# PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python

arXiv: https://arxiv.org/abs/2201.04040 · HTML: https://mlsanity.com/paper/2201.04040/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 11 Jan 2022.

## Builds on (most cited first, 21 of 21)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Acceleration Methods](https://mlsanity.com/paper/2101.09545/) (2021)
- [Extragradient Method: O(1/K) Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity](https://mlsanity.com/paper/2110.04261/) (2021)
- [Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods](https://mlsanity.com/paper/1502.05666/) (2015)
- [Analysis of Optimization Algorithms via Integral Quadratic Constraints: Nonstrongly Convex Problems](https://mlsanity.com/paper/1705.03615/) (2017)
- [Optimized first-order methods for smooth convex minimization](https://mlsanity.com/paper/1406.5468/) (2014)
- [A Simple Practical Accelerated Method for Finite Sums](https://mlsanity.com/paper/1602.02442/) (2016)
- [Exact Worst-case Performance of First-order Methods for Composite Convex Optimization](https://mlsanity.com/paper/1512.07516/) (2015)
- [Optimizing the Efficiency of First-Order Methods for Decreasing the Gradient of Smooth Convex Functions](https://mlsanity.com/paper/1803.06600/) (2018)
- [Operator Splitting Performance Estimation: Tight contraction factors and optimal parameter selection](https://mlsanity.com/paper/1812.00146/) (2018)
- [Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions](https://mlsanity.com/paper/1902.00947/) (2019)
- [Lyapunov Functions for First-Order Methods: Tight Automated Convergence Guarantees](https://mlsanity.com/paper/1803.06073/) (2018)
- [Analysis and Design of First-Order Distributed Optimization Algorithms over Time-Varying Graphs](https://mlsanity.com/paper/1907.05448/) (2019)
- [A frequency-domain analysis of inexact gradient methods](https://mlsanity.com/paper/1912.13494/) (2019)
- [An optimal gradient method for smooth strongly convex minimization](https://mlsanity.com/paper/2101.09741/) (2021)
- [Exact worst-case convergence rates of the proximal gradient method for composite convex minimization](https://mlsanity.com/paper/1705.04398/) (2017)
- [Douglas-Rachford Splitting: Complexity Estimates and Accelerated Variants](https://mlsanity.com/paper/1407.6723/) (2014)
- [The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions](https://mlsanity.com/paper/2104.05468/) (2021)
- [Dissipativity Theory for Accelerating Stochastic Variance Reduction: A Unified Analysis of SVRG and Katyusha Using Semidefinite Programs](https://mlsanity.com/paper/1806.03677/) (2018)
- [Automated Worst-Case Performance Analysis of Decentralized Gradient Descent](https://mlsanity.com/paper/2103.14396/) (2021)

## Built on by (most cited first, 11 of 11)

- [The Surprising Agreement Between Convex Optimization Theory and Learning-Rate Scheduling for Large Model Training](https://mlsanity.com/paper/2501.18965/) (2025)
- [Tight Last-Iterate Convergence of the Extragradient and the Optimistic Gradient Descent-Ascent Algorithm for Constrained Monotone Variational Inequalities](https://mlsanity.com/paper/2204.09228/) (2022)
- [Data-Driven Performance Guarantees for Classical and Learned Optimizers](https://mlsanity.com/paper/2404.13831/) (2024)
- [Optimized projection-free algorithms for online learning: construction and worst-case analysis](https://mlsanity.com/paper/2506.05855/) (2025)
- [Greedy Alignment Principle for Optimizer Selection](https://mlsanity.com/paper/2512.06370/) (2025)
- [Tight analyses of first-order methods with error feedback](https://mlsanity.com/paper/2506.05271/) (2025)
- [Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimization](https://mlsanity.com/paper/2506.09730/) (2025)
- [Reinforcement learning for adaptive interior point methods in convex quadratic programming](https://mlsanity.com/paper/2509.07404/) (2025)
- [Step-Size Stability in Stochastic Optimization: A Theoretical Perspective](https://mlsanity.com/paper/2602.09842/) (2026)
- [A Tight Theory of Error Feedback Algorithms in Distributed Optimization](https://mlsanity.com/paper/2605.31594/) (2026)
- [Last-Iterate Convergence Rate of Normalized Gradient Descent under H"older Smoothness](https://mlsanity.com/paper/2610.06070/) (2026)

## Abstract (arXiv)

PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. To do that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modeling parts, and the worst-case analysis is performed numerically via a standard solver.