Quadratic programming solvers in Python with a unified API.

# Quadratic Programming Solvers in Python

This library provides a one-stop shop solve_qp function to solve convex quadratic programs:

$$\begin{split} \begin{array}{ll} \underset{x}{\mbox{minimize}} & \frac{1}{2} x^T P x + q^T x \ \mbox{subject to} & G x \leq h \ & A x = b \ & lb \leq x \leq ub \end{array} \end{split}$$

Vector inequalities apply coordinate by coordinate. The function returns the primal solution $x^*$ found by the backend QP solver, or None in case of failure/unfeasible problem. All solvers require the problem to be convex, meaning the matrix $P$ should be positive semi-definite. Some solvers further require the problem to be strictly convex, meaning $P$ should be positive definite.

Dual multipliers: there is also a solve_problem function that returns not only the primal solution, but also its dual multipliers and all other relevant quantities computed by the backend solver.

## Example

To solve a quadratic program, build the matrices that define it and call solve_qp, selecting the backend QP solver via the solver keyword argument:

import numpy as np
from qpsolvers import solve_qp

M = np.array([[1.0, 2.0, 0.0], [-8.0, 3.0, 2.0], [0.0, 1.0, 1.0]])
P = M.T @ M  # this is a positive definite matrix
q = np.array([3.0, 2.0, 3.0]) @ M
G = np.array([[1.0, 2.0, 1.0], [2.0, 0.0, 1.0], [-1.0, 2.0, -1.0]])
h = np.array([3.0, 2.0, -2.0])
A = np.array([1.0, 1.0, 1.0])
b = np.array([1.0])

x = solve_qp(P, q, G, h, A, b, solver="proxqp")
print(f"QP solution: {x = }")


This example outputs the solution [0.30769231, -0.69230769, 1.38461538]. It is also possible to get dual multipliers at the solution, as shown in this example.

## Installation

### PyPI

To install the library with open source QP solvers:

pip install qpsolvers[open_source_solvers]


This one-size-fits-all installation may not work immediately on all systems (for instance if a solver tries to compile from source). If you run into any issue, check out the following variants:

• pip install qpsolvers[wheels_only] will only install solvers with pre-compiled binaries,
• pip install qpsolvers[clarabel,daqp,proxqp,scs] (for instance) will install the listed set of QP solvers,
• pip install qpsolvers will only install the library itself.

When imported, qpsolvers loads all the solvers it can find and lists them in qpsolvers.available_solvers.

### Conda

conda install -c conda-forge qpsolvers


## Solvers

Solver Keyword Algorithm API License Warm-start
Clarabel clarabel Interior point Sparse Apache-2.0 ✖️
CVXOPT cvxopt Interior point Dense GPL-3.0 ✔️
DAQP daqp Active set Dense MIT ✖️
ECOS ecos Interior point Sparse GPL-3.0 ✖️
Gurobi gurobi Interior point Sparse Commercial ✖️
HiGHS highs Active set Sparse MIT ✖️
HPIPM hpipm Interior point Dense BSD-2-Clause ✔️
MOSEK mosek Interior point Sparse Commercial ✔️
NPPro nppro Active set Dense Commercial ✔️
OSQP osqp Augmented Lagrangian Sparse Apache-2.0 ✔️
PIQP piqp Proximal Interior Point Dense & Sparse BSD-2-Clause ✖️
ProxQP proxqp Augmented Lagrangian Dense & Sparse BSD-2-Clause ✔️
QPALM qpalm Augmented Lagrangian Sparse LGPL-3.0 ✔️
qpOASES qpoases Active set Dense LGPL-2.1
qpSWIFT qpswift Interior point Sparse GPL-3.0 ✖️
quadprog quadprog Active set Dense GPL-2.0 ✖️
SCS scs Augmented Lagrangian Sparse MIT ✔️

Matrix arguments are NumPy arrays for dense solvers and SciPy Compressed Sparse Column (CSC) matrices for sparse ones.

## Benchmark

The results below come from qpbenchmark, a benchmark for QP solvers in Python. In the following tables, solvers are called with their default settings and compared over whole test sets by shifted geometric mean ("shm" for short). Lower is better and 1.0 corresponds to the best solver.

### Maros-Meszaros (hard problems)

Check out the full report for high- and low-accuracy solver settings.

Success rate (%) Runtime (shm) Primal residual (shm) Dual residual (shm) Duality gap (shm) Cost error (shm)
clarabel 89.9 1.0 1.0 1.9 1.0 1.0
cvxopt 53.6 13.8 5.3 2.6 22.9 6.6
gurobi 16.7 57.8 10.5 37.5 94.0 34.9
highs 53.6 11.3 5.3 2.6 21.2 6.1
osqp 41.3 1.8 58.7 22.6 1950.7 42.4
proxqp 77.5 4.6 2.0 1.0 11.5 2.2
scs 60.1 2.1 37.5 3.4 133.1 8.4

### Maros-Meszaros dense (subset of dense problems)

Check out the full report for high- and low-accuracy solver settings.

Success rate (%) Runtime (shm) Primal residual (shm) Dual residual (shm) Duality gap (shm) Cost error (shm)
clarabel 100.0 1.0 1.0 78.4 1.0 1.0
cvxopt 66.1 1267.4 292269757.0 268292.6 269.1 72.5
daqp 50.0 4163.4 1056090169.5 491187.7 351.8 280.0
ecos 12.9 27499.0 996322577.2 938191.8 197.6 1493.3
gurobi 37.1 3511.4 497416073.4 13585671.6 4964.0 190.6
highs 64.5 1008.4 255341695.6 235041.8 396.2 54.5
osqp 51.6 371.7 5481100037.5 3631889.3 24185.1 618.4
proxqp 91.9 14.1 1184.3 1.0 71.8 7.2
qpoases 24.2 3916.0 8020840724.2 23288184.8 102.2 778.7
qpswift 25.8 16109.1 860033995.1 789471.9 170.4 875.0
quadprog 62.9 1430.6 315885538.2 4734021.7 2200.0 192.3
scs 72.6 95.6 2817718628.1 369300.9 3303.2 152.5

## Citing qpsolvers

If you find this project useful, please consider giving it a :star: or citing it if your work is scientific:

@software{qpsolvers2024,
author = {Caron, Stéphane and Arnström, Daniel and Bonagiri, Suraj and Dechaume, Antoine and Flowers, Nikolai and Heins, Adam and Ishikawa, Takuma and Kenefake, Dustin and Mazzamuto, Giacomo and Meoli, Donato and O'Donoghue, Brendan and Oppenheimer, Adam A. and Pandala, Abhishek and Quiroz Omaña, Juan José and Rontsis, Nikitas and Shah, Paarth and St-Jean, Samuel and Vitucci, Nicola and Wolfers, Soeren and Yang, Fengyu and @bdelhaisse and @MeindertHH and @rimaddo and @urob and @shaoanlu},
month = mar,
title = {{qpsolvers: Quadratic Programming Solvers in Python}},
url = {https://github.com/qpsolvers/qpsolvers},
version = {4.3.2},
year = {2024}
}


## Contributing

We welcome contributions! The first step is to install the library and use it. Report any bug in the issue tracker. If you're a developer looking to hack on open source, check out the contribution guidelines for suggestions.

We are also looking forward to hearing about your use cases! Please share them in Show and tell 🙌

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