MATH 524 Fall 2017
Schedule  Location  

TTh  10:05 AM  11:20 AM  LeConte 303B 
Recommended reading
Click on each title for further information.
Important deadlines you need to know
General Dates  

Classes begin  August 24, 2017 
Labor Day Holiday  September 4, 2017 
Fall Break  October 1920, 2017 
Thanksgiving Recess  November 2226, 2017 
Last Day of Classes  December 8, 2017 
Academic Deadlines  
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
Your final grade will be computed as follows:

Homework
20% of the final grade.
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.

Inclass tests
40% of the final grade (10% each test)
There will be four inclass tests scheduled as follows:
Test # Date 1 Tu Sep 19 2 Th Oct 12 3 Tu Nov 07 4 Th Nov 30 The content and difficulty of inclass 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
40% of the final grade.
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.
 Participation in an authorized University activity (such as musical performances, academic competitions, or varsity athletic events in which the student plays a formal role in a University sanctioned event)
 Required participation in military duties
 Mandatory admission interviews for professional or graduate school which cannot be rescheduled
 Participation in legal proceedings or administrative duties that require a student’s presence
 Death or major illness in a student’s immediate family
 Illness of a dependent family member
 Religious holy day if listed on www.interfaithcalendar.org
 Illness that is too severe or contagious for the student to attend class
 Weatherrelated emergencies
The course grade will be determined as follows:
GRADE  RANGE 

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, signup through [this link] with your academic email address to receive a base 4GB storage, plus an extra 500MB, free of charge.

Grades Reported through Blackboard
Remember to change your email address in [blackboard.sc.edu] if necessary.

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
—Orthogonal Gradient, Lagrange Multipliers
HW Assignment: Problems 1.1 — 1.12

Tu, Sep 05
Existence and Characterization for Unconstrained Optimization.
—Continuous and Differentiable realvalued 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 inclass Exam. Chapters 1 and 2.

Th, Sep 21
Numerical Approximation for Unconstrained Optimization.
—Rates of convergence
—NewtonRaphson Method to solve nonlinear equations \( f(x) = 0 \)
for univariate realvalued functions \( f\colon \mathbb{R} \to \mathbb{R} \)HW Assignment: Problems 3.1 — 3.16
—Graduate Students: Submit all advanced problems.
Submit one Basic, one Intermediate and one CAS of your choice.—Undergraduate Students: Submit 5 problems of your choice.
You may submit more problems for extra credit in Exam #2 
Tu, Sep 26
Numerical Approximation for Unconstrained Optimization.
—NewtonRaphson Method to solve nonlinear equations \( \boldsymbol{g}(\boldsymbol{x}) = 0 \)
for multivariate functions \( \boldsymbol{g} \colon \mathbb{R}^d \to \mathbb{R}^d \)—NewtonRaphson Method to find local minimum values for multivariate realvalued functions \( f \colon \mathbb{R}^d \to \mathbb{R} \)
HW Assignment: Problems 3.17 — 3.22
—Graduate Students: Submit all advanced problems.
Submit one Basic, one Intermediate and one CAS of your choice.—Undergraduate Students: Submit 3 problems 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
—Graduate Students: Submit all basic and advanced problems.
Submit two CAS of your choice.—Undergraduate Students: Submit 3 problems 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 inclass 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 inclass 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: NewtonRaphson 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
—FrankWolfe method

Th, Nov 30
Numerical Methods for Constrained Optimization
—Applications

Tu, Dec 05
Review