 ``0`` : Optimization proceeding nominally.

            ``1`` : Iteration limit reached.

            ``2`` : Problem appears to be infeasible.

            ``3`` : Problem appears to be unbounded.

            ``4`` : Numerical difficulties encountered.

            nit : int
                The current iteration number.
            message : str
                A string descriptor of the algorithm status.

        Callback functions are not currently supported by the HiGHS methods.

    options : dict, optional
        A dictionary of solver options. All methods accept the following
        options:

        maxiter : int
            Maximum number of iterations to perform.
            Default: see method-specific documentation.
        disp : bool
            Set to ``True`` to print convergence messages.
            Default: ``False``.
        presolve : bool
            Set to ``False`` to disable automatic presolve.
            Default: ``True``.

        All methods except the HiGHS solvers also accept:

        tol : float
            A tolerance which determines when a residual is "close enough" to
            zero to be considered exactly zero.
        autoscale : bool
            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.
            Default: ``False``.
        rr : bool
            Set to ``False`` to disable automatic redundancy removal.
            Default: ``True``.
        rr_method : string
            Method used to identify and remove redundant rows from the
            equality constraint matrix after presolve. For problems with
            dense input, the available methods for redundancy removal are:

            "SVD":
                Repeatedly performs singular value decomposition on
                the matrix, detecting redundant rows based on nonzeros
                in the left singular vectors that correspond with
                zero singular values. May be fast when the matrix is
                nearly full rank.
            "pivot":
                Uses the algorithm presented in [5]_ to identify
                redundant rows.
            "ID":
                Uses a randomized interpolative decomposition.
                Identifies columns of the matrix transpose not used in
                a full-rank interpolative decomposition of the matrix.
            None:
                Uses "svd" if the matrix is nearly full rank, that is,
                the difference between the matrix rank and the number
                of rows is less than five. If not, uses "pivot". The
                behavior of this default is subject to change without
                prior notice.

            Default: None.
            For problems with sparse input, this option is ignored, and the
            pivot-based algorithm presented in [5]_ is used.

        For method-specific options, see
        :func:`show_options('linprog') <show_options>`.

    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.

    integrality : 1-D array or int, optional
        Indicates the type of integrality constraint on each decision variable.

        ``0`` : Continuous variable; no integrality constraint.

        ``1`` : Integer variable; decision variable must be an integer
        within `bounds`.

        ``2`` : Semi-continuous variable; decision variable must be within
        `bounds` or take value ``0``.

        ``3`` : Semi-integer variable; decision variable must be an integer
        within `bounds` or take value ``0``.

        By default, all variables are continuous.

        For mixed integrality constraints, supply an array of shape `c.shape`.
        To infer a constraint on each decision variable from shorter inputs,
        the argument will be broadcasted to `c.shape` using `np.broadcast_to`.

        This argument is currently used only by the ``'highs'`` method and
        ignored otherwise.

    Returns
    -------
    res : OptimizeResult
        A :class:`scipy.optimize.OptimizeResult` consisting of the fields
        below. Note that the return types of the fields may depend on whether
        the optimization was successful, therefore it is recommended to check
        `OptimizeResult.status` before relying on the other fields:

        x : 1-D array
            The values of the decision variables that minimizes the
            objective function while satisfying the constraints.
        fun : float
            The optimal value of the objective function ``c @ x``.
        slack : 1-D array
            The (nominally positive) values of the slack variables,
            ``b_ub - A_ub @ x``.
        con : 1-D array
            The (nominally zero) residuals of the equality constraints,
            ``b_eq - A_eq @ x``.
        success : bool
            ``True`` when the algorithm succeeds in finding an optimal
            solution.
        status : int
            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.

        nit : int
            The total number of iterations performed in all phases.
        message : str
            A string descriptor of the exit status of the algorithm.

    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.

    `'highs-ds'` and
    `'highs-ipm'` are interfaces to the
    HiGHS simplex and interior-point method solvers [13]_, respectively.
    `'highs'` (default) 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.
    The other solvers (`'interior-point'`, `'revised simplex'`, and
    `'simplex'`) are legacy methods and will be removed in SciPy 1.11.0.

    Method *highs-ds* is a wrapper of the C++ high performance dual
    revised simplex implementation (HSOL) [13]_, [14]_. Method *highs-ipm*
    is a wrapper of a C++ implementation of an **i**\ nterior-\ **p**\ oint
    **m**\ ethod [13]_; 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.

    .. versionadded:: 1.6.0

    Method *interior-point* uses the primal-dual path following algorithm
    as outlined in [4]_. 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.

    .. versionadded:: 1.0.0

    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.

    .. versionadded:: 1.3.0

    Method *simplex* uses a traditional, full-tableau implementation of
    Dantzig's simplex algorithm [1]_, [2]_ (*not* the
    Nelder-Mead simplex). This algorithm is included for backwards
    compatibility and educational purposes.

    .. versionadded:: 0.15.0

    Before applying *interior-point*, *revised simplex*, or *simplex*,
    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 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 [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.
    Optionally, the problem is automatically scaled via equilibration [12]_.
    The selected algorithm solves the standard form problem, and a
    postprocessing routine converts the result to a solution to the original
    problem.

    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.
    .. [11] Bartels, Richard H. "A stabilization of the simplex method."
            Journal in  Numerische Mathematik 16.5 (1971): 414-434.
    .. [12] Tomlin, J. A. "On scaling linear programming problems."
            Mathematical Programming Study 4 (1975): 146-166.
    .. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J.
            "HiGHS - high performance software for linear optimization."
            https://highs.dev/
    .. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised
            simplex method." Mathematical Programming Computation, 10 (1),
            119-142, 2018. DOI: 10.1007/s12532-017-0130-5

    Examples
    --------
    Consider the following problem:

    .. math::

        \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 :math:`-1`. Note also that the last
    constraint is really the simple bound :math:`-3 \leq x_1 \leq \infty`.
    Finally, since there are no bounds on :math:`x_0`, we must explicitly
    specify the bounds :math:`-\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:

    >>> from scipy.optimize import linprog
    >>> c = [-1, 4]
    >>> A = [[-3, 1], [1, 2]]
    >>> b = [6, 4]
    >>> x0_bounds = (None, None)
    >>> x1_bounds = (-3, None)
    >>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds])
    >>> res.fun
    -22.0
    >>> res.x
    array([10., -3.])
    >>> res.message
    'Optimization terminated successfully. (HiGHS Status 7: Optimal)'

    The marginals (AKA dual values / shadow prices / Lagrange multipliers)
    and residuals (slacks) are also available.

    >>> res.ineqlin
      residual: [ 3.900e+01  0.000e+00]
     marginals: [-0.000e+00 -1.000e+00]

    For example, because the marginal associated with the second inequality
    constraint is -1, we expect the optimal value of the objective function
    to decrease by ``eps`` if we add a small amount ``eps`` to the right hand
    side of the second inequality constraint:

    >>> eps = 0.05
    >>> b[1] += eps
    >>> linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun
    -22.05

    Also, because the residual on the first inequality constraint is 39, we
    can decrease the right hand side of the first constraint by 39 without
    affecting the optimal solution.

    >>> b = [6, 4]  # reset to original values
    >>> b[0] -= 39
    >>> linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun
    -22.0

    >