Least squares¶
To solve a linear leastsquares problem, simply build the matrices that define
it and call the solve_ls()
function:
from numpy import array, dot
from qpsolvers import solve_ls
R = array([[1., 2., 0.], [8., 3., 2.], [0., 1., 1.]])
s = array([3., 2., 3.])
G = array([[1., 2., 1.], [2., 0., 1.], [1., 2., 1.]])
h = array([3., 2., 2.]).reshape((3,))
x_sol = solve_ls(R, s, G, h, solver="osqp")
print(f"LS solution: x = {x}")
The backend QP solver is selected among supported solvers via the solver
keyword argument. This example outputs the
solution [0.0530504, 0.0265252, 2.1061008]
.

qpsolvers.
solve_ls
(R, s, G=None, h=None, A=None, b=None, lb=None, ub=None, W=None, solver=None, initvals=None, sym_proj=False, verbose=False, **kwargs)¶ Solve a constrained weighted linear Least Squares problem defined as:
\[\begin{split}\begin{split}\begin{array}{ll} \underset{x}{\mbox{minimize}} & \frac12 \ R x  s \^2_W = \frac12 (R x  s)^T W (R x  s) \\ \mbox{subject to} & G x \leq h \\ & A x = b \\ & lb \leq x \leq ub \end{array}\end{split}\end{split}\]using the QP solver selected by the
solver
keyword argument. Parameters
R (
Union
[ndarray
,csc_matrix
]) – Union[np.ndarray, spa.csc_matrix] factor of the cost function (most solvers require \(R^T W R\) to be positive definite, which means \(R\) should have full row rank).s (
ndarray
) – Vector term of the cost function.G (
Union
[ndarray
,csc_matrix
,None
]) – Linear inequality matrix.h (
Optional
[ndarray
]) – Linear inequality vector.A (
Union
[ndarray
,csc_matrix
,None
]) – Linear equality matrix.b (
Optional
[ndarray
]) – Linear equality vector.lb (
Optional
[ndarray
]) – Lower bound constraint vector.ub (
Optional
[ndarray
]) – Upper bound constraint vector.W (
Union
[ndarray
,csc_matrix
,None
]) – Definite symmetric weight matrix used to define the norm of the cost function. The standard L2 norm (W = Identity) is used by default.solver (
Optional
[str
]) – Name of the QP solver, to choose inqpsolvers.available_solvers
. This argument is mandatory.initvals (
Optional
[ndarray
]) – Vector of initial x values used to warmstart the solver.sym_proj (
bool
) – Set to True when the R matrix provided is not symmetric.verbose (
bool
) – Set to True to print out extra information.
 Return type
Optional
[ndarray
] Returns
Optimal solution if found, otherwise
None
.
Notes
Extra keyword arguments given to this function are forwarded to the underlying solvers. For example, OSQP has a setting eps_abs which we can provide by
solve_ls(R, s, G, h, solver='osqp', eps_abs=1e4)
.
See the examples/
folder in the repository for more advanced use cases. For
a more general introduction you can also check out this post on least squares
in Python.