MATH UN2500 (Section 002): Analysis and Optimization, Fall 2022

Announcement

Dec 7: Review session will be held on Tuesday December 13, 11:40am-12:55pm at 417 Math.
Oct 12: Office hours' location is changed to 614 Math.
Welcome to Analysis and Optimization (Section 002)! This is a one-semester course in mathematical analysis of continuous optimization, including unconstrained and contrained problems.

Course Information

Topics: Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010

Instructor: Chen-Chih Lai
Email: cl4205 at math dot columbia dot edu
Office: ~~307A~~ 614 Mathematics Building
Office Hours: Tue 2:40pm–3:55pm and Wed 10:10am–11:25am, 2:40pm–3:55pm, or by appointment (in-person or via Zoom)

Teaching assistant(s):
Heyuan Yao (Help room hours: Mon and Wed 9-10am in 406 Math Building.)

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

Grading: Homework (25%), Midterm 1 (15%), Midterm 2 (20%), Final Exam (40%).

Course Events

Lecture:
Days & Times: Tue, Thu 11:40am–12:55pm
Room: 326 Uris Hall
Class dates: Sep 6, 2022–Dec 12, 2022

Exam Dates:

Midterm 1: Tue Oct 4 (during class time)
Midterm 2: Thu Nov 10 (during class time)
(tentitive) Final Exam: Thu Dec 22 (4:10pm–7:00pm)

Homework

There will biweekly homework assignments.
Collaboration and discussion with your classmates is encouraged, but I encourage you to attempt the problems on your own before you discuss them with classmates, and you must write up assignments individually.
Make sure the writing in your submission is clear enough to the reader.
The lowest homework grade will be dropped.
Late homework will not be accepted under any circumstances. If you have a medical circumstance that prevents you from turning in homework, you will need a doctor's note to receive an extension.

Exams

There will be two midterm tests and one cumulative final exam.
The midterm tests will be taken during class time. The topics covered by each of the two tests will be announced in class and posted on the course website. If you are unable to take one of the midterms at the scheduled time, please let me know as soon as possible so that we can figure out a solution.
The final exam is provisionally scheduled by the University. The final exam is cumulative and covers all the material discussed during the semester.
Calculators, mobilde phones, computers or any other devices are not allows on exams.

Help Rooms

You are encouraged to take advantage of the math Help Rooms. The schedule is available at https://www.math.columbia.edu/general-information/help-rooms.

Tentative Schedule

This schedule will be updated as we go along.

Tuesday	Thursday
Sep 6: Brief overview, Review of Basic Linear Algebra (§1.1)	Sep 8: Linear Independence, The Rank of Matrix, Main Results on Linear Systems (§1.2, 1.3, 1.4)
Sep 13: Eigenvalues, Diagonalization (§1.5, 1.6)	Sep 15: Quadratic Forms (§1.7, 1.8)
Sep 20: Gradients and Directional Derivatives, Taylor’s Formula (§2.1, 2.6)/HW1 due	Sep 22: Unconstrained Optimization (§3.1, 3.2)
Sep 27: Envelope Theorem (§3.1)	Sep 29: Review/HW2 due
Oct 4: Miterm 1	Oct 6: Convex Sets, Convex Functions (§2.2, 2.3)
Oct 11: Concave and Convex Functions (§2.3, 2.4)	Oct 13: Point Set Topology in Euclidean Spaces (§13.1)
Oct 18: Topology and Convergence (§13.2), Theorem 3.1.2	Oct 20: Continuous Functions (§13.3)/HW3 due
Oct 25: Implicit and Inverse Function Theorems, The Lagrange Problem (§2.7, 3.3)	Oct 27: The Lagrange Problem (§3.3)
Nov 1: Local Second-Order Conditions (§3.4)	Nov 3: Review
Nov 8: No class/HW4 due	Nov 10: Midterm 2
Nov 15: Nonlinear Programming (§3.5)	Nov 17: Kuhn-Tucker Necessary Conditions (§3.8)
Nov 22: Kuhn-Tucker Sufficient Conditions (§3.6)	Nov 24: No class
Nov 29: Linear ODEs of First and Second Orders (§5.1-5.5, §6.1-6.3)/HW5 due	Dec 1: Calculus of Variations (§8.1, 8.2)
Dec 6: Euler-Lagrange Equation (§8.3)	Dec 8: Basic Optimal Control Problem (§9.1, 9.2)
Dec 13: Review (Optional) at 417 Mathematics Building	Dec 15: No class/HW6 due
Dec 20: No class	Dec 22: Final Exam (4:10pm-7:00pm)

Academic Accommodation of Students with Disabilities

Disability Services facilitates equal access for students with disabilities by coordinating accommodations and support services, and cultivating a campus culture that is sensitive and responsive to the needs of students.

In order to receive disability-related academic accommodations, CC and GS students must first be registered with Disability Services (DS). More information on the DS registration process is available online at https://health.columbia.edu/ods.

Faculty must be notified of registered students' accommodations before exams or other accommodations will be provided. Students who have, or think they may have, a disability are invited to contact Disability Services for a confidential discussion at (212) 854-2388 (Voice/TTY) or by email at disability@columbia.edu. Barnard students are invited to visit the Center for Accessibility Resources & Disability Services (CARDS).

Inclusivity

Columbia University is committed to an inclusive and respectful community. These principles and expectations extend to out of classroom and online activities, including electronic chat groups, video calls and other learning platforms.

If you are concerned about your classroom or out of classroom experiences, the Office of Equal Opportunity and Affirmative Action (EOAA) is available to advise and assist students who may be experiencing any equity, accessibility, inclusion, harassment, discrimination or sexual violence concerns. To make an appointment, or file a report, or find more information for students, visit the Student Conduct and Community Standards webpage.