MATH V2500 (Fall 2009)

General Course Information:
MW 09:10am - 10:25am
Mathematics Hall 417
Instructor: H. Pinkham


A semester of single variable calculus, a one-semester course in Linear Algebra, and one semester of Multivariable Calculus, unless you have taken Honors Math A and B). The Columbia courses you need are Calculus I (V1101), Linear Algebra (V2010), Calculus III (V1201). Calculus II (V1102) is highly recommended but not required.

Course Objectives

The first purpose of this course is to teach those parts of real analysis, linear algebra and convexity theory that are most relevant to solving optimization problems: finding maxima and minima. Two concepts will come to the fore: positivity (the variables in economics are often constrained to be positive) and convexity and concavity of functions.

The second purpose is to cover the principal results of continuous optimization theory, using examples from Economics.

While this course will not be as proof based as, for example, Introduction to Modern Analysis (V4601), you will be asked to master the most important proofs as well as the statements of all the theorems. You will also be asked to work out concrete applications of the main results. This can only be done by attending all the classes, reviewing your class notes, reading the lecture notes and the book carefully and doing (that is, struggling with) the homework.

Material Covered

Real numbers; metric spaces and elements of general topology; numerical sequences and series; continuity and differentiation; the Riemann Stieltjes integral; uniform convergence.

Method of Evaluation

35% In-class midterm.
45% Final exam. Cumulative
20% Weekly homework. No late homework will be accepted. On some weeks (unannounced), I will ask you to work one of the homework problems at the beginning of class on the day it is due.

Required Text

Introduction to Optimization, by Pablo Pedregal. We will not be using it at the very beginning of the semester. This site refers to it simply as "Pedregal". Reading and homework will be assigned from it, so you need access to a copy. Copies are on reserve in the Mathematics Library. Pedregal does not cover the parts of analysis and linear algebra we will be covering during the first few weeks of the class. Class notes will be distributed instead.


Outline of the class meetings:

Section I: The Geometry and Topology of Rn

1. Sep-9-09 Introduction to the problem
2. Sep-14-09 The Geometry of Rn
3. Sep-16-09 Open, closed and compact sets

4. Sep-21-09 Sequences and the Maximum Theorem
5. Sep-23-09 Taylor's Theorem in One and Several Variables

Section II: Linear Algebra

6. Sep-28-09 Determinants and Permutations
7. Sep-30-09 The Diagonalization of Symmetric Matrices
8. Oct-5-09 The Spectral Theorem
9. Oct-7-09 Unconstrained Optimization

Section III: Convexity

10. Oct-12-09 Convex Sets
11. Oct-14-09 Convex Functions
12. Oct-19-09 Unconstrained Minimization of Convex Functions
13. Oct-21-09 Midterm on the material of sections I, II, III.

Section IV: Linear Optimization

14. Oct-26-09 Cones and the Farkas Alternative
15. Oct-28-09 Linear Optimization
16. Nov-4-09 Duality
17. Nov-9-09 The Simplex Method

Section V: Nonlinear Optimization: Lagrange Multipliers

18. Nov-11-09 Equality Constrained Optimization: the Lagrangian
19. Nov-16-09 Second Order Tests for the Lagrangian
20. Nov-18-09 The Bordered Hessian
21. Nov-23-09 Quadratic Optimization
22. Nov-25-09 Convex Optimization
23. Nov-30-09 Inequality Constraints: Kuhn Tucker

Section VI: Numerical Methods

24. Dec-2-09 Iterative Techniques
25. Dec-7-09 Conjugate Direction Methods
26. Dec-9-09 Applications
27. Dec-14-09 Last day of class. Review.