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Fixed database typo and removed unnecessary class identifier.

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"""Simplex method for linear programming
The *simplex* method uses a traditional, full-tableau implementation of
Dantzig's simplex algorithm [1]_, [2]_ (*not* the Nelder-Mead simplex).
This algorithm is included for backwards compatibility and educational
purposes.
.. versionadded:: 0.15.0
Warnings
--------
The simplex method may encounter numerical difficulties when pivot
values are close to the specified tolerance. If encountered try
remove any redundant constraints, change the pivot strategy to Bland's
rule or increase the tolerance value.
Alternatively, more robust methods maybe be used. See
:ref:`'interior-point' <optimize.linprog-interior-point>` and
:ref:`'revised simplex' <optimize.linprog-revised_simplex>`.
References
----------
.. [1] Dantzig, George B., Linear programming and extensions. Rand
Corporation Research Study Princeton Univ. Press, Princeton, NJ,
1963
.. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to
Mathematical Programming", McGraw-Hill, Chapter 4.
"""
import numpy as np
from warnings import warn
from .optimize import OptimizeResult, OptimizeWarning, _check_unknown_options
from ._linprog_util import _postsolve
def _pivot_col(T, tol=1e-9, bland=False):
"""
Given a linear programming simplex tableau, determine the column
of the variable to enter the basis.
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
tol : float
Elements in the objective row larger than -tol will not be considered
for pivoting. Nominally this value is zero, but numerical issues
cause a tolerance about zero to be necessary.
bland : bool
If True, use Bland's rule for selection of the column (select the
first column with a negative coefficient in the objective row,
regardless of magnitude).
Returns
-------
status: bool
True if a suitable pivot column was found, otherwise False.
A return of False indicates that the linear programming simplex
algorithm is complete.
col: int
The index of the column of the pivot element.
If status is False, col will be returned as nan.
"""
ma = np.ma.masked_where(T[-1, :-1] >= -tol, T[-1, :-1], copy=False)
if ma.count() == 0:
return False, np.nan
if bland:
# ma.mask is sometimes 0d
return True, np.nonzero(np.logical_not(np.atleast_1d(ma.mask)))[0][0]
return True, np.ma.nonzero(ma == ma.min())[0][0]
def _pivot_row(T, basis, pivcol, phase, tol=1e-9, bland=False):
"""
Given a linear programming simplex tableau, determine the row for the
pivot operation.
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a Problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
basis : array
A list of the current basic variables.
pivcol : int
The index of the pivot column.
phase : int
The phase of the simplex algorithm (1 or 2).
tol : float
Elements in the pivot column smaller than tol will not be considered
for pivoting. Nominally this value is zero, but numerical issues
cause a tolerance about zero to be necessary.
bland : bool
If True, use Bland's rule for selection of the row (if more than one
row can be used, choose the one with the lowest variable index).
Returns
-------
status: bool
True if a suitable pivot row was found, otherwise False. A return
of False indicates that the linear programming problem is unbounded.
row: int
The index of the row of the pivot element. If status is False, row
will be returned as nan.
"""
if phase == 1:
k = 2
else:
k = 1
ma = np.ma.masked_where(T[:-k, pivcol] <= tol, T[:-k, pivcol], copy=False)
if ma.count() == 0:
return False, np.nan
mb = np.ma.masked_where(T[:-k, pivcol] <= tol, T[:-k, -1], copy=False)
q = mb / ma
min_rows = np.ma.nonzero(q == q.min())[0]
if bland:
return True, min_rows[np.argmin(np.take(basis, min_rows))]
return True, min_rows[0]
def _apply_pivot(T, basis, pivrow, pivcol, tol=1e-9):
"""
Pivot the simplex tableau inplace on the element given by (pivrow, pivol).
The entering variable corresponds to the column given by pivcol forcing
the variable basis[pivrow] to leave the basis.
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
basis : 1-D array
An array of the indices of the basic variables, such that basis[i]
contains the column corresponding to the basic variable for row i.
Basis is modified in place by _apply_pivot.
pivrow : int
Row index of the pivot.
pivcol : int
Column index of the pivot.
"""
basis[pivrow] = pivcol
pivval = T[pivrow, pivcol]
T[pivrow] = T[pivrow] / pivval
for irow in range(T.shape[0]):
if irow != pivrow:
T[irow] = T[irow] - T[pivrow] * T[irow, pivcol]
# The selected pivot should never lead to a pivot value less than the tol.
if np.isclose(pivval, tol, atol=0, rtol=1e4):
message = (
"The pivot operation produces a pivot value of:{0: .1e}, "
"which is only slightly greater than the specified "
"tolerance{1: .1e}. This may lead to issues regarding the "
"numerical stability of the simplex method. "
"Removing redundant constraints, changing the pivot strategy "
"via Bland's rule or increasing the tolerance may "
"help reduce the issue.".format(pivval, tol))
warn(message, OptimizeWarning, stacklevel=5)
def _solve_simplex(T, n, basis, callback, postsolve_args,
maxiter=1000, tol=1e-9, phase=2, bland=False, nit0=0,
):
"""
Solve a linear programming problem in "standard form" using the Simplex
Method. Linear Programming is intended to solve the following problem form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
n : int
The number of true variables in the problem.
basis : 1-D array
An array of the indices of the basic variables, such that basis[i]
contains the column corresponding to the basic variable for row i.
