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]

To install a subset of QP solvers:

pip install qpsolvers[clarabel,daqp,proxqp,scs]

To install only the library itself:

pip install qpsolvers

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.

Frequently Asked Questions

• Can I print the list of solvers available on my machine?
• Is it possible to solve a least squares rather than a quadratic program?
• I have a squared norm in my cost function, how can I apply a QP solver to my problem?
• I have a non-convex quadratic program, is there a solver I can use?
• I have quadratic equality constraints, is there a solver I can use?
• Error: Mircrosoft Visual C++ 14.0 or greater is required on Windows
• Can I add penalty terms as in ridge regression or LASSO?

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 ⭐ or citing it if your work is scientific:

@software{qpsolvers2023,
  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 @bdelhaisse and @MeindertHH and @rimaddo and @urob and @shaoanlu},
  license = {LGPL-3.0},
  month = dec,
  title = {{qpsolvers: Quadratic Programming Solvers in Python}},
  url = {https://github.com/qpsolvers/qpsolvers},
  version = {4.2.0},
  year = {2023}
}

A citation template is also available via the Cite this repository button on GitHub.

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 🙌