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 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.
Syllabus
Outline of the class meetings:
Section I: The Geometry and Topology of Rn
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1. Sep-9-09 | Introduction to the problem |
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2. Sep-14-09 | The Geometry of Rn |
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3. Sep-16-09 | Open, closed and compact sets |
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4. Sep-21-09 | Sequences and the Maximum Theorem |
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5. Sep-23-09 | Taylor's Theorem in One and Several Variables |
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Section II: Linear Algebra
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6. Sep-28-09 | Determinants and Permutations |
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7. Sep-30-09 | The Diagonalization of Symmetric Matrices |
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8. Oct-5-09 | The Spectral Theorem |
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9. Oct-7-09 | Unconstrained Optimization | |
Section III: Convexity |
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10. Oct-12-09 | Convex Sets |
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11. Oct-14-09 | Convex Functions |
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12. Oct-19-09 | Unconstrained Minimization of Convex Functions |
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13. Oct-21-09 | Midterm on the material of sections I, II, III. | |
Section IV: Linear Optimization
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14. Oct-26-09 | Cones and the Farkas Alternative |
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15. Oct-28-09 | Linear Optimization |
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16. Nov-4-09 | Duality | |
17. Nov-9-09 | The Simplex Method |
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Section V: Nonlinear Optimization: Lagrange Multipliers
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18. Nov-11-09 | Equality Constrained Optimization: the Lagrangian |
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19. Nov-16-09 | Second Order Tests for the Lagrangian |
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20. Nov-18-09 | The Bordered Hessian |
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21. Nov-23-09 | Quadratic Optimization |
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22. Nov-25-09 | Convex Optimization |
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23. Nov-30-09 | Inequality Constraints: Kuhn Tucker |
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Section VI: Numerical Methods
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24. Dec-2-09 | Iterative Techniques |
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25. Dec-7-09 | Conjugate Direction Methods |
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26. Dec-9-09 | Applications | |
27. Dec-14-09 | Last day of class. Review. |