linprog(method=’revised simplex’)

scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='revised_simplex', callback=None, options={'maxiter': 5000, 'disp': False, 'presolve': True, 'tol': 1e-12, 'autoscale': False, 'rr': True, 'maxupdate': 10, 'mast': False, 'pivot': 'mrc'}, x0=None)

Linear programming: minimize a linear objective function subject to linear equality and inequality constraints using the revised simplex method.

Linear programming solves problems of the following form:

\[\begin{split}\min_x \ & c^T x \\ \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ & A_{eq} x = b_{eq},\\ & l \leq x \leq u ,\end{split}\]

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.

Parameters

c1-D array

The coefficients of the linear objective function to be minimized.

A_ub2-D array, optional

The inequality constraint matrix. Each row of A_ub specifies the coefficients of a linear inequality constraint on x.

b_ub1-D array, optional

The inequality constraint vector. Each element represents an upper bound on the corresponding value of A_ub @ x.

A_eq2-D array, optional

The equality constraint matrix. Each row of A_eq specifies the coefficients of a linear equality constraint on x.

b_eq1-D array, optional

The equality constraint vector. Each element of A_eq @ x must equal the corresponding element of b_eq.

boundssequence, optional

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.

methodstr

This is the method-specific documentation for ‘revised simplex’. ‘highs’, ‘highs-ds’, ‘highs-ipm’, ‘interior-point’ (default), and ‘simplex’ (legacy) are also available.

callbackcallable, optional

Callback function to be executed once per iteration.

x01-D array, optional

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 x0 represents a basic feasible solution.

Returns

resOptimizeResult

A scipy.optimize.OptimizeResult consisting of the fields:

x1-D array

The values of the decision variables that minimizes the objective function while satisfying the constraints.

funfloat

The optimal value of the objective function c @ x.

slack1-D array

The (nominally positive) values of the slack variables, b_ub - A_ub @ x.

con1-D array

The (nominally zero) residuals of the equality constraints, b_eq - A_eq @ x.

successbool

True when the algorithm succeeds in finding an optimal solution.

statusint

An integer representing the exit status of the algorithm.

0 : Optimization terminated successfully.

1 : Iteration limit reached.

2 : Problem appears to be infeasible.

3 : Problem appears to be unbounded.

4 : Numerical difficulties encountered.

5 : Problem has no constraints; turn presolve on.

6 : Invalid guess provided.

messagestr

A string descriptor of the exit status of the algorithm.

nitint

The total number of iterations performed in all phases.

See also

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

Options

maxiterint (default: 5000)

The maximum number of iterations to perform in either phase.

dispbool (default: False)

Set to True if indicators of optimization status are to be printed to the console each iteration.

presolvebool (default: True)

Presolve attempts to identify trivial infeasibilities, identify trivial unboundedness, and simplify the problem before sending it to the main solver. It is generally recommended to keep the default setting True; set to False if presolve is to be disabled.

tolfloat (default: 1e-12)

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 serve as an optimal solution.

autoscalebool (default: False)

Set to True to automatically perform equilibration. Consider using this option if the numerical values in the constraints are separated by several orders of magnitude.

rrbool (default: True)

Set to False to disable automatic redundancy removal.

maxupdateint (default: 10)

The maximum number of updates performed on the LU factorization. After this many updates is reached, the basis matrix is factorized from scratch.

mastbool (default: False)

Minimize Amortized Solve Time. If enabled, the average time to solve a linear system using the basis factorization is measured. Typically, the average solve time will decrease with each successive solve after initial factorization, as factorization takes much more time than the solve operation (and updates). Eventually, however, the updated factorization becomes sufficiently complex that the average solve time begins to increase. When this is detected, the basis is refactorized from scratch. Enable this option to maximize speed at the risk of nondeterministic behavior. Ignored if maxupdate is 0.

pivot“mrc” or “bland” (default: “mrc”)

Pivot rule: Minimum Reduced Cost (“mrc”) or Bland’s rule (“bland”). Choose Bland’s rule if iteration limit is reached and cycling is suspected.

unknown_optionsdict

Optional arguments not used by this particular solver. If unknown_options is non-empty a warning is issued listing all unused options.

Notes

Method revised simplex uses the revised simplex method as described in [9], except that a factorization [11] of the basis matrix, rather than its inverse, is efficiently maintained and used to solve the linear systems at each iteration of the algorithm.

References

9

Bertsimas, Dimitris, and J. Tsitsiklis. “Introduction to linear programming.” Athena Scientific 1 (1997): 997.

11

Bartels, Richard H. “A stabilization of the simplex method.” Journal in Numerische Mathematik 16.5 (1971): 414-434.