optimize._linprog

A top-level linear programming interface. Currently this interface only solves linear programming problems via the Simplex Method.

New in version 0.15.0.

Module Contents

Functions

linprog_verbose_callback(xk,**kwargs) A sample callback function demonstrating the linprog callback interface.
linprog_terse_callback(xk,**kwargs) A sample callback function demonstrating the linprog callback interface.
_pivot_col(T,tol=1e-12,bland=False) Given a linear programming simplex tableau, determine the column
_pivot_row(T,pivcol,phase,tol=1e-12) Given a linear programming simplex tableau, determine the row for the
_solve_simplex(T,n,basis,maxiter=1000,phase=2,callback=None,tol=1e-12,nit0=0,bland=False) Solve a linear programming problem in “standard maximization form” using
_linprog_simplex(c,A_ub=None,b_ub=None,A_eq=None,b_eq=None,bounds=None,maxiter=1000,disp=False,callback=None,tol=1e-12,bland=False,**unknown_options) Solve the following linear programming problem via a two-phase
linprog(c,A_ub=None,b_ub=None,A_eq=None,b_eq=None,bounds=None,method=”simplex”,callback=None,options=None) Minimize a linear objective function subject to linear
linprog_verbose_callback(xk, **kwargs)

A sample callback function demonstrating the linprog callback interface. This callback produces detailed output to sys.stdout before each iteration and after the final iteration of the simplex algorithm.

xk : array_like
The current solution vector.
**kwargs : dict

A dictionary containing the following parameters:

tableau : array_like
The current tableau of the simplex algorithm. Its structure is defined in _solve_simplex.
phase : int
The current Phase of the simplex algorithm (1 or 2)
nit : int
The current iteration number.
pivot : tuple(int, int)
The index of the tableau selected as the next pivot, or nan if no pivot exists
basis : array(int)
A list of the current basic variables. Each element contains the name of a basic variable and its value.
complete : bool
True if the simplex algorithm has completed (and this is the final call to callback), otherwise False.
linprog_terse_callback(xk, **kwargs)

A sample callback function demonstrating the linprog callback interface. This callback produces brief output to sys.stdout before each iteration and after the final iteration of the simplex algorithm.

xk : array_like
The current solution vector.
**kwargs : dict

A dictionary containing the following parameters:

tableau : array_like
The current tableau of the simplex algorithm. Its structure is defined in _solve_simplex.
vars : tuple(str, …)
Column headers for each column in tableau. “x[i]” for actual variables, “s[i]” for slack surplus variables, “a[i]” for artificial variables, and “RHS” for the constraint RHS vector.
phase : int
The current Phase of the simplex algorithm (1 or 2)
nit : int
The current iteration number.
pivot : tuple(int, int)
The index of the tableau selected as the next pivot, or nan if no pivot exists
basics : list[tuple(int, float)]
A list of the current basic variables. Each element contains the index of a basic variable and its value.
complete : bool
True if the simplex algorithm has completed (and this is the final call to callback), otherwise False.
_pivot_col(T, tol=1e-12, bland=False)

Given a linear programming simplex tableau, determine the column of the variable to enter the basis.

T : 2D ndarray
The simplex tableau.
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).
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.
_pivot_row(T, pivcol, phase, tol=1e-12)

Given a linear programming simplex tableau, determine the row for the pivot operation.

T : 2D ndarray
The simplex tableau.
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.
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.
_solve_simplex(T, n, basis, maxiter=1000, phase=2, callback=None, tol=1e-12, nit0=0, bland=False)

Solve a linear programming problem in “standard maximization form” using the Simplex Method.

Minimize

subject to

T : array_like

A 2-D array representing the simplex T corresponding to the maximization 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 : 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 simplex algorithm. The callback must have the signature callback(xk, **kwargs) where xk is the current solution vector and kwargs is a dictionary containing the following:: “T” : The current Simplex algorithm T “nit” : The current iteration. “pivot” : The pivot (row, column) used for the next iteration. “phase” : Whether the algorithm is in Phase 1 or Phase 2. “basis” : The indices of the columns of the basic variables.
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.
res : OptimizeResult

The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array, success a Boolean flag indicating if the optimizer exited successfully and message which describes the cause of the termination. Possible values for the status attribute are:

0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded

See OptimizeResult for a description of other attributes.