Basis is modified in place by _solve_simplex
callback : callable, optional
If a callback function is provided, it will be called within each
iteration of the algorithm. The callback must accept a
`scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
Current solution vector
fun : float
Current value of the objective function
success : bool
True only when a phase has completed successfully. This
will be False for most iterations.
slack : 1-D array
The values of the slack variables. Each slack variable
corresponds to an inequality constraint. If the slack is zero,
the corresponding constraint is active.
con : 1-D array
The (nominally zero) residuals of the equality constraints,
that is, ``b - A_eq @ x``
phase : int
The phase of the optimization being executed. In phase 1 a basic
feasible solution is sought and the T has an additional row
representing an alternate objective function.
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.
postsolve_args : tuple
Data needed by _postsolve to convert the solution to the standard-form
problem into the solution to the original problem.
maxiter : int
The maximum number of iterations to perform before aborting the
optimization.
tol : float
The tolerance which determines when a solution is "close enough" to
zero in Phase 1 to be considered a basic feasible solution or close
enough to positive to serve as an optimal solution.
phase : int
The phase of the optimization being executed. In phase 1 a basic
feasible solution is sought and the T has an additional row
representing an alternate objective function.
bland : bool
If True, choose pivots using Bland's rule [3]_. In problems which
fail to converge due to cycling, using Bland's rule can provide
convergence at the expense of a less optimal path about the simplex.
nit0 : int
The initial iteration number used to keep an accurate iteration total
in a two-phase problem.
Returns
-------
nit : int
The number of iterations. Used to keep an accurate iteration total
in the two-phase problem.
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
"""
nit = nit0
status = 0
message = ''
complete = False
if phase == 1:
m = T.shape[1]-2
elif phase == 2:
m = T.shape[1]-1
else:
raise ValueError("Argument 'phase' to _solve_simplex must be 1 or 2")
if phase == 2:
# Check if any artificial variables are still in the basis.
# If yes, check if any coefficients from this row and a column
# corresponding to one of the non-artificial variable is non-zero.
# If found, pivot at this term. If not, start phase 2.
# Do this for all artificial variables in the basis.
# Ref: "An Introduction to Linear Programming and Game Theory"
# by Paul R. Thie, Gerard E. Keough, 3rd Ed,
# Chapter 3.7 Redundant Systems (pag 102)
for pivrow in [row for row in range(basis.size)
if basis[row] > T.shape[1] - 2]:
non_zero_row = [col for col in range(T.shape[1] - 1)
if abs(T[pivrow, col]) > tol]
if len(non_zero_row) > 0:
pivcol = non_zero_row[0]
_apply_pivot(T, basis, pivrow, pivcol, tol)
nit += 1
if len(basis[:m]) == 0:
solution = np.zeros(T.shape[1] - 1, dtype=np.float64)
else:
solution = np.zeros(max(T.shape[1] - 1, max(basis[:m]) + 1),
dtype=np.float64)
while not complete:
# Find the pivot column
pivcol_found, pivcol = _pivot_col(T, tol, bland)
if not pivcol_found:
pivcol = np.nan
pivrow = np.nan
status = 0
complete = True
else:
# Find the pivot row
pivrow_found, pivrow = _pivot_row(T, basis, pivcol, phase, tol, bland)
if not pivrow_found:
status = 3
complete = True
if callback is not None:
solution[:] = 0
solution[basis[:n]] = T[:n, -1]
x = solution[:m]
x, fun, slack, con, _ = _postsolve(
x, postsolve_args, tol=tol
)
res = OptimizeResult({
'x': x,
'fun': fun,
'slack': slack,
'con': con,
'status': status,
'message': message,
'nit': nit,
'success': status == 0 and complete,
'phase': phase,
'complete': complete,
})
callback(res)
if not complete:
if nit >= maxiter:
# Iteration limit exceeded
status = 1
complete = True
else:
_apply_pivot(T, basis, pivrow, pivcol, tol)
nit += 1
return nit, status
def _linprog_simplex(c, c0, A, b, callback, postsolve_args,
maxiter=1000, tol=1e-9, disp=False, bland=False,
**unknown_options):
"""
Minimize a linear objective function subject to linear equality and
non-negativity constraints using the two phase simplex method.
Linear programming is intended to solve problems of the following form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Parameters
----------
c : 1-D array
Coefficients of the linear objective function to be minimized.
c0 : float
Constant term in objective function due to fixed (and eliminated)
variables. (Purely for display.)
