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Vehicle-Anti-Theft-Face-Rec.../venv/Lib/site-packages/scipy/optimize/_linprog.py

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Fixed database typo and removed unnecessary class identifier.
2020-10-14 14:10:37 +00:00
"""
A top-level linear programming interface. Currently this interface solves
linear programming problems via the Simplex and Interior-Point methods.
.. versionadded:: 0.15.0
Functions
---------
.. autosummary::
:toctree: generated/
linprog
linprog_verbose_callback
linprog_terse_callback
"""
import numpy as np
from .optimize import OptimizeResult, OptimizeWarning
from warnings import warn
from ._linprog_ip import _linprog_ip
from ._linprog_simplex import _linprog_simplex
from ._linprog_rs import _linprog_rs
from ._linprog_util import (
_parse_linprog, _presolve, _get_Abc, _postprocess, _LPProblem, _autoscale)
from copy import deepcopy
__all__ = ['linprog', 'linprog_verbose_callback', 'linprog_terse_callback']
__docformat__ = "restructuredtext en"
def linprog_verbose_callback(res):
"""
A sample callback function demonstrating the linprog callback interface.
This callback produces detailed output to sys.stdout before each iteration
and after the final iteration of the simplex algorithm.
Parameters
----------
res : A `scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
The independent variable vector which optimizes the linear
programming problem.
fun : float
Value of the objective function.
success : bool
True if the algorithm succeeded in finding an optimal solution.
slack : 1-D array
The values of the slack variables. Each slack variable corresponds
to an inequality constraint. If the slack is zero, then the
corresponding constraint is active.
con : 1-D array
The (nominally zero) residuals of the equality constraints, that is,
``b - A_eq @ x``
phase : int
The phase of the optimization being executed. In phase 1 a basic
feasible solution is sought and the T has an additional row
representing an alternate objective function.
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.
"""
x = res['x']
fun = res['fun']
phase = res['phase']
status = res['status']
nit = res['nit']
message = res['message']
complete = res['complete']
saved_printoptions = np.get_printoptions()
np.set_printoptions(linewidth=500,
formatter={'float': lambda x: "{0: 12.4f}".format(x)})
if status:
print('--------- Simplex Early Exit -------\n'.format(nit))
print('The simplex method exited early with status {0:d}'.format(status))
print(message)
elif complete:
print('--------- Simplex Complete --------\n')
print('Iterations required: {}'.format(nit))
else:
print('--------- Iteration {0:d} ---------\n'.format(nit))
if nit > 0:
if phase == 1:
print('Current Pseudo-Objective Value:')
else:
print('Current Objective Value:')
print('f = ', fun)
print()
print('Current Solution Vector:')
print('x = ', x)
print()
np.set_printoptions(**saved_printoptions)
def linprog_terse_callback(res):
"""
A sample callback function demonstrating the linprog callback interface.
This callback produces brief output to sys.stdout before each iteration
and after the final iteration of the simplex algorithm.
Parameters
----------
res : A `scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
The independent variable vector which optimizes the linear
programming problem.
fun : float
Value of the objective function.
success : bool
True if the algorithm succeeded in finding an optimal solution.
slack : 1-D array
The values of the slack variables. Each slack variable corresponds
to an inequality constraint. If the slack is zero, then the
corresponding constraint is active.
con : 1-D array
The (nominally zero) residuals of the equality constraints, that is,
``b - A_eq @ x``.
phase : int
The phase of the optimization being executed. In phase 1 a basic
feasible solution is sought and the T has an additional row
representing an alternate objective function.
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.
"""
nit = res['nit']
x = res['x']
if nit == 0:
print("Iter: X:")
print("{0: <5d} ".format(nit), end="")
print(x)
def linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None,
bounds=None, method='interior-point', callback=None,
options=None, x0=None):
r"""
Linear programming: minimize a linear objective function subject to linear
equality and inequality constraints.
Linear programming solves problems of the following form:
.. math::
\min_x \ & c^T x \\
\mbox{such that} \ & A_{ub} x \leq b_{ub},\\
& A_{eq} x = b_{eq},\\
& l \leq x \leq u ,
where :math:`x` is a vector of decision variables; :math:`c`,
:math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and
:math:`A_{ub}` and :math:`A_{eq}` are matrices.
