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670 lines
27 KiB
Python
670 lines
27 KiB
Python
"""
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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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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_highs import _linprog_highs
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from ._linprog_doc import (_linprog_highs_doc, _linprog_ip_doc,
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_linprog_rs_doc, _linprog_simplex_doc,
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_linprog_highs_ipm_doc, _linprog_highs_ds_doc)
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from ._linprog_util import (
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_parse_linprog, _presolve, _get_Abc, _LPProblem, _autoscale,
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_postsolve, _check_result, _display_summary)
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from copy import deepcopy
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__all__ = ['linprog', 'linprog_verbose_callback', 'linprog_terse_callback']
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__docformat__ = "restructuredtext en"
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LINPROG_METHODS = ['simplex', 'revised simplex', 'interior-point', 'highs', 'highs-ds', 'highs-ipm']
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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 : 1-D 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 : 1-D 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 : 1-D 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 : 1-D 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 : 1-D 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 : 1-D 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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Alternatively, 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 : 1-D array
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The coefficients of the linear objective function to be minimized.
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A_ub : 2-D 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 : 1-D 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 : 2-D 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 : 1-D 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``
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to indicate that there is no bound. By default, bounds are
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``(0, None)`` (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 : str, optional
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The algorithm used to solve the standard form problem.
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:ref:`'highs-ds' <optimize.linprog-highs-ds>`,
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:ref:`'highs-ipm' <optimize.linprog-highs-ipm>`,
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:ref:`'highs' <optimize.linprog-highs>`,
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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 : 1-D 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 : 1-D 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 : 1-D 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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Callback functions are not currently supported by the HiGHS methods.
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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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presolve : bool
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Set to ``False`` to disable automatic presolve.
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Default: ``True``.
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All methods except the HiGHS solvers also accept:
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tol : float
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A tolerance which determines when a residual is "close enough" to
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zero to be considered exactly zero.
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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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rr : bool
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Set to ``False`` to disable automatic redundancy removal.
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Default: ``True``.
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rr_method : string
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Method used to identify and remove redundant rows from the
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equality constraint matrix after presolve. For problems with
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dense input, the available methods for redundancy removal are:
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"SVD":
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Repeatedly performs singular value decomposition on
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the matrix, detecting redundant rows based on nonzeros
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in the left singular vectors that correspond with
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zero singular values. May be fast when the matrix is
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nearly full rank.
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"pivot":
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Uses the algorithm presented in [5]_ to identify
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redundant rows.
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"ID":
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Uses a randomized interpolative decomposition.
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Identifies columns of the matrix transpose not used in
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a full-rank interpolative decomposition of the matrix.
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None:
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Uses "svd" if the matrix is nearly full rank, that is,
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the difference between the matrix rank and the number
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of rows is less than five. If not, uses "pivot". The
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behavior of this default is subject to change without
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prior notice.
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Default: None.
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For problems with sparse input, this option is ignored, and the
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pivot-based algorithm presented in [5]_ is used.
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For method-specific options, see
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:func:`show_options('linprog') <show_options>`.
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x0 : 1-D 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 : 1-D 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 : 1-D 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 : 1-D 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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`'highs-ds'` and
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`'highs-ipm'` are interfaces to the
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HiGHS simplex and interior-point method solvers [13]_, respectively.
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`'highs'` chooses between
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the two automatically. These are the fastest linear
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programming solvers in SciPy, especially for large, sparse problems;
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which of these two is faster is problem-dependent.
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`'interior-point'` is the default
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as it was the fastest and most robust method before the recent
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addition of the HiGHS solvers.
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`'revised simplex'` is more
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accurate than interior-point for the problems it solves.
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`'simplex'` is the legacy method and is
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included for backwards compatibility and educational purposes.
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Method *highs-ds* is a wrapper of the C++ high performance dual
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revised simplex implementation (HSOL) [13]_, [14]_. Method *highs-ipm*
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is a wrapper of a C++ implementation of an **i**\ nterior-\ **p**\ oint
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**m**\ ethod [13]_; it features a crossover routine, so it is as accurate
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as a simplex solver. Method *highs* chooses between the two automatically.
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For new code involving `linprog`, we recommend explicitly choosing one of
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these three method values.
