582 lines
23 KiB
Python
582 lines
23 KiB
Python
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"""
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A top-level linear programming interface. Currently this interface solves
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linear programming problems via the Simplex and Interior-Point methods.
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.. versionadded:: 0.15.0
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Functions
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---------
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.. autosummary::
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:toctree: generated/
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linprog
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linprog_verbose_callback
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linprog_terse_callback
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"""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from .optimize import OptimizeResult, OptimizeWarning
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from warnings import warn
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from ._linprog_ip import _linprog_ip
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from ._linprog_simplex import _linprog_simplex
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from ._linprog_rs import _linprog_rs
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from ._linprog_util import (
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_parse_linprog, _presolve, _get_Abc, _postprocess, _autoscale, _unscale
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)
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__all__ = ['linprog', 'linprog_verbose_callback', 'linprog_terse_callback']
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__docformat__ = "restructuredtext en"
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def linprog_verbose_callback(res):
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"""
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A sample callback function demonstrating the linprog callback interface.
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This callback produces detailed output to sys.stdout before each iteration
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and after the final iteration of the simplex algorithm.
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Parameters
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----------
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res : A `scipy.optimize.OptimizeResult` consisting of the following fields:
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x : 1D array
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The independent variable vector which optimizes the linear
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programming problem.
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fun : float
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Value of the objective function.
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success : bool
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True if the algorithm succeeded in finding an optimal solution.
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slack : 1D array
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The values of the slack variables. Each slack variable corresponds
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to an inequality constraint. If the slack is zero, then the
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corresponding constraint is active.
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con : 1D array
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The (nominally zero) residuals of the equality constraints, that is,
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``b - A_eq @ x``
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phase : int
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The phase of the optimization being executed. In phase 1 a basic
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feasible solution is sought and the T has an additional row
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representing an alternate objective function.
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status : int
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An integer representing the exit status of the optimization::
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0 : Optimization terminated successfully
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1 : Iteration limit reached
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2 : Problem appears to be infeasible
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3 : Problem appears to be unbounded
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4 : Serious numerical difficulties encountered
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nit : int
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The number of iterations performed.
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message : str
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A string descriptor of the exit status of the optimization.
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"""
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x = res['x']
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fun = res['fun']
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phase = res['phase']
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status = res['status']
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nit = res['nit']
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message = res['message']
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complete = res['complete']
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saved_printoptions = np.get_printoptions()
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np.set_printoptions(linewidth=500,
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formatter={'float': lambda x: "{0: 12.4f}".format(x)})
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if status:
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print('--------- Simplex Early Exit -------\n'.format(nit))
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print('The simplex method exited early with status {0:d}'.format(status))
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print(message)
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elif complete:
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print('--------- Simplex Complete --------\n')
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print('Iterations required: {}'.format(nit))
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else:
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print('--------- Iteration {0:d} ---------\n'.format(nit))
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if nit > 0:
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if phase == 1:
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print('Current Pseudo-Objective Value:')
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else:
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print('Current Objective Value:')
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print('f = ', fun)
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print()
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print('Current Solution Vector:')
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print('x = ', x)
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print()
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np.set_printoptions(**saved_printoptions)
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def linprog_terse_callback(res):
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"""
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A sample callback function demonstrating the linprog callback interface.
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This callback produces brief output to sys.stdout before each iteration
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and after the final iteration of the simplex algorithm.
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Parameters
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----------
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res : A `scipy.optimize.OptimizeResult` consisting of the following fields:
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x : 1D array
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The independent variable vector which optimizes the linear
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programming problem.
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fun : float
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Value of the objective function.
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success : bool
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True if the algorithm succeeded in finding an optimal solution.
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slack : 1D array
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The values of the slack variables. Each slack variable corresponds
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to an inequality constraint. If the slack is zero, then the
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corresponding constraint is active.
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con : 1D array
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The (nominally zero) residuals of the equality constraints, that is,
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``b - A_eq @ x``.
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phase : int
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The phase of the optimization being executed. In phase 1 a basic
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feasible solution is sought and the T has an additional row
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representing an alternate objective function.
