**Number of Credits:** 7 credits

**Hours:** 48 hours of Lectures and 36 hours of Tutorials including exam

**General Presentation:** This course introduces to Optimization in finite dimensional spaces (first part), and infinite dimensional spaces (second part). This is motivated by models in Economics, finance, Macroeconomics, Statistics, … where these tools are very important to study existence of Optimization problems, uniqueness, properties of the value, method to compute the solution, ...

**Part I: Optimization in Euclidean spaces.**

Presentation of Optimization, examples (micro, macro, statistics); vocabulary.

Convexity (definition of convexity, convex combination, examples (affine subset, simplex, convex cones). Convex function.

Topology in Euclidean spaces: norm, distance, continuity, close subset, compacity in Euclidean sets, open subset, boundary, interior points. Notion of level curves.

Unconstrained optimization problem. One-dimensional case. Basic differential calculus in R. First-order necessary condition, second-order sufficient condition, convex or concave optimization.

The Euclidean case: existence results (if compactness: Weierstrass; otherwise, coercivity). Applications.

Differentiable unconstrained optimization problem in Euclidean space: Reminders about differentiability (differential, first order development, Ck functions, Hessian, semi-definite matrices, second order development, ...). Existence results, first order necessary condition, second-order sufficient condition. Concave or Convex optimization. Examples.

Constrained optimization with equality constraints. Lagrange multipliers. first order necessary condition. Sufficient condition (convexity, concavity). Geometric interpretation.Constrained optimization problem: case of one several inequality or equality constraints. Karush Kuhn, Tucker. Qualification constraints (rank condition, affine case, …). First order necessary condition (KKT), second order necessary condition. Complementary slackness condition. Geometric interpretation.

**Part II: Dynamical optimization.**

Reminders about infinite dimensional normed spaces. Examples of norms in sequence spaces, functions spaces (lp, infinite norms). equivalence of norms in finite dimensional spaces. Consequence on the topology for a norm in finite dimensional space.Non compactness of a ball in infinite dimensional spaces (Riesz). Continuity of linear function in normed space, and link with lipschitz function.

Metric spaces. Compactness: definition with covering, definition with subsequence. Equivalence of these definitions. First properties (close subset of a compact is compact). Examples of sequence spaces. Basic example of a sequence space which is not compact and of another one which is compact. product metric on a infinite product of metric spaces, which induces product topology. Compactness of an infinite product of compact for this metric. Convergence for this metric (basically, convergence component by components). 3) Complete spaces (Cauchy sequences, link with convergence, examples for sequence spaces, function spaces,...). Banach fixed-point theorem. Blackwell fixed-point theorem.

Dynamic programming with a Finite horizon. Framework. decision variables. time consistency. Backward induction. Examples.

Dynamic programming with an Infinite horizon. Bellman equation. Notion of correspondence. General discounted optimization problem under some feasibility conditions (with correspondences). Existence of a solution, using the previous tools (Blackwell). Properties of the value function. Some applications, for example to growth models.

**Books: **Simon, C., Blume, L., *Mathematics for Economists,* (1994) Norton.De La Fuente, A., *Mathematical Methods and Models for Economists,* 2nd Ed. (2005) Cambridge University Press.

**Prerequisites:** Logic and Sets and Multivariable Calculus