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scipy.optimize.linprog¶

scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='simplex', callback=None, options=None)[source]¶

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

Parameters:

Parameters:	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')`.
Returns:	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.

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').

Returns:

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.

See also

show_options: Additional options accepted by the solvers

Notes

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) [R198], [R199]. 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 [R201]. 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.

References

[R198]

(1, 2) Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963

[R199]

(1, 2) Hillier, S.H. and Lieberman, G.J. (1995), “Introduction to Mathematical Programming”, McGraw-Hill, Chapter 4.

[R200]

Bland, Robert G. New finite pivoting rules for the simplex method. Mathematics of Operations Research (2), 1977: pp. 103-107.

[R201]

(1, 2) 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.

[R202]

Andersen, Erling D. “Finding all linearly dependent rows in large-scale linear programming.” Optimization Methods and Software 6.3 (1995): 219-227.

[R203]

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

[R204]

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

[R205]

Andersen, Erling D., and Knud D. Andersen. “Presolving in linear programming.” Mathematical Programming 71.2 (1995): 221-245.

[R206]

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.

Examples

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.