linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='interior-point', callback=None, options=None, x0=None)
Linear programming solves problems of the following form:
$$\min_x \ & c^T x \\ \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ & A_{eq} x = b_{eq},\\ & l \leq x \leq u ,$$where $x$ is a vector of decision variables; $c$ , $b_{ub}$ , $b_{eq}$ , $l$ , and $u$ are vectors; and $A_{ub}$ and $A_{eq}$ are matrices.
Alternatively, that's:
minimize:
c @ x
such that:
A_ub @ x <= b_ub A_eq @ x == b_eq lb <= x <= ub
Note that by default lb = 0
and ub = None
unless specified with bounds
.
This section describes the available solvers that can be selected by the 'method' parameter.
:None:None:`'highs-ds'` and :None:None:`'highs-ipm'` are interfaces to the HiGHS simplex and interior-point method solvers , respectively. :None:None:`'highs'` chooses between the two automatically. These are the fastest linear programming solvers in SciPy, especially for large, sparse problems; which of these two is faster is problem-dependent. :None:None:`'interior-point'` is the default as it was the fastest and most robust method before the recent addition of the HiGHS solvers. :None:None:`'revised simplex'` is more accurate than interior-point for the problems it solves. :None:None:`'simplex'` is the legacy method and is included for backwards compatibility and educational purposes.
Method highs-ds is a wrapper of the C++ high performance dual revised simplex implementation (HSOL) , . Method highs-ipm is a wrapper of a C++ implementation of an i\ nterior-\ p\ oint m\ ethod ; it features a crossover routine, so it is as accurate as a simplex solver. Method highs chooses between the two automatically. For new code involving linprog
, we recommend explicitly choosing one of these three method values.
Method interior-point uses the primal-dual path following algorithm as outlined in . This algorithm supports sparse constraint matrices and is typically faster than the simplex methods, especially for large, sparse problems. Note, however, that the solution returned may be slightly less accurate than those of the simplex methods and will not, in general, correspond with a vertex of the polytope defined by the constraints.
Method revised simplex uses the revised simplex method as described in , except that a factorization of the basis matrix, rather than its inverse, is efficiently maintained and used to solve the linear systems at each iteration of the algorithm.
Method simplex uses a traditional, full-tableau implementation of Dantzig's simplex algorithm , (not the Nelder-Mead simplex). This algorithm is included for backwards compatibility and educational purposes.
Before applying interior-point, revised simplex, or simplex, a presolve procedure based on 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
:None:None:`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 it is important to know whether the problem is actually infeasible, solve the problem again with option presolve=False
.
If neither infeasibility nor unboundedness are detected in a single pass of the presolve, 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 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 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. Optionally, the problem is automatically scaled via equilibration . The selected algorithm solves the standard form problem, and a postprocessing routine converts the result to a solution to the original problem.
The coefficients of the linear objective function to be minimized.
The inequality constraint matrix. Each row of A_ub
specifies the coefficients of a linear inequality constraint on x
.
The inequality constraint vector. Each element represents an upper bound on the corresponding value of A_ub @ x
.
The equality constraint matrix. Each row of A_eq
specifies the coefficients of a linear equality constraint on x
.
The equality constraint vector. Each element of A_eq @ x
must equal the corresponding element of b_eq
.
A sequence of (min, max)
pairs for each element in x
, defining the minimum and maximum values of that decision variable. Use None
to indicate that there is no bound. By default, bounds are (0, None)
(all decision variables are non-negative). If a single tuple (min, max)
is provided, then min
and max
will serve as bounds for all decision variables.
The algorithm used to solve the standard form problem. 'highs-ds' <optimize.linprog-highs-ds>
, 'highs-ipm' <optimize.linprog-highs-ipm>
, 'highs' <optimize.linprog-highs>
, 'interior-point' <optimize.linprog-interior-point>
(default), 'revised simplex' <optimize.linprog-revised_simplex>
, and 'simplex' <optimize.linprog-simplex>
(legacy) are supported.
If a callback function is provided, it will be called at least once per iteration of the algorithm. The callback function must accept a single scipy.optimize.OptimizeResult
consisting of the following fields:
x
x
fun
fun
success
success
slack
slack
con
con
phase
phase
status
status
Callback functions are not currently supported by the HiGHS methods.
A dictionary of solver options. All methods accept the following options:
maxiter
maxiter
disp
disp
presolve
presolve
All methods except the HiGHS solvers also accept:
tol
tol
autoscale
autoscale
rr
rr
rr_method
rr_method
For method-specific options, see show_options('linprog') <show_options>
.
Guess values of the decision variables, which will be refined by the optimization algorithm. This argument is currently used only by the 'revised simplex' method, and can only be used if :None:None:`x0` represents a basic feasible solution.
A scipy.optimize.OptimizeResult
consisting of the fields:
Linear programming: minimize a linear objective function subject to linear equality and inequality constraints.
show_options
Additional options accepted by the solvers.
Consider the following problem:
$$$$\min_{x_0, x_1} \ -x_0 + 4x_1 & \\ \mbox{such that} \ -3x_0 + x_1 & \leq 6,\\ -x_0 - 2x_1 & \geq -4,\\ x_1 & \geq -3.
The problem is not presented in the form accepted by linprog
. This is easily remedied by converting the "greater than" inequality constraint to a "less than" inequality constraint by multiplying both sides by a factor of $-1$
. Note also that the last constraint is really the simple bound $-3 \leq x_1 \leq \infty$
. Finally, since there are no bounds on $x_0$
, we must explicitly specify the bounds $-\infty \leq x_0 \leq \infty$
, as the default is for variables to be non-negative. After collecting coeffecients into arrays and tuples, the input for this problem is:
>>> 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])
Note that the default method for linprog
is 'interior-point', which is approximate by nature.
>>> print(res) con: array([], dtype=float64) fun: -21.99999984082494 # may vary message: 'Optimization terminated successfully.' nit: 6 # may vary slack: array([3.89999997e+01, 8.46872439e-08] # may vary status: 0 success: True x: array([ 9.99999989, -2.99999999]) # may vary
If you need greater accuracy, try 'revised simplex'.
>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds], method='revised simplex')
... print(res) con: array([], dtype=float64) fun: -22.0 # may vary message: 'Optimization terminated successfully.' nit: 1 # may vary slack: array([39., 0.]) # may vary status: 0 success: True x: array([10., -3.]) # may vary
You can use the options
parameter, e.g., to restrict the maximum number of iterations.
>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds],See :
... options={'maxiter': 4})
... print(res) con: array([], dtype=float64) fun: -21.35207150630407 # may vary message: 'The iteration limit was reached before the algorithm converged.' nit: 4 slack: array([37.19406046, 0.5727398 ]) status: 1 success: False x: array([ 9.4021973 , -2.98746855])
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.optimize._linprog.linprog
scipy.optimize
scipy.optimize._optimize.show_options
scipy.spatial._qhull.HalfspaceIntersection
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