linprog(method=’simplex’)¶

scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='simplex', callback=None, options={'maxiter': 1000, 'disp': False, 'tol': 1e-12, 'bland': False})

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

Parameters:

Parameters:	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.
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

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.

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:

: Optimization terminated successfully
: Iteration limit reached
: Problem appears to be infeasible
: 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

For documentation for the rest of the parameters, see scipy.optimize.linprog

Options:

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.

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.

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.

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:

>>> 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.])

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