Analysis and Optimization (Spring 2016)

Time	Mon, Wed 8:40–9:55am, 10:10–11:25am
Place	Math 312
Instructor	Anand Deopurkar (`anandrd at math`)
TAs	Sebastian (`spm2152`), Max (`mk3675`), Nawaz (`njs2155`), Peiran (`pf2317`), Peter (`pwl2107`), Yih-Jen (`yak2110`)
Office hours	Monday 12:30pm–2pm, Tuesday 6pm–7pm During reading week: Wednesday to Sunday 6pm–7pm

Announcements

Final practice problems are up. Solutions (in progress).
Homework 9 is up (due Apr 27).
I wrote some notes for calculus of variations, which I will update as we progress.
Midterm 2 statistics: mean 36, median 37, stdev 8, distribution.
Homework 8 is up (due Apr 20).
Here are solutions of midterm 2
Homework 7 is up (due Apr 13).
Midterm 2 practice problems are up along with solutions.
Homework 6 is up (due Mar 23).
Homework 5 is up (due Mar 9).
Here are solutions of midterm 1. The mean and median were about 40; the standard deviation was about 7.
Homework 4 is up (due Mar 2).
Homework 3 is up (due Feb 24).
Midterm 1 practice problems are up along with solutions.
Homework 2 is up (due Feb 10). Also fixed some typos.
Homework 1 is up (due Feb 3).

Course plan

This is a tentative plan of the course. It will be in flux, with some restructuring as we go along.

Date	Topic	Reading	Homework
Jan 20	Overview and review of optimization in one variable	Your favorite Calculus book
Jan 25	Linear inequalities and optimization in 2 variables	LEF 9.1 9.2
Jan 27	Review of linear algebra: matrices, linear dependence, rank	SHSS 1.1–1.4	Homework 1 (solutions).
Feb 1	Open, closed, compact sets, maximum theorem	SHSS 13.1, 13.2, 13.3
Feb 3	Convex sets	SHSS 2.2, 13.5	Homework 2 (solutions).
Feb 8	The simplex method	LEF 9.3
Feb 10	Duality	LEF 9.4
Feb 15	Simplex/duality continued	LEF 9.4, Chapter 2 from Tom Fergusson's notes, Lecture 25 from Pinkham.	Homework 3(solutions)
Feb 17	Midterm 1
Feb 22	Gradients and stationary points	SHSS 2.1, 3.1
Feb 24	Taylor's theorem	SHSS 2.6	Homework 4(solutions)
Feb 29	Quadratic forms, eigenvalues, spectral theorem	SHSS 1.7
Mar 2	Hessians, second order conditions, local min/max	SHSS 3.2	Homework 5(solutions).
Mar 7	Convex optimization	SHSS 2.2–2.4
Mar 9	Convex optimization continued	SHSS 2.2–2.4	Homework 6(solutions).
Mar 14	Spring break!
Mar 16	Spring break!
Mar 21	Implicit function theorem	SHSS 2.7
Mar 23	Constrained optimization: Lagrange multipliers	SHSS 3.3
Mar 28	Constrained optimization: Second order conditions	SHSS 3.4
Mar 30	Midterm 2
April 4	Constrained optimization: Second order conditions continued	SHSS 3.4
April 6	Inequality constraints: Kuhn-Tucker conditions	SHSS 3.5	Homework 7(solutions)
April 11	Convex optimization: Sufficient conditions	SHSS 3.6
April 13	Mixed constraints, envelope theorems	SHSS 3.8, 3.3	Homework 8(solutions)
April 18	Calculus of variations	SHSS 8.1
April 20	The Euler-Lagrange equation	SHSS 8.2–8.3	Homework 9(solutions)
April 25	Solving the EL equation	TBD
April 27	Constrained variational problems	TBD

Textbook and references

Our primary text will be

Sydsaeter, Hammond, Seierstand, and Strom, Further Mathematics for Economic Analysis, Second Edition.

We will complement it with the following:

Pinkham, Analysis, Convexity, and Optimization, and
Larson, Edwards, Falvo, (LEF) Elementary linear algebra.

The first book is available online for free. The chapters we will use from the second book are also available online for free. I have provided the links to the chapters in the course plan below.

Course outline

Many times in economics, science, and social science, we need to optimize a function under given constraints. That is, we must find the values of the inputs that make the function attain the maximum or the minimum value. The course will cover foundational topics from linear algebra, multivariable calculus, and mathematical analysis that are applicable to these questions. The techniques we will learn include:

Linear programming;
implicit function theorem, Lagrange multipliers;
convex optimization, Kuhn-Tucker conditions;
calculus of variations (if time permits).

The course will interweave theory and appliations. See the course plan for details.

Background and prerequisites

Calculus III and Linear Algebra are required. While we will spend some time refreshing ideas from these courses, it will not be a substitute for learning them for the first time.

Homework, exams, and grading

There will be weekly homework posted on this website. It will be usually due on Wednesday before 5pm in the homework boxes. Late homework will not be graded. To compensate for the inflexible late homework policy, I will drop the lowest 2 homework scores, so that you won't be penalized for not turning in homework in those really bad weeks.

There will be 2 in-class midterms and a final. The dates for the exams are as follows.

Midterm 1: February 17,
Midterm 2: March 30,
Final: May 11 for the 8:40 section, May 9 for the 10:10 section (both from 9am to 12pm).

If you have a conflict with the midterms, please write to me as soon as possible.

Your final grade will be determined by your performance on the the homework, the midterms, and the final. They will contribute in the following ratio:

Homework: 15%
Midterm 1: 20%
Midterm 2: 25%
Final: 40%

Collaboration policy

It is fine to work with your friends on homework but do spend time thinking about the problems on your own before you start discussing. Also, you must write up the solutions by yourself. Do not, under any circumstance, pass of someone else's work as your own! As a matter of academic honesty, write the names of your collaborators on top of your submission. This will not affect your grade.