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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=1.0E-12, bland=False): 

""" 

Given a linear programming simplex tableau, determine the column 

of the variable to enter the basis. 

 

Parameters 

---------- 

T : 2D array 

A 2D 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: 

return True, np.nonzero(ma.mask == False)[0][0] 

return True, np.ma.nonzero(ma == ma.min())[0][0] 

 

 

def _pivot_row(T, basis, pivcol, phase, tol=1.0E-12, bland=False): 

""" 

Given a linear programming simplex tableau, determine the row for the 

pivot operation. 

 

Parameters 

---------- 

T : 2D array 

A 2D 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-12): 

""" 

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 : 2D array 

A 2D 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 : 1D 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) 

 

 

def _solve_simplex(T, n, basis, maxiter=1000, phase=2, status=0, message='', 

callback=None, tol=1.0E-12, nit0=0, bland=False, _T_o=None): 

""" 

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 : 2D array 

A 2D 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 : 1D 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 

maxiter : int 

The maximum number of iterations to perform before aborting the 

optimization. 

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. 

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 : 1D 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 : 1D 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 : 1D 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. 

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. 

nit0 : int 

The initial iteration number used to keep an accurate iteration total 

in a two-phase problem. 

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. 

 

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 

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) 

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] 

c, A_ub, b_ub, A_eq, b_eq, bounds, undo = _T_o 

x, fun, slack, con, _, _ = _postsolve( 

x, c, A_ub, b_ub, A_eq, b_eq, bounds, undo=undo, 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) 

nit += 1 

return nit, status 

 

 

def _linprog_simplex(c, c0, A, b, maxiter=1000, disp=False, callback=None, 

tol=1.0E-12, bland=False, _T_o=None, **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 : 1D 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 : 2D array 

2D array such that ``A @ x``, gives the values of the equality 

constraints at ``x``. 

b : 1D array 

1D 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 : 1D array 

Current solution vector 

fun : float 

Current value of the objective function 

success : bool 

True when an algorithm has completed successfully. 

slack : 1D 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 : 1D 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. 

 

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. 

 

Returns 

------- 

x : 1D 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, phase=1, callback=callback, 

maxiter=maxiter, tol=tol, bland=bland, _T_o=_T_o) 

# 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, maxiter=maxiter, 

phase=2, callback=callback, tol=tol, 

nit0=nit1, bland=bland, _T_o=_T_o) 

 

solution = np.zeros(n + m) 

solution[basis[:n]] = T[:n, -1] 

x = solution[:m] 

 

return x, status, messages[status], int(nit2)