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status : int
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An integer representing the exit status of the optimization::
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0 : Optimization terminated successfully
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1 : Iteration limit reached
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2 : Problem appears to be infeasible
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3 : Problem appears to be unbounded
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4 : Serious numerical difficulties encountered
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nit : int
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The number of iterations performed.
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message : str
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A string descriptor of the exit status of the optimization.
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"""
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nit = res['nit']
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x = res['x']
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if nit == 0:
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print("Iter: X:")
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print("{0: <5d} ".format(nit), end="")
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print(x)
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def linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None,
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bounds=None, method='interior-point', callback=None,
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options=None, x0=None):
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r"""
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Linear programming: minimize a linear objective function subject to linear
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equality and inequality constraints.
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Linear programming solves problems of the following form:
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.. math::
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\min_x \ & c^T x \\
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\mbox{such that} \ & A_{ub} x \leq b_{ub},\\
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& A_{eq} x = b_{eq},\\
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& l \leq x \leq u ,
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where :math:`x` is a vector of decision variables; :math:`c`,
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:math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and
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:math:`A_{ub}` and :math:`A_{eq}` are matrices.
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Informally, that's:
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minimize::
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c @ x
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such that::
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A_ub @ x <= b_ub
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A_eq @ x == b_eq
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lb <= x <= ub
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Note that by default ``lb = 0`` and ``ub = None`` unless specified with
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``bounds``.
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Parameters
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----------
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c : 1D array
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The coefficients of the linear objective function to be minimized.
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A_ub : 2D array, optional
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The inequality constraint matrix. Each row of ``A_ub`` specifies the
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coefficients of a linear inequality constraint on ``x``.
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b_ub : 1D array, optional
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The inequality constraint vector. Each element represents an
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upper bound on the corresponding value of ``A_ub @ x``.
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A_eq : 2D array, optional
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The equality constraint matrix. Each row of ``A_eq`` specifies the
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coefficients of a linear equality constraint on ``x``.
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b_eq : 1D array, optional
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The equality constraint vector. Each element of ``A_eq @ x`` must equal
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the corresponding element of ``b_eq``.
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bounds : sequence, optional
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A sequence of ``(min, max)`` pairs for each element in ``x``, defining
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the minimum and maximum values of that decision variable. Use ``None`` to
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indicate that there is no bound. By default, bounds are ``(0, None)``
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(all decision variables are non-negative).
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If a single tuple ``(min, max)`` is provided, then ``min`` and
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``max`` will serve as bounds for all decision variables.
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method : {'interior-point', 'revised simplex', 'simplex'}, optional
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The algorithm used to solve the standard form problem.
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:ref:`'interior-point' <optimize.linprog-interior-point>` (default),
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:ref:`'revised simplex' <optimize.linprog-revised_simplex>`, and
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:ref:`'simplex' <optimize.linprog-simplex>` (legacy)
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are supported.
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callback : callable, optional
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If a callback function is provided, it will be called at least once per
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iteration of the algorithm. The callback function must accept a single
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`scipy.optimize.OptimizeResult` consisting of the following fields:
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x : 1D array
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The current solution vector.
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fun : float
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The current value of the objective function ``c @ x``.
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success : bool
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``True`` when the algorithm has completed successfully.
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slack : 1D array
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The (nominally positive) values of the slack,
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``b_ub - A_ub @ x``.
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con : 1D array
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The (nominally zero) residuals of the equality constraints,
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``b_eq - A_eq @ x``.
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phase : int
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The phase of the algorithm being executed.
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status : int
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An integer representing the status of the algorithm.
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``0`` : Optimization proceeding nominally.
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``1`` : Iteration limit reached.
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``2`` : Problem appears to be infeasible.
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``3`` : Problem appears to be unbounded.
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``4`` : Numerical difficulties encountered.
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nit : int
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The current iteration number.
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message : str
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A string descriptor of the algorithm status.
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options : dict, optional
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A dictionary of solver options. All methods accept the following
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options:
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maxiter : int
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Maximum number of iterations to perform.
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Default: see method-specific documentation.
