Convex Optimization – A.A. 2023/24
About
The aim of the course is to provide students with fundamental concepts in convexity and convex optimization, as well as their application to nonlinear optimization problems. The course will focus on how to recognize convexity, how to formulate convex relaxations of nonlinear optimization problems, and how to solve convex optimization problems. The course is addressed at an audience from all areas of mathematics. Planned topics include:

Convex sets, convex hulls, Caratheodory’s theorem, polyhedra and polytopes, extreme points, Minkowski’s theorem

Convexity of functions, subgradients, inequalities related to convexity, convex conjugate

Convex optimization problems, Lagrange duality, KarushKuhnTucker optimality conditions

Convex optimization algorithms, gradient and subgradient methods, mirror descent

Convex optimization modeling software (such as CVXPY), disciplined convex programming
Enrollment
Prospective students should express their interest by sending an email message to the lecturer (vincenzo.bonifaci@uniroma3.it) in order to be enrolled in the course, or if they seek additional information.
Schedule
When: November 15 — December 20, 2023
Every Wednesday 14.0016.00 and Friday 14.0016.00 (except December 8)
Where: Rooms M1 (Wednesday) and M3 (Friday), Dipartimento di Matematica e Fisica, Università Roma Tre
via Lungotevere Dante, 376 — also accessible by walking from Largo San Leonardo Murialdo, 1
Lectures will also be streamed on the Microsoft Teams platform.
Date  Room  Hours  Topics 

November 15 
M1 
14.0016.00 
Convex sets I — Optimization problems, convex combinations, Carathéodory’s theorem 
November 17 
M3 
14.0016.00 
Convex sets II — Convexitypreserving set transformations, separating and support hyperplanes, polytopes, Minkowski’s theorem 
November 22 
M1 
14.0016.00 
Convex functions I — Characterizations, threeslope inequality, local to global optimality, epigraph, subgradients 
November 24 
M3 
14.0016.00 
Convex functions II — Inequalities related to convexity, convexitypreserving function transformations, convex conjugate 
November 29 
M1 
14.0016.00 
Convex optimization problems — Optimality condition for a constrained convex problem, examples 
December 1 
M3 
14.0016.00 
Convex optimization problems — Lagrangian Duality, weak and strong duality, complementary slackness conditions 
December 6 
M1 
14.0016.00 
Convex optimization algorithms — KarushKuhnTucker conditions, Gradient Descent 
December 13 
F (in via Vasca Navale, 84) 
14.0016.00 
Convex optimization algorithms — Projected Gradient Descent, effects of smoothness and strong convexity 
December 15 
M3 
14.0016.00 
Convex optimization algorithms — Mirror Descent, Bregman projections, Hedge algorithm 
December 20 
M1 
14.0016.00 
Convex optimization software — Disciplined Convex Programming, CVXPY examples 
References

A. Barvinok. A Course in Convexity. American Mathematical Society, 2002.

A. Beck. FirstOrder Methods in Optimization. SIAM, 2017.

D. Bertsekas. Convex Optimization Theory. Athena Scientific, 2003.

J.M. Borwein, A.S. Lewis. Convex Analysis and Nonlinear Optimization. Canadian Mathematical Society, 2006.

J.M. Borwein, J.D. Vanderwerff. Convex Functions. Cambridge University Press, 2010.

S. Boyd, L. Vanderberghe. Convex Optimization. Cambridge University Press, 2004.

S. Bubeck. Convex Optimization: Algorithms and Complexity. Foundations and Trends in Machine Learning, Vol. 8, No. 34 (2015) 231–357

N. Vishnoi. Algorithms for Convex Optimization. Cambridge University Press, 2021.

Disciplined Convex Programming webpage.