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.. versionadded:: 1.5.0
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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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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 *interior-point*, *revised simplex*, or *simplex*,
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a presolve procedure based on [8]_ attempts
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to identify trivial infeasibilities, trivial unboundedness, and potential
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problem simplifications. Specifically, it checks for:
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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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- 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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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
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as soon as any sign of unboundedness is detected; consequently, a problem
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may be reported as unbounded when in reality the problem is infeasible
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(but infeasibility has not been detected yet). Therefore, if it is
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important to know whether the problem is actually infeasible, solve the
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problem again with option ``presolve=False``.
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If neither infeasibility nor unboundedness are detected in a single pass
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of the presolve, bounds are tightened where possible and fixed
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variables are removed from the problem. Then, linearly dependent rows
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of the ``A_eq`` matrix are removed, (unless they represent an
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infeasibility) to avoid numerical difficulties in the primary solve
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routine. Note that rows that are nearly linearly dependent (within a
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prescribed tolerance) may also be removed, which can change the optimal
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solution in rare cases. If this is a concern, eliminate redundancy from
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your problem formulation and run with option ``rr=False`` or
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``presolve=False``.
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Several potential improvements can be made here: additional presolve
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checks outlined in [8]_ should be implemented, the presolve routine should
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be run multiple times (until no further simplifications can be made), and
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more of the efficiency improvements from [5]_ should be implemented in the
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redundancy removal routines.
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After presolve, the problem is transformed to standard form by converting
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the (tightened) simple bounds to upper bound constraints, introducing
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non-negative slack variables for inequality constraints, and expressing
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unbounded variables as the difference between two non-negative variables.
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Optionally, the problem is automatically scaled via equilibration [12]_.
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The selected algorithm solves the standard form problem, and a
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postprocessing routine converts the result to a solution to the original
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problem.
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References
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----------
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.. [1] Dantzig, George B., Linear programming and extensions. Rand
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Corporation Research Study Princeton Univ. Press, Princeton, NJ,
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1963
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.. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to
|
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Mathematical Programming", McGraw-Hill, Chapter 4.
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.. [3] Bland, Robert G. New finite pivoting rules for the simplex method.
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Mathematics of Operations Research (2), 1977: pp. 103-107.
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.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
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optimizer for linear programming: an implementation of the
|
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homogeneous algorithm." High performance optimization. Springer US,
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2000. 197-232.
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.. [5] Andersen, Erling D. "Finding all linearly dependent rows in
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large-scale linear programming." Optimization Methods and Software
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|
6.3 (1995): 219-227.
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.. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear
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|
Programming based on Newton's Method." Unpublished Course Notes,
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|
March 2004. Available 2/25/2017 at
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|
https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf
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.. [7] Fourer, Robert. "Solving Linear Programs by Interior-Point Methods."
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Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at
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http://www.4er.org/CourseNotes/Book%20B/B-III.pdf
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.. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear
|
|
programming." Mathematical Programming 71.2 (1995): 221-245.
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.. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear
|
|
programming." Athena Scientific 1 (1997): 997.
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|
.. [10] Andersen, Erling D., et al. Implementation of interior point
|
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methods for large scale linear programming. HEC/Universite de
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|
Geneve, 1996.
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.. [11] Bartels, Richard H. "A stabilization of the simplex method."
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Journal in Numerische Mathematik 16.5 (1971): 414-434.
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.. [12] Tomlin, J. A. "On scaling linear programming problems."
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Mathematical Programming Study 4 (1975): 146-166.
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.. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J.
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"HiGHS - high performance software for linear optimization."
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Accessed 4/16/2020 at https://www.maths.ed.ac.uk/hall/HiGHS/#guide
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.. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised
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simplex method." Mathematical Programming Computation, 10 (1),
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119-142, 2018. DOI: 10.1007/s12532-017-0130-5
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Examples
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--------
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Consider the following problem:
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.. math::
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\min_{x_0, x_1} \ -x_0 + 4x_1 & \\
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\mbox{such that} \ -3x_0 + x_1 & \leq 6,\\
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-x_0 - 2x_1 & \geq -4,\\
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x_1 & \geq -3.
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The problem is not presented in the form accepted by `linprog`. This is
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easily remedied by converting the "greater than" inequality
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constraint to a "less than" inequality constraint by
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multiplying both sides by a factor of :math:`-1`. Note also that the last
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constraint is really the simple bound :math:`-3 \leq x_1 \leq \infty`.
