5 * ||A x - b||**2
    subject to lb <= x <= ub

This optimization problem is convex, hence a found minimum (if iterations
have converged) is guaranteed to be global.

Parameters
----------
A : array_like, sparse matrix of LinearOperator, shape (m, n)
    Design matrix. Can be `scipy.sparse.linalg.LinearOperator`.
b : array_like, shape (m,)
    Target vector.
bounds : 2-tuple of array_like or `Bounds`, optional
    Lower and upper bounds on parameters. Defaults to no bounds.
    There are two ways to specify the bounds:

    - Instance of `Bounds` class.
    - 2-tuple of array_like: Each element of the tuple must be either
      an array with the length equal to the number of parameters, or a
      scalar (in which case the bound is taken to be the same for all
      parameters). Use ``np.inf`` with an appropriate sign to disable
      bounds on all or some parameters.

method : 'trf' or 'bvls', optional
    Method to perform minimization.

    * 'trf' : Trust Region Reflective algorithm adapted for a linear
      least-squares problem. This is an interior-point-like method
      and the required number of iterations is weakly correlated with
      the number of variables.
    * 'bvls' : Bounded-variable least-squares algorithm. This is
      an active set method, which requires the number of iterations
      comparable to the number of variables. Can't be used when `A` is
      sparse or LinearOperator.

    Default is 'trf'.
tol : float, optional
    Tolerance parameter. The algorithm terminates if a relative change
    of the cost function is less than `tol` on the last iteration.
    Additionally, the first-order optimality measure is considered:

    * ``method='trf'`` terminates if the uniform norm of the gradient,
      scaled to account for the presence of the bounds, is less than
      `tol`.
    * ``method='bvls'`` terminates if Karush-Kuhn-Tucker conditions
      are satisfied within `tol` tolerance.

lsq_solver : {None, 'exact', 'lsmr'}, optional
    Method of solving unbounded least-squares problems throughout
    iterations:

    * 'exact' : Use dense QR or SVD decomposition approach. Can't be
      used when `A` is sparse or LinearOperator.
    * 'lsmr' : Use `scipy.sparse.linalg.lsmr` iterative procedure
      which requires only matrix-vector product evaluations. Can't
      be used with ``method='bvls'``.

    If None (default), the solver is chosen based on type of `A`.
lsmr_tol : None, float or 'auto', optional
    Tolerance parameters 'atol' and 'btol' for `scipy.sparse.linalg.lsmr`
    If None (default), it is set to ``1e-2 * tol``. If 'auto', the
    tolerance will be adjusted based on the optimality of the current
    iterate, which can speed up the optimization process, but is not always
    reliable.
max_iter : None or int, optional
    Maximum number of iterations before termination. If None (default), it
    is set to 100 for ``method='trf'`` or to the number of variables for
    ``method='bvls'`` (not counting iterations for 'bvls' initialization).
verbose : {0, 1, 2}, optional
    Level of algorithm's verbosity:

    * 0 : work silently (default).
    * 1 : display a termination report.
    * 2 : display progress during iterations.

lsmr_maxiter : None or int, optional
    Maximum number of iterations for the lsmr least squares solver,
    if it is used (by setting ``lsq_solver='lsmr'``). If None (default), it
    uses lsmr's default of ``min(m, n)`` where ``m`` and ``n`` are the
    number of rows and columns of `A`, respectively. Has no effect if
    ``lsq_solver='exact'``.

Returns
-------
OptimizeResult with the following fields defined:
x : ndarray, shape (n,)
    Solution found.
cost : float
    Value of the cost function at the solution.
fun : ndarray, shape (m,)
    Vector of residuals at the solution.
optimality : float
    First-order optimality measure. The exact meaning depends on `method`,
    refer to the description of `tol` parameter.
active_mask : ndarray of int, shape (n,)
    Each component shows whether a corresponding constraint is active
    (that is, whether a variable is at the bound):

    *  0 : a constraint is not active.
    * -1 : a lower bound is active.
    *  1 : an upper bound is active.

