optsuiteoptlib

optlib

We aim to formalize the broad area of mathematical optimization including convex analysis, convex optimization, nonlinear programming, integer programming and etc in Lean4. Related topics include but are not limited to the definition and properties of convex and nonconvex functions, optimality conditions, convergence of various algorithms.

More topics related to computational mathematics such as numerical linear algebra and numerical analysis will be included in the future.

Our GitHub web page corresponding to this work can be found at here .

Lean4 Toolchain Installation

How to use the code in this repository

If anything goes wrong, please feel free to contact Chenyi Li through email (lichenyi@stu.pku.edu.cn).

The version of Lean4 that used by this repository can be checked here.

Use the Optlib library as a Lean4 project dependency

In a Lean4 project, add these lines to your lakefile.lean:

require optlib from git
  "https://github.com/optsuite/optlib"

or in new lakefile.lean Lake DSL:

require "optsuite" / "optlib" @ "git#master"

The optlib library uses mathlib4 as a dependency, command lake exe cache get can be used to fetch mathlib4 cache.

Contribute to the Optlib library

The command

git clone https://github.com/optsuite/optlib.git && cd optlib && code .

will download the source of the optlib library. After editing those files, you can fork this project on GitHub and file a pull request.

Code Explanations

Differential

Convex

Function

Optimality

Algorithm

  • GradientDescent.lean: convergence rate of gradient descent algorithm for smooth convex functions.
  • GradientDescentStronglyConvex.lean: convergence rate of gradient descent algorithm for smooth strongly convex functions.
  • NesterovSmooth.lean: convergence rate of Nesterov accelerated gradient descent algorithm for smooth convex functions.
  • SubgradientMethod.lean: convergence rate of subgradient method with different choices of stepsize for nonsmooth convex functions.
  • ProximalGradient.lean: convergence rate of the proximal gradient method for composite optimization problems.
  • NesterovAccelerationFirst.lean: convergence rate of the first version of Nesterov acceleration method for composite optimization problems.
  • NesterovAccelerationSecond.lean: convergence rate of the second version of Nesterov acceleration method for composite optimization problems.
  • LASSO.lean: convergence rate of the LASSO algorithm for L1-regularized least squares problem.
  • the BCD method: convergence analysis of the block coordinate descent method.
  • the ADMM method: convergence analysis of the alternating direction method of multipliers.

What we have done

Convex Analysis

  • First and Second Order Conditions for Convex Functions
  • Definition and Properties of Strongly Convex Functions
  • Definition and Properties of L-smooth Functions
  • Definition and Properties of Proper Functions and Conjugate Functions
  • ......

Optimality Conditions

  • First and Second Order Conditions for Constrained and Unconstrained Problems
  • KKT Conditions for Constrained Problems under constraint qualifications
  • Slater Condition and KKT Conditions for convex optimization problems (Ongoing)
  • ......

Convergence of Optimization Algorithms

  • Gradient Descent for Convex and Strongly Convex Functions
  • Proximal Gradient Method and Nesterov's Acceleration
  • Block Coordinate Descent Method
  • Alternating Direction Method of Multipliers
  • ......

Mathematics Textbook Formalization Project

Objectives

The project aims to systematically formalize core areas of mathematics, including convex analysis, numerical linear algebra, and high-dimensional probability using Lean. By formalizing classical textbooks and building a machine-readable, verifiable knowledge network, the initiative will bridge traditional human-written mathematics with AI-assisted reasoning and scientific computing. The ultimate goal is to create a replicable paradigm for textbook formalization, foster integration between mathematics and artificial intelligence, and establish a foundation for next-generation mathematical infrastructure.

Recruitment

The project will build a collaborative research team centered on Lean-based formalization. We welcome outstanding undergraduate seniors, master’s students, and PhD students with strong mathematical backgrounds and programming skills. Team members will be organized into groups under the guidance of group leaders, focusing respectively on convex optimization, numerical algebra, and high-dimensional probability.

Compensation and Benefits

  • Group Members: Base stipend of RMB 1,000 per month, rising up to RMB 3,000 depending on the number and quality of formalized theorems.

  • Group Leaders: Base stipend of RMB 1,500 per month, rising up to RMB 4,000 depending on individual and team contributions.

Topics

Convex Analysis

We mainly follow the book Convex Analysis (R. T. Rockafellar).

  • Topological Properties
  • Duality Correspondences
  • Representation and Inequalities
  • Differential Theory
  • ...
Nonlinear Programming

We mainly follow the books Numerical Optimization (Jorge Nocedal, Stephen J. Wright) and Optimization: Modeling, Algorithm and Theory

  • Line Search Methods
  • Quasi-Newton Methods
  • Theory of Constrained Optimization
  • ...
Integer Programming

We mainly follow the book Integer Programming (Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli).

  • Linear Inequlities and Polyhedra
  • Perfect Formulations
  • Intersection Cuts and Corner Polyhedra
  • ...
Numerical Linear Algebra

We mainly follow the book Matrix Computations (Golub & Van Loan).

  • Matrix Factorizations (LU/Cholesky, QR, Schur, SVD; Jordan canonical form)
  • Matrix Functions
  • Matrix Differential Calculus
  • ...
High Dimensional Probability

We mainly follow the book High-Dimensional Probability (Roman Vershynin).

  • Concentration Inequalities
  • Random Vectors
  • Random Matrices
  • Random Processes
  • ...

Sponsor

  • Beijing International Center for Mathematical Research, Peking University
  • Sino-Russian Mathematics Center
  • Great Bay University
  • National Natural Science Foundation of China

Publications

Formalization of Optimization

Autoformalization and Automatic Theorem Proving

  • Ziyu Wang, Bowen Yang, Shihao Zhou, Chenyi Li, Yuan Zhang, Bin Dong, Zaiwen Wen, Translating Informal Proofs into Formal Proofs Using a Chain of States
  • Chenyi Li, Wanli Ma, Zichen Wang, Zaiwen Wen, SITA: A Framework for Structure-to-Instance Theorem Autoformalization

Premise Selection

  • Zichen Wang, Anjie Dong, Zaiwen Wen, Tree-Based Premise Selection for Lean4

References

Training Sessions

There will be a weekly online session on formalization every Wednesday from 6:15 p.m. to 7:15 p.m throughout the Spring semester of 2025. Join via Tencent Meeting (ID: 659-2856-6723).

The Team

We are a group of scholars and students with a keen interest in mathematical formalization.

Members

Other Contributors

  • Undergraduate students from Peking University:

    Hongjia Chen, Wanyi He, Yuxuan Wu, Shengyang Xu, Junda Ying, Penghao Yu, ...

  • Undergraduate students from Summer Seminar on Mathematical Formalization and Theorem Proving, BICMR, Peking University, 2023:

    Zhipeng Cao, Yiyuan Chen, Heying Wang, Zuokai Wen, Mingquan Zhang, Ruichong Zhang, ...

  • Undergraduate and graduate students from Summer Seminar on Mathematical Formalization and Theorem Proving, BICMR, Peking University, 2024:

    Yifan Bai, Yunxi Duan, Anqing Shen, Yuqing Gao, Pengfei Hao

  • Other collaborators:

    Anjie Dong, ...

Copyright

Copyright (c) 2025 Chenyi Li, Zichen Wang, Ziyu Wang, Zaiwen Wen. All rights reserved.

Released under Apache 2.0 license as described in the file LICENSE.