# 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
    c 1-D array

      The coefficients of the linear objective function to be minimized.

    A_ub 2-D array, optional

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

    b_ub 1-D array, optional

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

    A_eq 2-D array, optional

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

    b_eq 1-D array, optional

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

    bounds sequence, 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.

    method str

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

    callback callable, optional

      Callback function to be executed once per iteration.

    x0 1-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
    res OptimizeResult

      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
    maxiter int (default: 5000)

      The maximum number of iterations to perform in either phase.

    disp bool (default: False)

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

    presolve bool (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.

    tol float (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.

    autoscale bool (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.

    rr bool (default: True)

      Set to `False` to disable automatic redundancy removal.

    maxupdate int (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.

    mast bool (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_options dict

      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.

linprog(method=’interior-point’) linprog(method=’highs-ipm’)