# MATH 524 Fall 2017

Schedule Location
TTh 10:05 AM - 11:20 AM LeConte 303B

Click on each title for further information.

Peressini, Sullivan, Uhl

Dimitri Bertsekas

Walter Gautschi

Walter Rudin

FJ Blanco-Silva

## Important deadlines you need to know

General Dates
Classes begin August 24, 2017
Labor Day Holiday September 4, 2017
Fall Break October 19--20, 2017
Thanksgiving Recess November 22--26, 2017
Last Day of Classes December 8, 2017
Last Day to Change/Drop without W August 30, 2017
First Day W Grade Assigned August 31, 2017
Last Day to Drop/Withdraw without WF October 16, 2017
First Day WF Grade Assigned October 17, 2017

## Prerequisites

• #### Vector Calculus

A grade of C or better in MATH 241

• #### Linear Algebra

A grade of C or better in MATH 344 or MATH 544

• Otherwise, you must obtain consent from the Undergraduate Director.

## Course Structure and Grading Policies

• #### Homework

There will be HW assigned at the end of each lecture. The assignments are usually due one week later (except if there is no class on that day).

Each problem is labeled as Basic, Intermediate, Advanced or CAS (Computer Algebra System required). Advanced problems usually require the proof of a result stated in class, and are thus beyond the reach of most undergraduate students. CAS problems usually require students to create their own code and perform experiments. Students are encouraged to use their preferred computer language, or preferred CAS (Maple, Matlab, Octave, Mathematica, etc). Examples in class will be performed using the scipy stack (Scientific Computing libraries for the Python language).

The score of each assignment for any undergraduate students will be averaged over the set of Basic, Intermediate and CAS problems. Undergraduate students are encouraged to work Advanced problems, but this will not grant them extra credit.

The score of each assignment for any graduate students will be averaged over the set of all problems listed.

• #### In-class tests

40% of the final grade (10% each test)

There will be four in-class tests scheduled as follows:

Test # Date
1 Tu Sep 19
2 Tu Oct 10
3 TBA
4 TBA

The content and difficulty of in-class tests is equivalent to the corresponding HW assignments. There will be two versions of each test: one for Undergraduate students, and one for Graduate students.

• #### Final Exam

There will be a comprehensive final exam scheduled on Tuesday December 12th, 2017, at 9:00 AM.

The content and difficulty of the final exam is equivalent to the corresponding HW assignments. There will be two versions of the final exam: one for Undergraduate students, and one for Graduate students.

The course grade will be determined as follows:

A 90%-100%
B+ 85%-89%
B 80%-84%
C+ 75%-79%
C 70%-74%
D+ 65%-69%
D 60%-64%
F below 60%

## Further Information

• #### Honor Code

The Honor Code applies to all work for this course. Please review the Honor Code and watch the Academic Integrity Tutorial at [this link]. Students found violating the Honor Code will be subject to discipline.

• #### Class Material

Class notes will be accessible via Overleaf. Click on the image below to access the last draft

Other additional material will be stored in Dropbox. In that case, you may need an account to retrieve it. If you do not have one already, sign-up through [this link] with your academic e-mail address to receive a base 4GB storage, plus an extra 500MB, free of charge.

• #### Student Disability Resource Center

If you have special needs as addressed by the Americans with Disabilities Act and need any assistance, please notify the instructor immediately.

## Learning Outcomes

Students will learn to address problems of finding the maximum or minimum value of functions under possible constraints. The focus is mainly:

• The Development of existence and characterization of extrema.
• The Development and study of different algorithms to track extrema.

## Lesson Plan

• #### Th, Aug 24

Review of Course content, policies, etc.

Review of Optimization in the setting of Vector Calculus

—Local minima

• #### Tu, Aug 29

Review of Optimization in the setting of Vector Calculus

—Global minima

• #### Th, Aug 31

Review of Optimization in the setting of Vector Calculus

HW Assignment: Problems 1.1 — 1.12

• #### Tu, Sep 05

Existence and Characterization for Unconstrained Optimization.

