# 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={'bland': False, 'tol': 1e-12, 'disp': False, 'maxiter': 1000})

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: 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 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. “bv” : A structured array containing a string representation of each basic variable and its current value. 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. 

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

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 to serve as an optimal solution. bland : bool If True, use Bland’s anti-cycling rule  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

 [R806] Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963
 [R807] Hillier, S.H. and Lieberman, G.J. (1995), “Introduction to Mathematical Programming”, McGraw-Hill, Chapter 4.
 [R808] 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 + 4*x

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

where: -inf <= x <= 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.])