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disp : bool
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Set to ``True`` to print convergence messages.
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Default: ``False``.
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autoscale : bool
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Set to ``True`` to automatically perform equilibration.
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Consider using this option if the numerical values in the
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constraints are separated by several orders of magnitude.
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Default: ``False``.
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presolve : bool
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Set to ``False`` to disable automatic presolve.
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Default: ``True``.
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rr : bool
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Set to ``False`` to disable automatic redundancy removal.
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Default: ``True``.
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For method-specific options, see
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:func:`show_options('linprog') <show_options>`.
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x0 : 1D array, optional
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Guess values of the decision variables, which will be refined by
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the optimization algorithm. This argument is currently used only by the
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'revised simplex' method, and can only be used if `x0` represents a
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basic feasible solution.
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Returns
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-------
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res : OptimizeResult
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A :class:`scipy.optimize.OptimizeResult` consisting of the fields:
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x : 1D array
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The values of the decision variables that minimizes the
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objective function while satisfying the constraints.
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fun : float
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The optimal value of the objective function ``c @ x``.
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slack : 1D array
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The (nominally positive) values of the slack variables,
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``b_ub - A_ub @ x``.
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con : 1D array
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The (nominally zero) residuals of the equality constraints,
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``b_eq - A_eq @ x``.
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success : bool
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``True`` when the algorithm succeeds in finding an optimal
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solution.
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status : int
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An integer representing the exit status of the algorithm.
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``0`` : Optimization terminated successfully.
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``1`` : Iteration limit reached.
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``2`` : Problem appears to be infeasible.
|
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``3`` : Problem appears to be unbounded.
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``4`` : Numerical difficulties encountered.
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nit : int
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The total number of iterations performed in all phases.
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message : str
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A string descriptor of the exit status of the algorithm.
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See Also
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--------
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show_options : Additional options accepted by the solvers.
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Notes
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-----
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This section describes the available solvers that can be selected by the
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'method' parameter.
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:ref:`'interior-point' <optimize.linprog-interior-point>` is the default
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as it is typically the fastest and most robust method.
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:ref:`'revised simplex' <optimize.linprog-revised_simplex>` is more
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accurate for the problems it solves.
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:ref:`'simplex' <optimize.linprog-simplex>` is the legacy method and is
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included for backwards compatibility and educational purposes.
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Method *interior-point* uses the primal-dual path following algorithm
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as outlined in [4]_. This algorithm supports sparse constraint matrices and
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is typically faster than the simplex methods, especially for large, sparse
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problems. Note, however, that the solution returned may be slightly less
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accurate than those of the simplex methods and will not, in general,
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correspond with a vertex of the polytope defined by the constraints.
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.. versionadded:: 1.0.0
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Method *revised simplex* uses the revised simplex method as described in
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[9]_, except that a factorization [11]_ of the basis matrix, rather than
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||
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its inverse, is efficiently maintained and used to solve the linear systems
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at each iteration of the algorithm.
|
||
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.. versionadded:: 1.3.0
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Method *simplex* uses a traditional, full-tableau implementation of
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Dantzig's simplex algorithm [1]_, [2]_ (*not* the
|
||
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Nelder-Mead simplex). This algorithm is included for backwards
|
||
|
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compatibility and educational purposes.
|
||
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.. versionadded:: 0.15.0
|
||
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|
||
|
|
Before applying any method, a presolve procedure based on [8]_ attempts
|
||
|
|
to identify trivial infeasibilities, trivial unboundedness, and potential
|
||
|
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problem simplifications. Specifically, it checks for:
|
||
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||
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- rows of zeros in ``A_eq`` or ``A_ub``, representing trivial constraints;
|
||
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- columns of zeros in ``A_eq`` `and` ``A_ub``, representing unconstrained
|
||
|
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variables;
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||
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- column singletons in ``A_eq``, representing fixed variables; and
|
||
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- column singletons in ``A_ub``, representing simple bounds.
|
||
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If presolve reveals that the problem is unbounded (e.g. an unconstrained
|
||
|
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and unbounded variable has negative cost) or infeasible (e.g. a row of
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||
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zeros in ``A_eq`` corresponds with a nonzero in ``b_eq``), the solver
|
||
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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.
