code: def linprog(
      c,
      A_ub=None,
      b_ub=None,
      A_eq=None,
      b_eq=None,
      bounds=None,
      method="simplex",
      callback=None,
      options=None,
  ):
      """
      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`` and ``ub = None`` unless set in ``bounds``.
  
      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 upper-bound
          inequality constraints at ``x``.
      b_ub : 1D array, optional
          1D array of values representing the upper-bound 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 at ``x``.
      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 in ``x``, defining
          the bounds on that parameter. Use None for one of ``min`` or
          ``max`` when there is no bound in that direction. By default
          bounds are ``(0, None)`` (non-negative).
          If a sequence containing a single tuple is provided, then ``min`` and
          ``max`` will be applied to all variables in the problem.
      method : str, optional
          Type of solver.  :ref:`'simplex' <optimize.linprog-simplex>`
          and :ref:`'interior-point' <optimize.linprog-interior-point>`
          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 method-specific options, see :func:`show_options('linprog')`.
  
      Returns
      -------
      res : OptimizeResult
          A :class:`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 :ref:`Simplex <optimize.linprog-simplex>`.
      :ref:`Interior point <optimize.linprog-interior-point>` is also available.
  
      Method *simplex* uses the simplex algorithm (as it relates to linear
      programming, NOT the Nelder-Mead simplex) [1]_, [2]_. This algorithm
      should be reasonably reliable and fast for small problems.
  
      .. versionadded:: 0.15.0
  
      Method *interior-point* uses the primal-dual 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`` or ``A_ub``, representing trivial constraints;
      - columns of zeros in ``A_eq`` `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 the output
      message states that unboundedness is detected in presolve and it is
      necessary to know whether the problem is actually infeasible, set option
      ``presolve=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 option ``rr=False`` or
      ``presolve=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
      non-negative slack variables for inequality constraints, and expressing
      unbounded variables as the difference between two non-negative variables.
  
      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.
      .. [4] 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. 197-232.
      .. [5] Andersen, Erling D. "Finding all linearly dependent rows in
             large-scale linear programming." Optimization Methods and Software
             6.3 (1995): 219-227.
      .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear
             Programming based on Newton's Method." Unpublished Course Notes,
             March 2004. Available 2/25/2017 at
             https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf
      .. [7] Fourer, Robert. "Solving Linear Programs by Interior-Point Methods."
             Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at
             http://www.4er.org/CourseNotes/Book%20B/B-III.pdf
      .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear
             programming." Mathematical Programming 71.2 (1995): 221-245.
      .. [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
      non-negative. Since the problem variables don't have the standard bounds of
      ``(0, None)``, the variable bounds must be set using ``bounds`` explicitly.
  
      There are two upper-bound 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.])
  
      """
      meth = method.lower()
      default_tol = 1e-12 if meth == "simplex" else 1e-9
  
      (
          c,
          A_ub,
          b_ub,
          A_eq,
          b_eq,
          bounds,
          solver_options,
      ) = _parse_linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, options)
      tol = solver_options.get("tol", default_tol)
  
      iteration = 0
      complete = False  # will become True if solved in presolve
      undo = []
  
      # Keep the original arrays to calculate slack/residuals for original
      # problem.
      c_o, A_ub_o, b_ub_o, A_eq_o, b_eq_o = (
          c.copy(),
          A_ub.copy(),
          b_ub.copy(),
          A_eq.copy(),
          b_eq.copy(),
      )
  
      # Solve trivial problem, eliminate variables, tighten bounds, etc...
      c0 = 0  # we might get a constant term in the objective
      if solver_options.pop("presolve", True):
          rr = solver_options.pop("rr", True)
          (
              c,
              c0,
              A_ub,
              b_ub,
              A_eq,
              b_eq,
              bounds,
              x,
              undo,
              complete,
              status,
              message,
          ) = _presolve(c, A_ub, b_ub, A_eq, b_eq, bounds, rr, tol)
  
      if not complete:
          A, b, c, c0 = _get_Abc(
              c, c0, A_ub, b_ub, A_eq, b_eq, bounds, undo
          )
          T_o = (c_o, A_ub_o, b_ub_o, A_eq_o, b_eq_o, bounds, undo)
          if meth == "simplex":
              x, status, message, iteration = _linprog_simplex(
                  c,
                  c0=c0,
                  A=A,
                  b=b,
                  callback=callback,
                  _T_o=T_o,
                  **solver_options,
              )
          elif meth == "interior-point":
              x, status, message, iteration = _linprog_ip(
                  c,
                  c0=c0,
                  A=A,
                  b=b,
                  callback=callback,
                  **solver_options,
              )
          else:
              raise ValueError("Unknown solver %s" % method)
  
      # Eliminate artificial variables, re-introduce presolved variables, etc...
      # need modified bounds here to translate variables appropriately
      disp = solver_options.get("disp", False)
      x, fun, slack, con, status, message = _postprocess(
          x,
          c_o,
          A_ub_o,
          b_ub_o,
          A_eq_o,
          b_eq_o,
          bounds,
          complete,
          undo,
          status,
          message,
          tol,
          iteration,
          disp,
      )
  
      sol = {
          "x": x,
          "fun": fun,
          "slack": slack,
          "con": con,
          "status": status,
          "message": message,
          "nit": iteration,
          "success": status == 0,
      }
  
      return OptimizeResult(sol)
  
file_path: catboost/contrib/python/scipy/py2/scipy/optimize/_linprog.py
imports: from __future__ import division, print_function, absolute_import
  import numpy
  from .optimize import OptimizeResult
  from ._linprog_ip import _linprog_ip
  from ._linprog_simplex import _linprog_simplex
  from ._linprog_util import _parse_linprog, _presolve, _get_Abc, _postprocess
label_desc: Use image node_id 3 for calling a global function with example usage: linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, method, callback, options) and returns: OptimizeResult