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:  c : array_like
Coefficients of the linear objective function to be minimized.
 A_ub : array_like, optional
2D array which, when matrixmultiplied by
x
, gives the values of the upperbound inequality constraints atx
. b_ub : array_like, optional
1D array of values representing the upperbound of each inequality constraint (row) in
A_ub
. A_eq : array_like, optional
2D array which, when matrixmultiplied by
x
, gives the values of the equality constraints atx
. b_eq : array_like, optional
1D array of values representing the RHS of each equality constraint (row) in
A_eq
. bounds : sequence, optional
(min, max)
pairs for each element inx
, defining the bounds on that parameter. Use None for one ofmin
ormax
when there is no bound in that direction. By default bounds are(0, None)
(nonnegative) If a sequence containing a single tuple is provided, thenmin
andmax
will be applied to all variables in the problem. method : str, optional
Type of solver. ‘simplex’ and ‘interiorpoint’ 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)
wherexk
is the current solution vector andkwargs
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 methodspecific 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 NelderMead simplex) [1], [2]. This algorithm should be reasonably reliable and fast for small problems.
New in version 0.15.0.
Method interiorpoint uses the primaldual 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.
References
[1] (1, 2) Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963 [2] (1, 2) Hillier, S.H. and Lieberman, G.J. (1995), “Introduction to Mathematical Programming”, McGrawHill, Chapter 4. [3] Bland, Robert G. New finite pivoting rules for the simplex method. Mathematics of Operations Research (2), 1977: pp. 103107. [4] (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. 197232. [5] Andersen, Erling D. “Finding all linearly dependent rows in largescale linear programming.” Optimization Methods and Software 6.3 (1995): 219227. [6] Freund, Robert M. “PrimalDual InteriorPoint Methods for Linear Programming based on Newton’s Method.” Unpublished Course Notes, March 2004. Available 2/25/2017 at https://ocw.mit.edu/courses/sloanschoolofmanagement/15084jnonlinearprogrammingspring2004/lecturenotes/lec14_int_pt_mthd.pdf [7] Fourer, Robert. “Solving Linear Programs by InteriorPoint Methods.” Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at http://www.4er.org/CourseNotes/Book%20B/BIII.pdf [8] Andersen, Erling D., and Knud D. Andersen. “Presolving in linear programming.” Mathematical Programming 71.2 (1995): 221245. [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. 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 nonnegative. Since the variables don’t have standard bounds where 0 <= x <= inf, the bounds of the variables must be explicitly set.
There are two upperbound 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.