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Finally, since there are no bounds on :math:`x_0`, we must explicitly
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specify the bounds :math:`-\infty \leq x_0 \leq \infty`, as the
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default is for variables to be non-negative. After collecting coeffecients
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into arrays and tuples, the input for this problem is:
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>>> c = [-1, 4]
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>>> A = [[-3, 1], [1, 2]]
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>>> b = [6, 4]
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>>> x0_bounds = (None, None)
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>>> x1_bounds = (-3, None)
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>>> from scipy.optimize import linprog
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>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds])
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Note that the default method for `linprog` is 'interior-point', which is
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approximate by nature.
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>>> print(res)
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con: array([], dtype=float64)
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fun: -21.99999984082494 # may vary
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message: 'Optimization terminated successfully.'
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nit: 6 # may vary
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slack: array([3.89999997e+01, 8.46872439e-08] # may vary
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status: 0
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success: True
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x: array([ 9.99999989, -2.99999999]) # may vary
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If you need greater accuracy, try 'revised simplex'.
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>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds], method='revised simplex')
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>>> print(res)
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con: array([], dtype=float64)
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fun: -22.0 # may vary
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message: 'Optimization terminated successfully.'
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nit: 1 # may vary
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slack: array([39., 0.]) # may vary
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status: 0
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success: True
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x: array([10., -3.]) # may vary
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"""
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meth = method.lower()
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methods = {"simplex", "revised simplex", "interior-point",
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"highs", "highs-ds", "highs-ipm"}
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if meth not in methods:
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raise ValueError(f"Unknown solver '{method}'")
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if x0 is not None and meth != "revised simplex":
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warning_message = "x0 is used only when method is 'revised simplex'. "
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warn(warning_message, OptimizeWarning)
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lp = _LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0)
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lp, solver_options = _parse_linprog(lp, options)
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tol = solver_options.get('tol', 1e-9)
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# Give unmodified problem to HiGHS
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if meth.startswith('highs'):
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if callback is not None:
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raise NotImplementedError("HiGHS solvers do not support the "
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"callback interface.")
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highs_solvers = {'highs-ipm': 'ipm', 'highs-ds': 'simplex',
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'highs': None}
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sol = _linprog_highs(lp, solver=highs_solvers[meth],
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**solver_options)
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sol['status'], sol['message'] = (
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_check_result(sol['x'], sol['fun'], sol['status'], sol['slack'],
|
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sol['con'], lp.bounds, tol, sol['message']))
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sol['success'] = sol['status'] == 0
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return OptimizeResult(sol)
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|
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iteration = 0
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complete = False # will become True if solved in presolve
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undo = []
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|
|
|
# Keep the original arrays to calculate slack/residuals for original
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# problem.
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|
lp_o = deepcopy(lp)
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|
|
|
# Solve trivial problem, eliminate variables, tighten bounds, etc.
|
|
rr_method = solver_options.pop('rr_method', None) # need to pop these;
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rr = solver_options.pop('rr', True) # they're not passed to methods
|
|
c0 = 0 # we might get a constant term in the objective
|
|
if solver_options.pop('presolve', True):
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|
(lp, c0, x, undo, complete, status, message) = _presolve(lp, rr,
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rr_method,
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tol)
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|
|
|
C, b_scale = 1, 1 # for trivial unscaling if autoscale is not used
|
|
postsolve_args = (lp_o._replace(bounds=lp.bounds), undo, C, b_scale)
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|
|
|
if not complete:
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|
A, b, c, c0, x0 = _get_Abc(lp, c0)
|
|
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)
|
|
|
|
# Eliminate artificial variables, re-introduce presolved variables, etc.
|
|
disp = solver_options.get('disp', False)
|
|
|
|
x, fun, slack, con = _postsolve(x, postsolve_args, complete)
|
|
|
|
status, message = _check_result(x, fun, status, slack, con, lp_o.bounds, tol, message)
|
|
|
|
if disp:
|
|
_display_summary(message, status, fun, iteration)
|
|
|
|
sol = {
|
|
'x': x,
|
|
'fun': fun,
|
|
'slack': slack,
|
|
'con': con,
|
|
'status': status,
|
|
'message': message,
|
|
'nit': iteration,
|
|
'success': status == 0}
|
|
|
|
return OptimizeResult(sol)
|