linprog(method=’simplex’)

scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='highs', callback=None, options=None, x0=None, integrality=None)

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

Deprecated since version 1.9.0: method=’simplex’ will be removed in SciPy 1.11.0. It is replaced by method=’highs’ because the latter is faster and more robust.

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 ‘simplex’. ‘highs’, ‘highs-ds’, ‘highs-ipm’, ‘interior-point’ (default), and ‘revised simplex’ are also available.

callbackcallable, optional

Callback function to be executed once per iteration.

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.

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.

blandbool

If True, use Bland’s anti-cycling rule [3] to choose pivots to prevent cycling. If False, choose pivots which should lead to a converged solution more quickly. The latter method is subject to cycling (non-convergence) in rare instances.

unknown_optionsdict

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

References

[1]

Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963

[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.