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)

Minimize a linear objective function subject to linear equality and inequality constraints. Linear Programming is intended to solve the following problem form:

Minimize:

c @ x

Subject to:

A_ub @ x <= b_ub
A_eq @ x == b_eq
 lb <= x <= ub

where lb = 0 and ub = None unless set in bounds.

Parameters:

c : 1D array

Coefficients of the linear objective function to be minimized.

A_ub : 2D array, optional

2D array such that A_ub @ x gives the values of the upper-bound inequality constraints at x.

b_ub : 1D array, optional

1D array of values representing the upper-bound of each inequality constraint (row) in A_ub.

A_eq : 2D, optional

2D array such that A_eq @ x gives the values of the equality constraints at x.

b_eq : 1D array, 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 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 provided, it will be called within each iteration of the simplex algorithm. The callback must require a scipy.optimize.OptimizeResult consisting of the following fields:

x : 1D array

The independent variable vector which optimizes the linear programming problem.

fun : float

Value of the objective function.

success : bool

True if the algorithm succeeded in finding an optimal solution.

slack : 1D array

The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, the corresponding constraint is active.

con : 1D array

The (nominally zero) residuals of the equality constraints that is, b - A_eq @ x

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.

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
4 : Serious numerical difficulties encountered

nit : int

The number of iterations performed.

message : str

A string descriptor of the exit status of the optimization.

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:

res : OptimizeResult

A scipy.optimize.OptimizeResult consisting of the fields:

x : 1D array

The independent variable vector which optimizes the linear programming problem.

fun : float

Value of the objective function.

slack : 1D array

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.

con : 1D array

The (nominally zero) residuals of the equality constraints, that is, b - A_eq @ x

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
4 : Serious numerical difficulties encountered

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) [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.

Before applying either method a presolve procedure based on [8] attempts to identify trivial infeasibilities, trivial unboundedness, and potential problem simplifications. Specifically, it checks for:

• rows of zeros in A_eq or A_ub, representing trivial constraints;
• columns of zeros in A_eq and A_ub, representing unconstrained variables;
• column singletons in A_eq, representing fixed variables; and
• column singletons in A_ub, representing simple bounds.

If presolve reveals that the problem is unbounded (e.g. an unconstrained and unbounded variable has negative cost) or infeasible (e.g. a row of zeros in A_eq corresponds with a nonzero in b_eq), the solver terminates with the appropriate status code. Note that presolve terminates as soon as any sign of unboundedness is detected; consequently, a problem may be reported as unbounded when in reality the problem is infeasible (but infeasibility has not been detected yet). Therefore, if the output message states that unboundedness is detected in presolve and it is necessary to know whether the problem is actually infeasible, set option presolve=False.

If neither infeasibility nor unboundedness are detected in a single pass of the presolve check, bounds are tightened where possible and fixed variables are removed from the problem. Then, linearly dependent rows of the A_eq matrix are removed, (unless they represent an infeasibility) to avoid numerical difficulties in the primary solve routine. Note that rows that are nearly linearly dependent (within a prescribed tolerance) may also be removed, which can change the optimal solution in rare cases. If this is a concern, eliminate redundancy from your problem formulation and run with option rr=False or presolve=False.

Several potential improvements can be made here: additional presolve checks outlined in [8] should be implemented, the presolve routine should be run multiple times (until no further simplifications can be made), and more of the efficiency improvements from [5] should be implemented in the redundancy removal routines.

After presolve, the problem is transformed to standard form by converting the (tightened) simple bounds to upper bound constraints, introducing non-negative slack variables for inequality constraints, and expressing unbounded variables as the difference between two non-negative variables.

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”, 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]	(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.

[5]	(1, 2) 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]	(1, 2, 3) 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.

Examples

Consider the following problem:

Minimize:

f = -1x[0] + 4x[1]

Subject to:

-3x[0] + 1x[1] <= 6
 1x[0] + 2x[1] <= 4
          x[1] >= -3
  -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 problem variables don’t have the standard bounds of (0, None), the variable bounds must be set using bounds explicitly.

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: 5 # may vary
>>> print(res)
     con: array([], dtype=float64)
     fun: -22.0
 message: 'Optimization terminated successfully.'
     nit: 5 # may vary
   slack: array([39.,  0.]) # may vary
  status: 0
 success: True
       x: array([10., -3.])