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� Z.d � Z/ dd�Z0dS )a^ HiGHS Linear Optimization Methods
Interface to HiGHS linear optimization software.
https://highs.dev/
.. versionadded:: 1.5.0
References
----------
.. [1] Q. Huangfu and J.A.J. Hall. "Parallelizing the dual revised simplex
method." Mathematical Programming Computation, 10 (1), 119-142,
2018. DOI: 10.1007/s12532-017-0130-5
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Solve the following linear programming problem using one of the HiGHS
solvers:
User-facing documentation is in _linprog_doc.py.
Parameters
----------
lp : _LPProblem
A ``scipy.optimize._linprog_util._LPProblem`` ``namedtuple``.
solver : "ipm" or "simplex" or None
Which HiGHS solver to use. If ``None``, "simplex" will be used.
Options
-------
maxiter : int
The maximum number of iterations to perform in either phase. For
``solver='ipm'``, this does not include the number of crossover
iterations. Default is the largest possible value for an ``int``
on the platform.
disp : bool
Set to ``True`` if indicators of optimization status are to be printed
to the console each iteration; default ``False``.
time_limit : float
The maximum time in seconds allotted to solve the problem; default is
the largest possible value for a ``double`` on the platform.
presolve : bool
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.
dual_feasibility_tolerance : double
Dual feasibility tolerance. Default is 1e-07.
The minimum of this and ``primal_feasibility_tolerance``
is used for the feasibility tolerance when ``solver='ipm'``.
primal_feasibility_tolerance : double
Primal feasibility tolerance. Default is 1e-07.
The minimum of this and ``dual_feasibility_tolerance``
is used for the feasibility tolerance when ``solver='ipm'``.
ipm_optimality_tolerance : double
Optimality tolerance for ``solver='ipm'``. Default is 1e-08.
Minimum possible value is 1e-12 and must be smaller than the largest
possible value for a ``double`` on the platform.
simplex_dual_edge_weight_strategy : str (default: None)
Strategy for simplex dual edge weights. The default, ``None``,
automatically selects one of the following.
``'dantzig'`` uses Dantzig's original strategy of choosing the most
negative reduced cost.
``'devex'`` uses the strategy described in [15]_.
``steepest`` uses the exact steepest edge strategy as described in
[16]_.
``'steepest-devex'`` begins with the exact steepest edge strategy
until the computation is too costly or inexact and then switches to
the devex method.
Curently, using ``None`` always selects ``'steepest-devex'``, but this
may change as new options become available.
mip_max_nodes : int
The maximum number of nodes allotted to solve the problem; default is
the largest possible value for a ``HighsInt`` on the platform.
Ignored if not using the MIP solver.
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.
Returns
-------
sol : dict
A dictionary consisting of the fields:
x : 1D 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 : 1D array
The (nominally positive) values of the slack,
``b_ub - A_ub @ x``.
con : 1D 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 or time limit reached.
``2`` : Problem appears to be infeasible.
``3`` : Problem appears to be unbounded.
``4`` : The HiGHS solver ran into a problem.
message : str
A string descriptor of the exit status of the algorithm.
nit : int
The total number of iterations performed.
For ``solver='simplex'``, this includes iterations in all
phases. For ``solver='ipm'``, this does not include
crossover iterations.
crossover_nit : int
The number of primal/dual pushes performed during the
crossover routine for ``solver='ipm'``. This is ``0``
for ``solver='simplex'``.
ineqlin : OptimizeResult
Solution and sensitivity information corresponding to the
inequality constraints, `b_ub`. A dictionary consisting of the
fields:
residual : np.ndnarray
The (nominally positive) values of the slack variables,
``b_ub - A_ub @ x``. This quantity is also commonly
referred to as "slack".
marginals : np.ndarray
The sensitivity (partial derivative) of the objective
function with respect to the right-hand side of the
inequality constraints, `b_ub`.
eqlin : OptimizeResult
Solution and sensitivity information corresponding to the
equality constraints, `b_eq`. A dictionary consisting of the
fields:
residual : np.ndarray
The (nominally zero) residuals of the equality constraints,
``b_eq - A_eq @ x``.
marginals : np.ndarray
The sensitivity (partial derivative) of the objective
function with respect to the right-hand side of the
equality constraints, `b_eq`.
lower, upper : OptimizeResult
Solution and sensitivity information corresponding to the
lower and upper bounds on decision variables, `bounds`.
residual : np.ndarray
The (nominally positive) values of the quantity
``x - lb`` (lower) or ``ub - x`` (upper).
marginals : np.ndarray
The sensitivity (partial derivative) of the objective
function with respect to the lower and upper
`bounds`.
mip_node_count : int
The number of subproblems or "nodes" solved by the MILP
solver. Only present when `integrality` is not `None`.
mip_dual_bound : float
The MILP solver's final estimate of the lower bound on the
optimal solution. Only present when `integrality` is not
`None`.
mip_gap : float
The difference between the final objective function value
and the final dual bound, scaled by the final objective
function value. Only present when `integrality` is not
`None`.
Notes
-----
The result fields `ineqlin`, `eqlin`, `lower`, and `upper` all contain
`marginals`, or partial derivatives of the objective function with respect
to the right-hand side of each constraint. These partial derivatives are
also referred to as "Lagrange multipliers", "dual values", and
"shadow prices". The sign convention of `marginals` is opposite that
of Lagrange multipliers produced by many nonlinear solvers.
References
----------
.. [15] Harris, Paula MJ. "Pivot selection methods of the Devex LP code."
Mathematical programming 5.1 (1973): 1-28.
.. [16] Goldfarb, Donald, and John Ker Reid. "A practicable steepest-edge
simplex algorithm." Mathematical Programming 12.1 (1977): 361-371.
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