_linprog_simplex(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, maxiter=1000, disp=False, callback=None, tol=1e-12, bland=False, **unknown_options)

Solve the following linear programming problem via a two-phase simplex algorithm.:

minimize:     c^T * x

subject to:   A_ub * x <= b_ub
              A_eq * x == b_eq
c : array_like
Coefficients of the linear objective function to be minimized.
A_ub : array_like
2-D array which, when matrix-multiplied by x, gives the values of the upper-bound inequality constraints at x.
b_ub : array_like
1-D array of values representing the upper-bound of each inequality constraint (row) in A_ub.
A_eq : array_like
2-D array which, when matrix-multiplied by x, gives the values of the equality constraints at x.
b_eq : array_like
1-D array of values representing the RHS of each equality constraint (row) in A_eq.
bounds : array_like

The bounds for each independent variable in the solution, which can take one of three forms:

None : The default bounds, all variables are non-negative. (lb, ub) : If a 2-element sequence is provided, the same

lower bound (lb) and upper bound (ub) will be applied to all variables.
[(lb_0, ub_0), (lb_1, ub_1), …] : If an n x 2 sequence is provided,
each variable x_i will be bounded by lb[i] and ub[i].

Infinite bounds are specified using -np.inf (negative) or np.inf (positive).

callback : callable

If a callback function is provide, it will be called within each iteration of the simplex algorithm. The callback must have the signature callback(xk, **kwargs) where xk is the current s olution vector and kwargs is a dictionary containing the following:

"tableau" : The current Simplex algorithm tableau
"nit" : The current iteration.
"pivot" : The pivot (row, column) used for the next iteration.
"phase" : Whether the algorithm is in Phase 1 or Phase 2.
"bv" : A structured array containing a string representation of each
basic variable and its current value.
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.

A scipy.optimize.OptimizeResult consisting of the following fields:

x : ndarray
The independent variable vector which optimizes the linear programming problem.
fun : float
Value of the objective function.
slack : ndarray
The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, then the corresponding constraint is active.
success : bool
Returns True if the algorithm succeeded in finding an optimal solution.
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
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.

Consider the following problem:

Minimize: f = -1*x[0] + 4*x[1]

Subject to: -3*x[0] + 1*x[1] <= 6
1*x[0] + 2*x[1] <= 4
x[1] >= -3

where: -inf <= x[0] <= inf

This problem deviates from the standard linear programming problem. In standard form, linear programming problems assume the variables x are non-negative. Since the variables don’t have standard bounds where 0 <= x <= inf, the bounds of the variables must be explicitly set.

There are two upper-bound constraints, which can be expressed as

dot(A_ub, x) <= b_ub

The input for this problem is as follows:

>>> from scipy.optimize import linprog
>>> c = [-1, 4]
>>> A = [[-3, 1], [1, 2]]
>>> b = [6, 4]
>>> x0_bnds = (None, None)
>>> x1_bnds = (-3, None)
>>> res = linprog(c, A, b, bounds=(x0_bnds, x1_bnds))
>>> print(res)
     fun: -22.0
 message: 'Optimization terminated successfully.'
     nit: 1
   slack: array([ 39.,   0.])
  status: 0
 success: True
       x: array([ 10.,  -3.])
[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.
linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method="simplex", callback=None, options=None)

Minimize a linear objective function subject to linear equality and inequality constraints.

Linear Programming is intended to solve the following problem form:

Minimize:     c^T * x

Subject to:   A_ub * x <= b_ub
              A_eq * x == b_eq
c : array_like
Coefficients of the linear objective function to be minimized.
A_ub : array_like, optional
2-D array which, when matrix-multiplied by x, gives the values of the upper-bound inequality constraints at x.
b_ub : array_like, optional
1-D array of values representing the upper-bound of each inequality constraint (row) in A_ub.
A_eq : array_like, optional
2-D array which, when matrix-multiplied by x, gives the values of the equality constraints at x.
b_eq : array_like, optional
1-D array of values representing the RHS of each equality constraint (row) in A_eq.
bounds : sequence, optional
(min, max) pairs for each element in x, defining the bounds on that parameter. Use None for one of min or max when there is no bound in that direction. By default bounds are (0, None) (non-negative) If a sequence containing a single tuple is provided, then min and max will be applied to all variables in the problem.
method : str, optional
Type of solver. ‘simplex’ and ‘interior-point’ are supported.
callback : callable, optional (simplex only)

