qpsolvers 3.3.0

Quadratic programming solvers in Python with a unified API.

QP Solvers for Python

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

Installation

Using PyPI

pip install qpsolvers

Using

conda install qpsolvers -c conda-forge

Check out the documentation for Windows instructions.

Usage

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.

Example

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.

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	✖️
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?
  • You can cast squared norms to QP matrices and feed the result to solve_qp.
• I have a non-convex quadratic program. Is there a solver I can use?
  • Unfortunately most available QP solvers are designed for convex problems (i.e. problems for which P is positive semidefinite). That's in a way expected, as solving non-convex QP problems is NP hard.
  • CPLEX has methods for solving non-convex quadratic problems to either local or global optimality. Notice that finding global solutions can be significantly slower than finding local solutions.
  • Gurobi can find global solutions to non-convex quadratic problems.
  • For a free non-convex solver, you can try the popular nonlinear solver IPOPT e.g. using CasADi.
  • A list of (convex/non-convex) quadratic programming software (not necessarily in Python) was compiled by Nick Gould and Phillip Toint.
• I get the following build error on Windows when running pip install qpsolvers.
  • You will need to install the Visual C++ Build Tools to build all package dependencies.
• 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.

Benchmark

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.

Contributing

We welcome contributions, see Contributing for details.