Proximal Methods in Numerical Optimization

Seminar series

• Sapienza University of Rome, May 4–7, 2026 - Rome, Italy.
  Invited by Veronica Piccialli and Marco Sciandrone. [link]
• University of Trento, March 2–6, 2026 - Povo, Italy.
  Invited by Enrico Bertolazzi. [link] Handouts [1, 2, 3, 4, 5]

Course description

This course offers a (biased) introduction to proximal methods, a family of algorithms at the forefront of modern numerical optimization. Starting from foundational concepts, we study the proximal point algorithm as a unifying theoretical framework. Attention then shifts to proximal-gradient methods, covering recent advances and convergence analysis under relaxed assumptions in nonconvex settings. For nonsmooth constrained problems, we move beyond classical penalty and barrier approaches to explore contemporary developments in augmented Lagrangian methods and proximal methods. The course prepares students to engage with current research literature and tackle large-scale optimization challenges.

Lecture handouts

• Motivation and Background [pdf]
• Proximal Points Methods [pdf]
• Proximal Gradient Methods [pdf]
• Nonsmooth Problems with Explicit Constraints [pdf]
• Convex Problems [pdf]

Tutorials

Interactive Julia notebooks using Pluto.jl:

• Proximal point algorithm [PPA]
• Equality-constrained quadratic programming [EQP]
• Proximal-gradient methods [PG]

References

• R. T. Rockafellar and R.J.B. Wets: Variational Analysis, Springer, 1998. [link]
• N. Parikh and S. Boyd: Foundations and Trends in Optimization, 1(3):123-231, 2014. [link]
• A. Beck: First-order Methods in Optimization, vol. 25. SIAM, 2017. [link]