A : 2-D array
2-D array such that ``A @ x``, gives the values of the equality
constraints at ``x``.
b : 1-D array
1-D array of values representing the right hand side of each equality
constraint (row) in ``A``.
callback : callable, optional
If a callback function is provided, it will be called within each
iteration of the algorithm. The callback function must accept a single
`scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
Current solution vector
fun : float
Current value of the objective function
success : bool
True when an algorithm has completed successfully.
slack : 1-D array
The values of the slack variables. Each slack variable
corresponds to an inequality constraint. If the slack is zero,
the corresponding constraint is active.
con : 1-D array
The (nominally zero) residuals of the equality constraints,
that is, ``b - A_eq @ x``
phase : int
The phase of the algorithm being executed.
status : int
An integer representing the status of the optimization::
0 : Algorithm proceeding nominally
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.
postsolve_args : tuple
Data needed by _postsolve to convert the solution to the standard-form
problem into the solution to the original problem.
Options
-------
maxiter : int
The maximum number of iterations to perform.
disp : bool
If True, print exit status message to sys.stdout
tol : float
The tolerance which determines when a solution is "close enough" to
zero in Phase 1 to be considered a basic feasible solution or close
enough to positive to serve as an optimal solution.
bland : bool
If True, use Bland's anti-cycling rule [3]_ to choose pivots to
prevent cycling. If False, choose pivots which should lead to a
converged solution more quickly. The latter method is subject to
cycling (non-convergence) in rare instances.
unknown_options : dict
Optional arguments not used by this particular solver. If
`unknown_options` is non-empty a warning is issued listing all
unused options.
Returns
-------
x : 1-D array
Solution vector.
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
message : str
A string descriptor of the exit status of the optimization.
iteration : int
The number of iterations taken to solve the problem.
References
----------
.. [1] Dantzig, George B., Linear programming and extensions. Rand
Corporation Research Study Princeton Univ. Press, Princeton, NJ,
1963
.. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to
Mathematical Programming", McGraw-Hill, Chapter 4.
.. [3] Bland, Robert G. New finite pivoting rules for the simplex method.
Mathematics of Operations Research (2), 1977: pp. 103-107.
Notes
-----
The expected problem formulation differs between the top level ``linprog``
module and the method specific solvers. The method specific solvers expect a
problem in standard form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Whereas the top level ``linprog`` module expects a problem of form:
Minimize::
c @ x
Subject to::
A_ub @ x <= b_ub
A_eq @ x == b_eq
lb <= x <= ub
where ``lb = 0`` and ``ub = None`` unless set in ``bounds``.
The original problem contains equality, upper-bound and variable constraints
whereas the method specific solver requires equality constraints and
variable non-negativity.
``linprog`` module converts the original problem to standard form by
converting the simple bounds to upper bound constraints, introducing
non-negative slack variables for inequality constraints, and expressing
unbounded variables as the difference between two non-negative variables.
"""
_check_unknown_options(unknown_options)
status = 0
messages = {0: "Optimization terminated successfully.",
1: "Iteration limit reached.",
2: "Optimization failed. Unable to find a feasible"
" starting point.",
3: "Optimization failed. The problem appears to be unbounded.",
4: "Optimization failed. Singular matrix encountered."}
n, m = A.shape
# All constraints must have b >= 0.
is_negative_constraint = np.less(b, 0)
A[is_negative_constraint] *= -1
b[is_negative_constraint] *= -1
# As all constraints are equality constraints the artificial variables
# will also be basic variables.
av = np.arange(n) + m
basis = av.copy()
# Format the phase one tableau by adding artificial variables and stacking
# the constraints, the objective row and pseudo-objective row.
row_constraints = np.hstack((A, np.eye(n), b[:, np.newaxis]))
row_objective = np.hstack((c, np.zeros(n), c0))
row_pseudo_objective = -row_constraints.sum(axis=0)
row_pseudo_objective[av] = 0
T = np.vstack((row_constraints, row_objective, row_pseudo_objective))
nit1, status = _solve_simplex(T, n, basis, callback=callback,
postsolve_args=postsolve_args,
maxiter=maxiter, tol=tol, phase=1,
bland=bland
)
# if pseudo objective is zero, remove the last row from the tableau and
# proceed to phase 2
nit2 = nit1
if abs(T[-1, -1]) < tol:
# Remove the pseudo-objective row from the tableau
T = T[:-1, :]
# Remove the artificial variable columns from the tableau
T = np.delete(T, av, 1)
else:
# Failure to find a feasible starting point
status = 2
messages[status] = (
"Phase 1 of the simplex method failed to find a feasible "
"solution. The pseudo-objective function evaluates to {0:.1e} "
"which exceeds the required tolerance of {1} for a solution to be "
"considered 'close enough' to zero to be a basic solution. "
"Consider increasing the tolerance to be greater than {0:.1e}. "
"If this tolerance is unacceptably large the problem may be "
"infeasible.".format(abs(T[-1, -1]), tol)
)
if status == 0:
# Phase 2
nit2, status = _solve_simplex(T, n, basis, callback=callback,
postsolve_args=postsolve_args,
maxiter=maxiter, tol=tol, phase=2,
bland=bland, nit0=nit1
)
solution = np.zeros(n + m)
solution[basis[:n]] = T[:n, -1]
x = solution[:m]
return x, status, messages[status], int(nit2)