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 @ x
Subject to:
A_ub @ x <= b_ub A_eq @ x == b_eq lb <= x <= ub
where
lb = 0
andub = None
unless set inbounds
.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 upperbound inequality constraints atx
. b_ub : 1D array, optional
1D array of values representing the upperbound 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 atx
. 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 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 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 methodspecific 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 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.
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
orA_ub
, representing trivial constraints;  columns of zeros in
A_eq
andA_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 inb_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 optionpresolve=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 optionrr=False
orpresolve=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 nonnegative slack variables for inequality constraints, and expressing unbounded variables as the difference between two nonnegative 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”, 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] (1, 2) 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] (1, 2, 3) 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 = 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 nonnegative. Since the problem variables don’t have the standard bounds of
(0, None)
, the variable bounds must be set usingbounds
explicitly.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: 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.])