scipy.optimize.lsq_linear — SciPy v1.13.0 Manual (2024)

Solve a linear least-squares problem with bounds on the variables.

Given a m-by-n design matrix A and a target vector b with m elements,lsq_linear solves the following optimization problem:

minimize 0.5 * ||A x - b||**2subject to lb <= x <= ub

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

Parameters:

Aarray_like, sparse matrix of LinearOperator, shape (m, n)

Design matrix. Can be scipy.sparse.linalg.LinearOperator.

barray_like, shape (m,)

Target vector.

bounds2-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 eitheran array with the length equal to the number of parameters, or ascalar (in which case the bound is taken to be the same for allparameters). Use np.inf with an appropriate sign to disablebounds on all or some parameters.

method‘trf’ or ‘bvls’, optional

Method to perform minimization.

• ‘trf’ : Trust Region Reflective algorithm adapted for a linearleast-squares problem. This is an interior-point-like methodand the required number of iterations is weakly correlated withthe number of variables.

• ‘bvls’ : Bounded-variable least-squares algorithm. This isan active set method, which requires the number of iterationscomparable to the number of variables. Can’t be used when A issparse or LinearOperator.

Default is ‘trf’.

tolfloat, optional

Tolerance parameter. The algorithm terminates if a relative changeof 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 thantol.

• method='bvls' terminates if Karush-Kuhn-Tucker conditionsare satisfied within tol tolerance.

  See Also

  The following data are provided: You want to use least-squares regression to fit these data with the following model: y=a+b x+(c)/(x) Determine the coefficients by setting up and solving Eq. (17.25). | NumeradeLeast Squares on LinkedIn: #generativeai #ml #dl #transformers #leastsquares #aiBMI (Body Mass Index) Calculator in Python

lsq_solver{None, ‘exact’, ‘lsmr’}, optional

Method of solving unbounded least-squares problems throughoutiterations:

• ‘exact’ : Use dense QR or SVD decomposition approach. Can’t beused when A is sparse or LinearOperator.

• ‘lsmr’ : Use scipy.sparse.linalg.lsmr iterative procedurewhich requires only matrix-vector product evaluations. Can’tbe used with method='bvls'.

If None (default), the solver is chosen based on type of A.

lsmr_tolNone, float or ‘auto’, optional

Tolerance parameters ‘atol’ and ‘btol’ for scipy.sparse.linalg.lsmrIf None (default), it is set to 1e-2 * tol. If ‘auto’, thetolerance will be adjusted based on the optimality of the currentiterate, which can speed up the optimization process, but is not alwaysreliable.

max_iterNone or int, optional

Maximum number of iterations before termination. If None (default), itis set to 100 for method='trf' or to the number of variables formethod='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_maxiterNone 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), ituses lsmr’s default of min(m, n) where m and n are thenumber of rows and columns of A, respectively. Has no effect iflsq_solver='exact'.

Returns:

OptimizeResult with the following fields defined:

xndarray, shape (n,)

Solution found.

costfloat

Value of the cost function at the solution.

funndarray, shape (m,)

Vector of residuals at the solution.

optimalityfloat

First-order optimality measure. The exact meaning depends on method,refer to the description of tol parameter.

active_maskndarray 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 asequence of strictly feasible iterates and active_mask is determinedwithin a tolerance threshold.

unbounded_soltuple

Unbounded least squares solution tuple returned by the least squaressolver (set with lsq_solver option). If lsq_solver is not set or isset to 'exact', the tuple contains an ndarray of shape (n,) withthe unbounded solution, an ndarray with the sum of squared residuals,an int with the rank of A, and an ndarray with the singular valuesof A (see NumPy’s linalg.lstsq for more information). Iflsq_solver is set to 'lsmr', the tuple contains an ndarray ofshape (n,) with the unbounded solution, an int with the exit code,an int with the number of iterations, and five floats withvarious norms and the condition number of A (see SciPy’ssparse.linalg.lsmr for more information). This output can beuseful for determining the convergence of the least squares solver,particularly the iterative 'lsmr' solver. The unbounded leastsquares problem is to minimize 0.5 * ||A x - b||**2.

nitint

Number of iterations. Zero if the unconstrained solution is optimal.

statusint

Reason for algorithm termination:

• -1 : the algorithm was not able to make progress on the lastiteration.

• 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.

messagestr

Verbal description of the termination reason.

successbool

True if one of the convergence criteria is satisfied (status > 0).

Notes

The algorithm first computes the unconstrained least-squares solution bynumpy.linalg.lstsq or scipy.sparse.linalg.lsmr depending onlsq_solver. This solution is returned as optimal if it lies within thebounds.

Method ‘trf’ runs the adaptation of the algorithm described in [STIR] fora linear least-squares problem. The iterations are essentially the same asin the nonlinear least-squares algorithm, but as the quadratic functionmodel is always accurate, we don’t need to track or modify the radius ofa trust region. The line search (backtracking) is used as a safety netwhen a selected step does not decrease the cost function. Read moredetailed 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, oneach iteration chooses a new variable to move from the active set to thefree set and then solves the unconstrained least-squares problem on freevariables. This algorithm is guaranteed to give an accurate solutioneventually, but may require up to n iterations for a problem with nvariables. Additionally, an ad-hoc initialization procedure isimplemented, that determines which variables to set free or activeinitially. It takes some number of iterations before actual BVLS starts,but can significantly reduce the number of further iterations.

References

Examples

>>> import numpy as np>>> from scipy.sparse import rand>>> from scipy.optimize import lsq_linear>>> rng = np.random.default_rng()...>>> m = 20000>>> n = 10000...>>> 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)# may varyThe relative change of the cost function is less than `tol`.Number of iterations 16, initial cost 1.5039e+04, final cost 1.1112e+04,first-order optimality 4.66e-08.