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.

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 = {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.

To install the library with open source QP solvers:

pip install qpsolvers[open_source_solvers]

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 install -c conda-forge qpsolvers

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

• 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?
  • 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.

The results below come from qpsolvers_benchmark, a benchmark for QP solvers in Python.

You can run the benchmark on your machine via a command-line tool (pip install qpsolvers_benchmark). Check out the benchmark repository for details. 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). We don't report the GitHub free-for-all test set yet, as it is still too small to be representative.

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

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

We welcome contributions, see the contribution guidelines for details. We are also looking forward to hearing about your use cases! Please share them in Show and tell.

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