pylppinv 3.5.0

Linear Programming via Regularized Least Squares

Linear Programming via Regularized Least Squares

The Linear Programming via Regularized Least Squares (LPPinv) is a two-stage estimation method that reformulates linear programs as structured least-squares problems. Based on the Convex Least Squares Programming (CLSP) framework, LPPinv solves linear inequality, equality, and bound constraints by (1) constructing a canonical constraint system and computing a pseudoinverse projection, followed by (2) a convex-programming correction stage to refine the solution under additional regularization (e.g., Lasso, Ridge, or Elastic Net). LPPinv is intended for underdetermined and ill-posed linear problems, for which standard solvers fail. All calculations are performed in numpy.float64 precision.

Installation

pip install pylppinv

Quick Example

import numpy  as     np
from   lppinv import lppinv

# LPRLS/QPRLS, based on an underdetermined and potentially infeasible problem

seed   = 123456789
rng    = np.random.default_rng(seed)

# sample (dataset)
A_ub   = rng.normal(size=(50, 500))                          # underdetermined LP/QP matrix
A_eq   = rng.normal(size=(25, 500))                          # underdetermined LP/QP matrix
b_ub   = rng.normal(size=(50,   1))                          # may be inconsistent with A_ub
b_eq   = rng.normal(size=(25,   1))                          # may be inconsistent with A_eq
model  = lppinv(
             A_ub=A_ub, A_eq=A_eq, b_ub=b_ub, b_eq=b_eq,
             non_negative=False,                             # allow negative values
             r=1,                                            # a solution without refinement
             alpha=1.0                                       # a unique MNBLUE estimator
         )

# results
# (for a summary, use result.summary(display=True))
print("x hat (x_M hat):")
print(np.round(model.x.flatten(), 4))

print("\nNumerical stability:")
print("  kappaC :", round(model.kappaC, 4))
print("  kappaB :", round(model.kappaB, 4))
print("  kappaA :", round(model.kappaA, 4))

print("\nGoodness-of-fit:")
print("  NRMSE                :", round(model.nrmse,              6))
print("  Diagnostic band (min):", np.round(np.min(model.x_lower), 4))
print("  Diagnostic band (max):", np.round(np.max(model.x_upper), 4))
print("  Bootstrap t-test:")
for kw, val in model.ttest(sample_size=30,                   # NRMSE sample
                           seed=seed, distribution="normal", # seed and distribution
                          ).items():
    print(f"    {kw}: {float(val):.6f}")

User Reference

For comprehensive information on the estimator's capabilities, advanced configuration options, and implementation details, please refer to the pyclsp module, on which LPPinv is based.

To ensure cross-platform reproducibility, all CLSP implementations use a modified condition number function based on singular values, with a relative cutoff equal to cond_tolerance * the largest singular value.

LPPINV Parameters:

c : array_like of shape (p,), optional
Objective function coefficients. Accepted for API parity; not used by CLSP.

A_ub : array_like of shape (i, p), optional
Matrix for inequality constraints A_ub @ x <= b_ub.

b_ub : array_like of shape (i,), optional
Right-hand side vector for inequality constraints.

A_eq : array_like of shape (j, p), optional
Matrix for equality constraints A_eq @ x = b_eq.

b_eq : array_like of shape (j,), optional
Right-hand side vector for equality constraints.

non_negative : bool, default = True
If False, no default nonnegativity is applied.

bounds : sequence of (low, high), optional
Bounds on variables. If a single tuple (low, high) is given, it is applied to all variables. If None, defaults to (0, None) for each variable (non-negativity).

replace_value : float or None, default = np.nan
Final replacement value for any cell in the returned CLSP.x that violates the specified bounds by more than the given tolerance.

tolerance : float, default = square root of machine epsilon
Convergence tolerance for bounds.

Please note that either A_ub and b_ub or A_eq and b_eq must be provided.

CLSP Parameters: final : bool, default = True
If True, a convex programming problem is solved to refine zhat. The resulting solution z minimizes a weighted L1/L2 norm around zhat subject to Az = b.

alpha : float, list[float] or None, default = None
Regularization parameter (weight) in the final convex program:
- α = 0: Lasso (L1 norm)
- α = 1: Tikhonov Regularization/Ridge (L2 norm)
- 0 < α < 1: Elastic Net
If a scalar float is provided, that value is used after clipping to [0, 1].
If a list/iterable of floats is provided, each candidate is evaluated via a full solve, and the α with the smallest NRMSE is selected.
If None, α is chosen, based on an error rule: α = min(1.0, NRMSE_{α = 0} / (NRMSE_{α = 0} + NRMSE_{α = 1} + tolerance))

cond_tolerance : float or None, default = None
Singular-value cutoff for the custom condition number function.
If None, the implementation uses an internal relative cutoff of 1e-14.

*args, **kwargs : optional
CVXPY arguments passed to the CVXPY solver.

Returns: self

self.A : np.ndarray
Design matrix A = [C | S; M | Q], where Q is either a zero matrix or S_residual.

self.b : np.ndarray
Vector of the right-hand side.

self.zhat : np.ndarray
Vector of the first-step estimate.

self.r : int
Number of refinement iterations performed in the first step.

self.z : np.ndarray
Vector of the final solution. If the second step is disabled, it equals self.zhat.

self.x : np.ndarray
m × p matrix or vector containing the variable component of z.

self.y : np.ndarray
Vector containing the slack component of z.

self.kappaC : float
Spectral κ() for C_canon.

self.kappaB : float
Spectral κ() for B = A C_canon^+.

self.kappaA : float
Spectral κ() for A.

self.rmsa : float
Total root mean square alignment (RMSA).

self.r2_partial : float
R² for the M block in A.

self.nrmse : float
Mean square error calculated from A and normalized by standard deviation (NRMSE).

self.nrmse_partial : float
Mean square error from the M block in A and normalized by standard deviation (NRMSE).

self.z_lower : np.ndarray
Lower bound of the diagnostic interval (confidence band) based on κ(A).

self.z_upper : np.ndarray
Upper bound of the diagnostic interval (confidence band) based on κ(A).

self.x_lower : np.ndarray
Lower bound of the diagnostic interval (confidence band) based on κ(A).

self.x_upper : np.ndarray
Upper bound of the diagnostic interval (confidence band) based on κ(A).

self.y_lower : np.ndarray
Lower bound of the diagnostic interval (confidence band) based on κ(A).

self.y_upper : np.ndarray
Upper bound of the diagnostic interval (confidence band) based on κ(A).

Bibliography

Bolotov, I. (2025). CLSP: Linear Algebra Foundations of a Modular Two-Step Convex Optimization-Based Estimator for Ill-Posed Problems. Mathematics, 13(21), 3476. https://doi.org/10.3390/math13213476

License

MIT License — see the LICENSE file.