai descrittori 1 e 2 Closed book exam Propedeuticità consigliate Calculus I. Metodi Didattici Lecture, tutorial, and lab. Lingua di insegnamento English.