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Quadratic programming solvers in Python with a unified API.

Project description

Quadratic Programming Solvers in Python

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Unified interface to convex Quadratic Programming (QP) solvers available in Python.


Using PyPI

pip install qpsolvers


conda install qpsolvers -c conda-forge

Check out the documentation for Windows instructions.


The library provides a one-stop shop solve_qp function with a solver keyword argument to select the backend solver. It solves convex quadratic programs in standard form:

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

Vector inequalities apply coordinate by coordinate. The function returns the solution $x^*$ found by the 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: alternatively, the solve_problem function returns a more complete solution object containing both the primal solution and its corresponding dual multipliers.


To solve a quadratic program, build the matrices that define it and call the solve_qp function:

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 = {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.


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 ✖️
MOSEK mosek Interior point Sparse Commercial ✔️
NPPro nppro Active set Dense Commercial ✔️
OSQP osqp Augmented Lagrangian Sparse Apache-2.0 ✔️
ProxQP proxqp Augmented Lagrangian Dense & Sparse BSD-2-Clause ✔️
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?
    • Absolutely: print(qpsolvers.available_solvers)
  • Is it possible to solve a least squares rather than a quadratic program?
    • Yes, there is also a solve_ls function.
  • 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 get the following build error on Windows when running pip install qpsolvers.
  • Can I help?
    • Absolutely! 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.


On a dense problem, the performance of all solvers (as measured by IPython's %timeit on an Intel(R) Core(TM) i7-6700K CPU @ 4.00GHz) is:

Solver Type Time (ms)
qpswift Dense 0.008
quadprog Dense 0.01
qpoases Dense 0.02
osqp Sparse 0.03
scs Sparse 0.03
ecos Sparse 0.27
cvxopt Dense 0.44
gurobi Sparse 1.74
mosek Sparse 7.17

On a sparse problem with n = 500 optimization variables, these performances become:

Solver Type Time (ms)
osqp Sparse 1
qpswift Dense 2
scs Sparse 4
mosek Sparse 17
ecos Sparse 33
cvxopt Dense 51
gurobi Sparse 221
quadprog Dense 427
qpoases Dense 1560

On a model predictive control problem for robot locomotion, we get:

Solver Type Time (ms)
quadprog Dense 0.03
qpswift Dense 0.08
qpoases Dense 0.36
osqp Sparse 0.48
ecos Sparse 0.69
scs Sparse 0.76
cvxopt Dense 2.75

Finally, here is a small benchmark of random dense problems (each data point corresponds to an average over 10 runs):

Note that performances of QP solvers largely depend on the problem solved. For instance, MOSEK performs an automatic conversion to Second-Order Cone Programming (SOCP) which the documentation advises bypassing for better performance. Similarly, ECOS reformulates from QP to SOCP and works best on small problems.


We welcome contributions, see Contributing for details.

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

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