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Math 707: Optimization – Convex Analysis and Continuous Optimization
Spring 2018 Graduate Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

Course Information

Course Description: Topics in the qualitative behavior of solutions of ordinary differential equations with applications to engineering problems. Includes phase plane analysis, stability, dynamical systems, and chaos.

Number of Credits: 3

Prerequisites: Math 614 (numerical methods), 631 (linear algebra), 613 (math modeling/partial differential equations) or permission by instructor.

Course-Section and Instructors

Course-Section Instructor
Math 707-004 Professor D. Shirokoff

Office Hours for All Math Instructors: Spring 2018 Office Hours and Emails

Required Textbooks:

Title Convex Optimization
Author S. Boyd and L. Vandenberghe
Edition 1st
Publisher Cambridge University Press
ISBN # 978-0521833783
Website Textbook is freely available at the authors website:

Secondary Textbook:

Title Convex Optimization Theory
Author D. P. Bertsekas
Edition 1st
Publisher Athena Scientific
ISBN # 978-1-521-886529-31-1

University-wide Withdrawal Date:The last day to withdraw with a w is Monday, April 2, 2018. It will be strictly enforced.


DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

Homework Assignments 50%
Exam 20%
Project/ Presentation 30%

Your final letter grade will be based on the following tentative curve.

A 85 - 100 C 65 - 69
B+ 80 - 84 D 50 - 64
B 75 - 79 F 0 - 49
C+ 70 - 74    

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced.

Course Policies: See course website.

Project: This course requires submission of a term project (written document), to be handed in at the end of the term. The purpose of the project is to allow students to focus on a topic that closely relates to their own research.  In other words, the project will help students leverage the ideas on convex analysis/optimization in their own work.

Makeup Exam Policy: To properly report your absence from a midterm or final exam, please review and follow the required steps under the DMS Examination Policy found here:

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

Additional Resources

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at The office is located in Fenster Hall, Room 260.  A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Spring 2018 Academic Calendar, Registrar)

Date Day Event
January 16, 2018 T First Day of Classes
January 22, 2018 M Last Day to Add/Drop Classes
March 11 - 18, 2018 Su - Su Spring Recess - No Classes/ University Closed
March 30, 2018 F Good Friday - No Classes/ University Closed
April 2, 2018 M Last Day to Withdraw
May 1, 2018 T Friday Classes Meet - Last Day of Classes
May 2 - 3, 2018 W - R Reading Days
May 4 - 10, 2018 F - R Final Exam Period

Course Outline

Section/ Chapter Topic Number of Lectures
1 Introduction to convex analysis: definitions and properties of convex sets and (strictly) convex functions.  Caratheodory’s theorem.  Norms are convex functions, both finite and infinite dimensional. L2 versus L1 norms. Separating hyperplanes. Convex envelopes. 6
2 Conditions for optimality: Development of the (necessary) KKT conditions for optimality in convex and non-convex problems from a geometric viewpoint.  Discussion on when KKT conditions are sufficient. Includes both equality (Lagrange multipliers) and inequality constraints. 4
3 Introduction to Numerical Algorithms: Interior point methods as a way to handle inequality constraints.  Discussion of Newton’s method and gradient descent as approaches for solving the interior point (penalty) equations. Examples from PDE based problems, including Stokes equations, obstacle problems, elasticity, image processing.  4
4 Introduction to Duality: Advantages of duality (for non-convex problems). Definition of Lagrangian and geometric interpretation of duality.  3
5 Focus on Special Topics Including: (i) Optimal control problems, (ii) PDE constrained optimization, (iii) Obstacle problems, (iv) Optimal transport, (v) Non-convex problems and DC functions, (vi) Semi-definite and conic programing.  2 per topic.

General Notes on Outline: The first two chapters/sections of the course focus on fundamentals. Convex analysis provides the theoretical underpinnings and terminology for the optimization of continuous problems, as well as many areas in applied mathematics. The section on numerical algorithms provides a working knowledge in at least one prominent optimization technique. Students should be able to use this technique in practice.

Updated by Professor D. Shirokoff - 1/18/2018
Department of Mathematical Sciences Course Syllabus, Spring 2018