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PIQP is a Proximal Interior Point Quadratic Programming solver, which can solve dense and sparse quadratic programs of the form

\[\begin{aligned} \min_{x} \quad & \frac{1}{2} x^\top P x + c^\top x \\ \text {s.t.}\quad & Ax=b, \\ & h_l \leq Gx \leq h_u, \\ & x_l \leq x \leq x_u, \end{aligned}\]

with primal decision variables \(x \in \mathbb{R}^n\), matrices \(P\in \mathbb{S}_+^n\), \(A \in \mathbb{R}^{p \times n}\), \(G \in \mathbb{R}^{m \times n}\), and vectors \(c \in \mathbb{R}^n\), \(b \in \mathbb{R}^p\), \(h_l \in \mathbb{R}^m\), \(h_u \in \mathbb{R}^m\), \(x_l \in \mathbb{R}^n\), and \(x_u \in \mathbb{R}^n\). Combining an infeasible interior point method with the proximal method of multipliers, the algorithm can handle ill-conditioned convex QP problems without the need for linear independence of the constraints.

For more detailed technical results see our papers:

PIQP: A Proximal Interior-Point Quadratic Programming Solver
R. Schwan, Y. Jiang, D. Kuhn, C.N. Jones
IEEE Conference on Decision and Control (CDC), 2023

Exploiting Multistage Optimization Structure in Proximal Solvers
R. Schwan, D. Kuhn, C.N. Jones
ArXiv, 2025

Features

• PIQP is written in header only C++ 14 leveraging the Eigen library for vectorized linear algebra.
• Dense and sparse problem formulations are supported. For small dense problems, vectorized instructions and cache locality can be exploited more efficiently.
• Special backend for multistage optimization problems.
• Allocation free problem updates and re-solves.
• Open source under the BSD 2-Clause License.

Interfaces

PIQP support a wide range of interfaces including

• C/C++ (with Eigen support)
• Python
• Matlab/Octave
• R

Credits

PIQP is developed by the following people:

• Roland Schwan (main developer)
• Yuning Jiang (methods and maths)
• Daniel Kuhn (methods and maths)
• Colin N. Jones (methods and maths)

All contributors are affiliated with the Laboratoire d’Automatique and/or the Risk Analytics and Optimization Chair at EPFL, Switzerland.

This work was supported by the Swiss National Science Foundation under the NCCR Automation (grant agreement 51NF40_180545).

PIQP is an adapted implementation of work by Spyridon Pougkakiotis and Jacek Gondzio, and is built on the following open-source libraries:

• Eigen: It’s the work horse under the hood, responsible for producing optimized numerical linear algebra code.
• Blasfeo: Used in the sparse_multistage KKT solver backend.
• ProxSuite: The code structure (folder/namespace structure, etc.), some utility functions/helper macros, and the instruction set optimized python bindings are based on ProxSuite.
• SuiteSparse - LDL (modified version): Used for solving linear systems in the sparse solver.
• pybind11: Used for generating the python bindings.
• cpu_features: Used for run-time instruction set detection in the interface bindings.
• OSQP: The C and Matlab interface is inspired by OSQP.
• Clarabel: Parts of the iterative refinement scheme are inspired by Clarabel’s implementation.

License

PIQP is licensed under the BSD 2-Clause License.