|
||
|
|
|
||
|
|
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:
|
||
|
|
|
||
|
|
>>> 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])
|
||
|
|
|
||
|
|
Note that the default method for `linprog` is 'interior-point', which is
|
||
|
|
approximate by nature.
|
||
|
|
|
||
|
|
>>> print(res)
|
||
|
|
con: array([], dtype=float64)
|
||
|
|
fun: -21.99999984082494 # may vary
|
||
|
|
message: 'Optimization terminated successfully.'
|
||
|
|
nit: 6 # may vary
|
||
|
|
slack: array([3.89999997e+01, 8.46872439e-08] # may vary
|
||
|
|
status: 0
|
||
|
|
success: True
|
||
|
|
x: array([ 9.99999989, -2.99999999]) # may vary
|
||
|
|
|
||
|
|
If you need greater accuracy, try 'revised simplex'.
|
||
|
|
|
||
|
|
>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds], method='revised simplex')
|
||
|
|
>>> print(res)
|
||
|
|
con: array([], dtype=float64)
|
||
|
|
fun: -22.0 # may vary
|
||
|
|
message: 'Optimization terminated successfully.'
|
||
|
|
nit: 1 # may vary
|
||
|
|
slack: array([39., 0.]) # may vary
|
||
|
|
status: 0
|
||
|
|
success: True
|
||
|
|
x: array([10., -3.]) # may vary
|
||
|
|
|
||
|
|
"""
|
||
|
|
meth = method.lower()
|
||
|
|
|
||
|
|
if x0 is not None and meth != "revised simplex":
|
||
|
|
warning_message = "x0 is used only when method is 'revised simplex'. "
|
||
|
|
warn(warning_message, OptimizeWarning)
|
||
|
|
|
||
|
|
c, A_ub, b_ub, A_eq, b_eq, bounds, solver_options, x0 = _parse_linprog(
|
||
|
|
c, A_ub, b_ub, A_eq, b_eq, bounds, options, x0)
|
||
|
|
tol = solver_options.get('tol', 1e-9)
|
||
|
|
|
||
|
|
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, x0, undo, complete, status,
|
||
|
|
message) = _presolve(c, A_ub, b_ub, A_eq, b_eq, bounds, x0, rr, tol)
|
||
|
|
|
||
|
|
C, b_scale = 1, 1 # for trivial unscaling if autoscale is not used
|
||
|
|
postsolve_args = (c_o, A_ub_o, b_ub_o, A_eq_o, b_eq_o, bounds, undo,
|
||
|
|
C, b_scale)
|
||
|
|
|
||
|
|
if not complete:
|
||
|
|
A, b, c, c0, x0 = _get_Abc(c, c0, A_ub, b_ub, A_eq,
|
||
|
|
b_eq, bounds, x0, undo)
|
||
|
|
if solver_options.pop('autoscale', False):
|
||
|
|
A, b, c, x0, C, b_scale = _autoscale(A, b, c, x0)
|
||
|
|
postsolve_args = postsolve_args[:-2] + (C, b_scale)
|
||
|
|
|
||
|
|
if meth == 'simplex':
|
||
|
|
x, status, message, iteration = _linprog_simplex(
|
||
|
|
c, c0=c0, A=A, b=b, callback=callback,
|
||
|
|
postsolve_args=postsolve_args, **solver_options)
|
||
|
|
elif meth == 'interior-point':
|
||
|
|
x, status, message, iteration = _linprog_ip(
|
||
|
|
c, c0=c0, A=A, b=b, callback=callback,
|
||
|
|
postsolve_args=postsolve_args, **solver_options)
|
||
|
|
elif meth == 'revised simplex':
|
||
|
|
x, status, message, iteration = _linprog_rs(
|
||
|
|
c, c0=c0, A=A, b=b, x0=x0, callback=callback,
|
||
|
|
postsolve_args=postsolve_args, **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, postsolve_args,
|
||
|
|
complete, 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)
|