—Continuous and Differentiable real-valued functions $$f \colon \mathbb{R}^d \to \mathbb{R}$$

HW Assignment: Problems 2.1 — 2.6

• #### Th, Sep 07

Existence and Characterization for Unconstrained Optimization.

—Coercive functions

HW Assignment: Problems 2.7 — 2.8

• #### Tu, Sep 12

Existence and Characterization for Unconstrained Optimization.

—Convex sets and Convex Functions

HW Assignment: Problems 2.9 — 2.11

• #### Th, Sep 14

Existence and Characterization for Unconstrained Optimization.

—Existence Theorems for Continuous functions

—Differentiability and Characterization

HW Assignment: Problems 2.12 — 2.19

• #### Tu, Sep 19

First in-class Exam. Chapters 1 and 2.

• #### Th, Sep 21

Numerical Approximation for Unconstrained Optimization.

—Rates of convergence

—Newton-Raphson Method to solve nonlinear equations $$f(x) = 0$$
for univariate real-valued functions $$f\colon \mathbb{R} \to \mathbb{R}$$

HW Assignment: Problems 3.1 — 3.16

Submit one Basic, one Intermediate and one CAS of your choice.

You may submit more problems for extra credit in Exam #2

• #### Tu, Sep 26

Numerical Approximation for Unconstrained Optimization.

—Newton-Raphson Method to solve nonlinear equations $$\boldsymbol{g}(\boldsymbol{x}) = 0$$
for multivariate functions $$\boldsymbol{g} \colon \mathbb{R}^d \to \mathbb{R}^d$$

—Newton-Raphson Method to find local minimum values for multivariate real-valued functions $$f \colon \mathbb{R}^d \to \mathbb{R}$$

HW Assignment: Problems 3.17 — 3.22

Submit one Basic, one Intermediate and one CAS of your choice.

You may submit more problems for extra credit in Exam #2

• #### Th, Sep 28

Numerical Approximation for Unconstrained Optimization.

—Secant Methods

HW Assignment: Problems 3.23 — 3.25

• #### Tu, Oct 03

Numerical Approximation for Unconstrained Optimization.

—Steepest Descent, part I

HW Assignment: Problems 3.26 — 3.31

Submit two CAS of your choice.

You may submit more problems for extra credit in Exam #2

• #### Th, Oct 05

Classes Canceled

• #### Tu, Oct 10

Numerical Approximation for Unconstrained Optimization.

—Steepest Descent, part II

• #### Th, Oct 12

Second in-class Exam. Chapter 3.

• #### Tu, Oct 17

Numerical Approximation for Unconstrained Optimization.

—Effective methods: Wolfe's Theorem

• #### Tu, Oct 24

Numerical Approximation for Unconstrained Optimization.

—Effective methods: DFP

—Effective methods: BFGS

Existence and Characterization for Constrained Optimization

—Intro: Notation and terminology

• #### Th, Oct 26

Existence and Characterization for Constrained Optimization

—Necessary conditions (Geometric and Fritz John conditions)

HW Assignment: Problems 4.1 — 4.5

• #### Tu, Oct 31

Existence and Characterization for Constrained Optimization

—Necessary conditions (KKT conditions)

• #### Th, Nov 02

Existence and Characterization for Constrained Optimization

—Sufficient conditions

—Key Examples

• #### Tu, Nov 07

Third in-class Exam. Chapter 4.

• #### Th, Nov 09

Numerical Methods for Constrained Optimization

—Intro

• #### Tu, Nov 14

Numerical Methods for Constrained Optimization

—Projection Methods: Steepest Descent method

• #### Th, Nov 16

Numerical Methods for Constrained Optimization

—Projection Methods: Newton-Raphson method

• #### Tu, Nov 21

Numerical Methods for Constrained Optimization

—Linear Programming: the Simplex method

HW Assignment: Problems 5.1 — 5.7

• #### Tu, Nov 28

Numerical Methods for Constrained Optimization

—Frank-Wolfe method