Optimization
Number of Credits: 7 (in Bielefeld) and 8 (in Paris) ECTS Credits
Hours: 4 hours per week, about 60 hours total including tests
General Presentation:
This course introduces the student to the mathematical foundations of (i) convergence : continuity of functions and the underlying topological structure in metric spaces and (ii) convexity and optimization and their use in economic models. It covers
A. Convergence in metric spaces
- Metric spaces, distance, norm on a vector space, open and closed sets, sequences in a metric space, continuity, uniform continuity.
Compact sets in a metric space.
Complete spaces, Contractions
Finite dimensional vector spaces.
Complement to calculus :Frechet differentiability
and the Implicit function theorem
B. Convexity and optimization
B.1. Convexity of sets and functions.
Convex sets. Examples : budget sets, balls, production sets.
Convex and concave functions, graph, epigraph and hypograph.
Quasiconvex and quasiconcave functions.
Strictly convex and quasi convex functions.
Characterization of a convex funtion with its first order derivative
Characterization of a convex funtion with its second order derivative.
Topological properties of convex sets.
Projection on a closed convex set.
Separation theorems.
Orthogonality and polarity.
The bipolar theorem. Farkas lemma.
B.2. Optimization under constraints
B.2.1. Unconstrained optimization.
Global and local maximum (minimum).
First order necessary conditions.
Second order necessary condition and second order sufficient condition.
Global maxima for concave (convex) functions. Examples.
B.2.2. Constrained optimization.
Convexity conditions and Slater condition.
The Kuhn-Tucker problem in convex programming (statement without proof)
Applications of Kuhn-Tucker Theorem in consumer theory and producer theory
More examples of Applications of Kuhn-Tucker Theorem
Linear programming
Quadratic programming
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 .
Teacher and syllabus: For specific information for this year, please click
here for Bielefeld and
here for Optimization A: Real Analysis -Paris or
here for Optimization B: Convexity & Optimization -Paris