    Might be somewhat arbitrary for the `trf` method as it generates a
    sequence of strictly feasible iterates and active_mask is determined
    within a tolerance threshold.
unbounded_sol : tuple
    Unbounded least squares solution tuple returned by the least squares
    solver (set with `lsq_solver` option). If `lsq_solver` is not set or is
    set to ``'exact'``, the tuple contains an ndarray of shape (n,) with
    the unbounded solution, an ndarray with the sum of squared residuals,
    an int with the rank of `A`, and an ndarray with the singular values
    of `A` (see NumPy's ``linalg.lstsq`` for more information). If
    `lsq_solver` is set to ``'lsmr'``, the tuple contains an ndarray of
    shape (n,) with the unbounded solution, an int with the exit code,
    an int with the number of iterations, and five floats with
    various norms and the condition number of `A` (see SciPy's
    ``sparse.linalg.lsmr`` for more information). This output can be
    useful for determining the convergence of the least squares solver,
    particularly the iterative ``'lsmr'`` solver. The unbounded least
    squares problem is to minimize ``0.5 * ||A x - b||**2``.
nit : int
    Number of iterations. Zero if the unconstrained solution is optimal.
status : int
    Reason for algorithm termination:

    * -1 : the algorithm was not able to make progress on the last
      iteration.
    *  0 : the maximum number of iterations is exceeded.
    *  1 : the first-order optimality measure is less than `tol`.
    *  2 : the relative change of the cost function is less than `tol`.
    *  3 : the unconstrained solution is optimal.

message : str
    Verbal description of the termination reason.
success : bool
    True if one of the convergence criteria is satisfied (`status` > 0).

See Also
--------
nnls : Linear least squares with non-negativity constraint.
least_squares : Nonlinear least squares with bounds on the variables.

Notes
-----
The algorithm first computes the unconstrained least-squares solution by
`numpy.linalg.lstsq` or `scipy.sparse.linalg.lsmr` depending on
`lsq_solver`. This solution is returned as optimal if it lies within the
bounds.

Method 'trf' runs the adaptation of the algorithm described in [STIR]_ for
a linear least-squares problem. The iterations are essentially the same as
in the nonlinear least-squares algorithm, but as the quadratic function
model is always accurate, we don't need to track or modify the radius of
a trust region. The line search (backtracking) is used as a safety net
when a selected step does not decrease the cost function. Read more
detailed description of the algorithm in `scipy.optimize.least_squares`.

Method 'bvls' runs a Python implementation of the algorithm described in
[BVLS]_. The algorithm maintains active and free sets of variables, on
each iteration chooses a new variable to move from the active set to the
free set and then solves the unconstrained least-squares problem on free
variables. This algorithm is guaranteed to give an accurate solution
eventually, but may require up to n iterations for a problem with n
variables. Additionally, an ad-hoc initialization procedure is
implemented, that determines which variables to set free or active
initially. It takes some number of iterations before actual BVLS starts,
but can significantly reduce the number of further iterations.

References
----------
.. [STIR] M. A. Branch, T. F. Coleman, and Y. Li, "A Subspace, Interior,
          and Conjugate Gradient Method for Large-Scale Bound-Constrained
          Minimization Problems," SIAM Journal on Scientific Computing,
          Vol. 21, Number 1, pp 1-23, 1999.
.. [BVLS] P. B. Start and R. L. Parker, "Bounded-Variable Least-Squares:
          an Algorithm and Applications", Computational Statistics, 10,
          129-141, 1995.

Examples
--------
In this example, a problem with a large sparse matrix and bounds on the
variables is solved.

>>> import numpy as np
>>> from scipy.sparse import rand
>>> from scipy.optimize import lsq_linear
>>> rng = np.random.default_rng()
...
>>> m = 2000
>>> n = 1000
...
>>> A = rand(m, n, density=1e-4, random_state=rng)
>>> b = rng.standard_normal(m)
...
>>> lb = rng.standard_normal(n)
>>> ub = lb + 1
...
>>> res = lsq_linear(A, b, bounds=(lb, ub), lsmr_tol='auto', verbose=1)
The relative change of the cost function is less than `tol`.
Number of iterations 10, initial cost 1.0070e+03, final cost 9.6602e+02,
first-order optimality 2.21e-09.        # may vary
)