625.694.81 - Introduction to Convexity

Applied and Computational Mathematics
Fall 2023


Convexity is a simple mathematical concept that has become central in a diverse range of applications in engineering, science, and business applications. Our main focus, from the applications perspective, will be the use of convexity within optimization problems, where convexity plays a key role in identifying the "easy" problems from the "hard" ones. The course will have an equal emphasis on expositing the rich mathematical structure of the field itself (properties of convex sets, convex functions, polarity/duality, subdifferential calculus, polyhedral theory, sublinearity), and demonstrating how these ideas can be used to model and solve optimization problems.The course requires basic familiarity with concepts like sequences, convergence and limits at the level of a rigorous multivariate calculus course (a course in real analysis such as EN.625.601, will be more than sufficient, but only the most basic ideas from real analysis are needed; a formal course is not required). The course also needs background in basic linear algebra at the level of EN.625.252 (EN.625.609 will be more than sufficient).Prerequisites: Multivariable calculus, linear algebra

Expanded Course Description


Linear algebra. Some familiarity with basic real analysis is useful, but not strictly required. Chapter 1 of the lecture notes will give an idea of which topics from linear algebra and real analysis will be useful for the course.


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Amitabh Basu

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Beryl Castello


Course Structure

The course materials are divided into modules which can be accessed by clicking Modules on the course menu. A module will have several sections including the overview, content, readings, discussions, and assignments. You are encouraged to preview all sections of the module before starting. Most modules run for a period of seven (7) days, starting on MONDAYS; exceptions are noted in the Course Outline. You should regularly check the Calendar and Announcements for assignment due dates.

Course Topics

Course Goals

The main goal of this course is to introduce students to the structural and algorithmic aspects of convex geometry and analysis, and their use in optimization. The course will have roughly 3 interrelated parts:

  1. Structure and properties of convex sets
  2. Structure and properties of convex functions
  3. Introduction to convex optimization

Course Learning Outcomes (CLOs)


The textbook is provided in Canvas as part of the course:

Basu, A. (2023). Convexity and its use in discrete and continuous optimization. Unpublished manuscript.

The textbook will also serve as the accessible version of the video lecture material.

Other Materials & Online Resources

Other useful textbooks (but not required):

Convex Analysis and Minimization Algorithms, vol. I by Hirriat-Urruty and Lemarechal; Springer-Verlag, ISBN 0-387-56852-2.

A Course in Convexity by Alexander Barvinok; AMS Graduate Studies in Mathematics, ISBN 0-8218-2968-8.

Convex Analysis by R. T. Rockafeller; Princeton Landmarks in Mathematics, ISBN 0-691-01586-4.

Convex and Discrete Geometry by Peter Gruber; Springer, ISBN 978-3-540-71132-2.

Some of these books may be available as ebooks through the JHU library.

Required Software


You will need access to a recent version of MATLAB. A license is provided at no cost to you, through JHU.

Visit the JHU IT Services Portal. Log in with your JHED ID and type “Matlab” in the search bar. Click on “Matlab for Students” in the search results and follow the instructions provided.

However, you may use any programming language of your choice for the coding assignments.

Student Coursework Requirements

It is expected that each module will take approximately 7–10 hours per week to complete. Here is an approximate breakdown: reading the assigned sections of the texts (approximately 3–4 hours per week) as well as some outside reading, listening to the audio annotated slide presentations (approximately 2–3 hours per week), and writing assignments (approximately 2–3 hours per week).

This course will consist of the following basic student requirements:

Preparation and Participation - Discussions (10% of Final Grade Calculation)

You are responsible for carefully reading all assigned material and being prepared for discussion. The majority of readings are from the course text. Additional reading may be assigned to supplement text readings.

Post your initial response to the discussion questions by 11:59pm of Day 4 (typically THURSDAY) for that module week. Posting a response to the discussion question is part one of your grade for module discussions (i.e., Timeliness).

Part two of your grade for module discussion is your interaction (i.e., responding to classmate postings with thoughtful responses) with at least two classmates (i.e., Critical Thinking). These types of posts should be made by 11:59pm of Day 7 (typically SUNDAY) for that module week.  Just posting your response to a discussion question is not sufficient; we want you to interact with your classmates. Be detailed in your postings and in your responses to your classmates' postings. Feel free to agree or disagree with your classmates. Please ensure that your postings are civil and constructive.

We will monitor module discussions and will respond to some of the discussions as discussions are posted. In some instances, we will summarize the overall discussions and post the summary for the module.

Evaluation of preparation and participation is based on contribution to Discussions.

Preparation and participation is evaluated by the following grading elements:

Preparation and participation is graded as follows:

Assignments (35% of Final Grade Calculation)

Assignments will generally be in the form of quantitative problem sets. On each submission, include your name and assignment identifier. Also include your name and a page number indicator (i.e., page x of y) on each page of your submissions. Each problem should have the assumptions, computations, and conclusions/discussion delineated. Any Figures or Tables should be captioned and labeled appropriately.

All assignments are due according to the dates in the Calendar.

Late submissions will not be accepted without approval.  You are allowed to drop your two lowest homework scores (provided that such grades are not the results of violations of academic integrity).

If, after reviewing your graded assignment you are not satisfied with the grade received, you are encouraged to contact us for a regrade.  When requesting a regrade, please be very specific about why you believe the grade you received was inappropriate.  Solutions will not be posted for assignments so you are also encouraged to come to office hours to review any problems you did not understand.

You are encouraged to collaborate (within the JHU guidelines on academic integrity) on these assignments but you should write up your own final solution. Direct copying of someone else’s written work or computer code is considered to be cheating and will result in a grade of zero on the assignment and a possible F in the course.

Quantitative assignments are graded as follows:

Exams (55% of Final Grade Calculation, combined from 25% for Midterm and 30% for Final)

The midterm exam will be available in Module 7 and due in Module 8. The final exam will be available in Module 13 and due in Module 14. You will have two weeks to complete the exams. You may use the course text to complete the exams.

The exams are evaluated by the following grading elements:

  1. Each part of question is answered (20%)
  2. Writing quality and technical accuracy (30%) (Writing is expected to meet or exceed accepted graduate-level English and scholarship standards. That is, all assignments will be graded on grammar and style as well as content.)
  3. Rationale for answer is provided (20%)
  4. Examples are included to illustrate rationale (15%) (If a student does not have direct experience related to a particular question, then the student is to provide analogies versus examples.)
  5. Outside references are included (15%)

Each question will be graded as follows:

Grading Policy

Assignments are due according to the dates posted in your Blackboard course site. You may check these due dates in the Course Calendar or the Assignments in the corresponding modules. I/We will post grades one week after assignment due dates.

We generally do not directly grade spelling and grammar. However, egregious violations of the rules of the English language will be noted without comment. Consistently poor performance in either spelling or grammar is taken as an indication of poor written communication ability that may detract from your grade.

A grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussion in every week.

A grade of B indicates work that meets all course requirements on a level appropriate for graduate academic work. These criteria apply to both undergraduates and graduate students taking the course.

EP uses a +/- grading system (see “Grading System”, Graduate Programs catalog, p. 10).

Score RangeLetter Grade
100-97= A+
96-93= A
92-90= A−
89-87= B+
86-83= B
82-80= B−
79-77= C+
76-73= C
72-70= C−
69-67= D+
66-63= D
<63= F

Final grades will be determined by the following weighting:


% of Grade

Preparation and Participation (Discussions)




Exams (Midterm + Final)

55% (25% + 30%)

Course Policies

Instructor Comments on Academic Dishonesty

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.  Ignorance of these rules is not an excuse.

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.