If a callback function is provide, it will be called within each iteration of the simplex algorithm. The callback must have the signature callback(xk, **kwargs) where xk is the current solution vector and kwargs is a dictionary containing the following:

"tableau" : The current Simplex algorithm tableau
"nit" : The current iteration.
"pivot" : The pivot (row, column) used for the next iteration.
"phase" : Whether the algorithm is in Phase 1 or Phase 2.
"basis" : The indices of the columns of the basic variables.
options : dict, optional

A dictionary of solver options. All methods accept the following generic options:

maxiter : int
Maximum number of iterations to perform.
disp : bool
Set to True to print convergence messages.

For method-specific options, see show_options('linprog')().

A scipy.optimize.OptimizeResult consisting of the following fields:

x : ndarray
The independent variable vector which optimizes the linear programming problem.
fun : float
Value of the objective function.
slack : ndarray
The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, then the corresponding constraint is active.
success : bool
Returns True if the algorithm succeeded in finding an optimal solution.
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
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.

show_options : Additional options accepted by the solvers

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is Simplex. Interior point is also available.

Method simplex uses the simplex algorithm (as it relates to linear programming, NOT the Nelder-Mead simplex) [1]_, [2]_. This algorithm should be reasonably reliable and fast for small problems.

New in version 0.15.0.

Method interior-point uses the primal-dual path following algorithm as outlined in [4]. This algorithm is intended to provide a faster and more reliable alternative to simplex, especially for large, sparse problems. Note, however, that the solution returned may be slightly less accurate than that of the simplex method and may not correspond with a vertex of the polytope defined by the constraints.

[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.
[4]Andersen, Erling D., and Knud D. Andersen. “The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm.” High performance optimization. Springer US, 2000. 197-232.
[5]Andersen, Erling D. “Finding all linearly dependent rows in large-scale linear programming.” Optimization Methods and Software 6.3 (1995): 219-227.
[6]Freund, Robert M. “Primal-Dual Interior-Point Methods for Linear Programming based on Newton’s Method.” Unpublished Course Notes, March 2004. Available 2/25/2017 at https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf
[7]Fourer, Robert. “Solving Linear Programs by Interior-Point Methods.” Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at http://www.4er.org/CourseNotes/Book%20B/B-III.pdf
[8]Andersen, Erling D., and Knud D. Andersen. “Presolving in linear programming.” Mathematical Programming 71.2 (1995): 221-245.
[9]Bertsimas, Dimitris, and J. Tsitsiklis. “Introduction to linear programming.” Athena Scientific 1 (1997): 997.
[10]Andersen, Erling D., et al. Implementation of interior point methods for large scale linear programming. HEC/Universite de Geneve, 1996.

Consider the following problem:

Minimize: f = -1*x[0] + 4*x[1]

Subject to: -3*x[0] + 1*x[1] <= 6
1*x[0] + 2*x[1] <= 4
x[1] >= -3

where: -inf <= x[0] <= inf

This problem deviates from the standard linear programming problem. In standard form, linear programming problems assume the variables x are non-negative. Since the variables don’t have standard bounds where 0 <= x <= inf, the bounds of the variables must be explicitly set.

There are two upper-bound constraints, which can be expressed as

dot(A_ub, x) <= b_ub

The input for this problem is as follows:

>>> c = [-1, 4]
>>> A = [[-3, 1], [1, 2]]
>>> b = [6, 4]
>>> x0_bounds = (None, None)
>>> x1_bounds = (-3, None)
>>> from scipy.optimize import linprog
>>> res = linprog(c, A_ub=A, b_ub=b, bounds=(x0_bounds, x1_bounds),
...               options={"disp": True})
Optimization terminated successfully.
     Current function value: -22.000000
     Iterations: 1
>>> print(res)
     fun: -22.0
 message: 'Optimization terminated successfully.'
     nit: 1
   slack: array([ 39.,   0.])
  status: 0
 success: True
       x: array([ 10.,  -3.])

Note the actual objective value is 11.428571. In this case we minimized the negative of the objective function.