MATH V2500 (Fall 2009)
ANALYSIS AND OPTIMIZATION
General Course Information:
MW 09:10am  10:25am
Mathematics Hall 417
Instructor: H. Pinkham
Prerequisites
A semester of single variable calculus, a onesemester 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% Inclass 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.
Syllabus
Outline of the class meetings:
Section I: The Geometry and Topology of Rn


1. Sep909  Introduction to the problem 

2. Sep1409  The Geometry of Rn 

3. Sep1609  Open, closed and compact sets 

4. Sep2109  Sequences and the Maximum Theorem 

5. Sep2309  Taylor's Theorem in One and Several Variables 

Section II: Linear Algebra


6. Sep2809  Determinants and Permutations 

7. Sep3009  The Diagonalization of Symmetric Matrices 

8. Oct509  The Spectral Theorem 

9. Oct709  Unconstrained Optimization  
Section III: Convexity 

10. Oct1209  Convex Sets 

11. Oct1409  Convex Functions 

12. Oct1909  Unconstrained Minimization of Convex Functions 

13. Oct2109  Midterm on the material of sections I, II, III.  
Section IV: Linear Optimization


14. Oct2609  Cones and the Farkas Alternative 

15. Oct2809  Linear Optimization 

16. Nov409  Duality  
17. Nov909  The Simplex Method 

Section V: Nonlinear Optimization: Lagrange Multipliers


18. Nov1109  Equality Constrained Optimization: the Lagrangian 

19. Nov1609  Second Order Tests for the Lagrangian 

20. Nov1809  The Bordered Hessian 

21. Nov2309  Quadratic Optimization 

22. Nov2509  Convex Optimization 

23. Nov3009  Inequality Constraints: Kuhn Tucker 

Section VI: Numerical Methods


24. Dec209  Iterative Techniques 

25. Dec709  Conjugate Direction Methods 

26. Dec909  Applications  
27. Dec1409  Last day of class. Review. 