Informally, that's:
minimize::
c @ x
such that::
A_ub @ x <= b_ub
A_eq @ x == b_eq
lb <= x <= ub
Note that by default ``lb = 0`` and ``ub = None`` unless specified with
``bounds``.
Parameters
----------
c : 1-D array
The coefficients of the linear objective function to be minimized.
A_ub : 2-D array, optional
The inequality constraint matrix. Each row of ``A_ub`` specifies the
coefficients of a linear inequality constraint on ``x``.
b_ub : 1-D array, optional
The inequality constraint vector. Each element represents an
upper bound on the corresponding value of ``A_ub @ x``.
A_eq : 2-D array, optional
The equality constraint matrix. Each row of ``A_eq`` specifies the
coefficients of a linear equality constraint on ``x``.
b_eq : 1-D array, optional
The equality constraint vector. Each element of ``A_eq @ x`` must equal
the corresponding element of ``b_eq``.
bounds : sequence, optional
A sequence of ``(min, max)`` pairs for each element in ``x``, defining
the minimum and maximum values of that decision variable. Use ``None`` to
indicate that there is no bound. By default, bounds are ``(0, None)``
(all decision variables are non-negative).
If a single tuple ``(min, max)`` is provided, then ``min`` and
``max`` will serve as bounds for all decision variables.
method : {'interior-point', 'revised simplex', 'simplex'}, optional
The algorithm used to solve the standard form problem.
:ref:`'interior-point' <optimize.linprog-interior-point>` (default),
:ref:`'revised simplex' <optimize.linprog-revised_simplex>`, and
:ref:`'simplex' <optimize.linprog-simplex>` (legacy)
are supported.
callback : callable, optional
If a callback function is provided, it will be called at least once per
iteration of the algorithm. The callback function must accept a single
`scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
The current solution vector.
fun : float
The current value of the objective function ``c @ x``.
success : bool
``True`` when the algorithm has completed successfully.
slack : 1-D array
The (nominally positive) values of the slack,
``b_ub - A_ub @ x``.
con : 1-D array
The (nominally zero) residuals of the equality constraints,
``b_eq - A_eq @ x``.
phase : int
The phase of the algorithm being executed.
status : int
An integer representing the status of the algorithm.
``0`` : Optimization proceeding nominally.
``1`` : Iteration limit reached.
``2`` : Problem appears to be infeasible.
``3`` : Problem appears to be unbounded.
``4`` : Numerical difficulties encountered.
nit : int
The current iteration number.
message : str
A string descriptor of the algorithm status.
options : dict, optional
A dictionary of solver options. All methods accept the following
options:
maxiter : int
Maximum number of iterations to perform.
Default: see method-specific documentation.
disp : bool
Set to ``True`` to print convergence messages.
Default: ``False``.
autoscale : bool
Set to ``True`` to automatically perform equilibration.
Consider using this option if the numerical values in the
constraints are separated by several orders of magnitude.
Default: ``False``.
presolve : bool
Set to ``False`` to disable automatic presolve.
Default: ``True``.
rr : bool
Set to ``False`` to disable automatic redundancy removal.
Default: ``True``.
For method-specific options, see
:func:`show_options('linprog') <show_options>`.
x0 : 1-D array, optional
Guess values of the decision variables, which will be refined by
the optimization algorithm. This argument is currently used only by the
'revised simplex' method, and can only be used if `x0` represents a
basic feasible solution.
Returns
-------
res : OptimizeResult
A :class:`scipy.optimize.OptimizeResult` consisting of the fields:
x : 1-D array
The values of the decision variables that minimizes the
objective function while satisfying the constraints.
fun : float
The optimal value of the objective function ``c @ x``.
slack : 1-D array
The (nominally positive) values of the slack variables,
``b_ub - A_ub @ x``.
con : 1-D array
The (nominally zero) residuals of the equality constraints,
``b_eq - A_eq @ x``.
success : bool
``True`` when the algorithm succeeds in finding an optimal
solution.
status : int
An integer representing the exit status of the algorithm.
``0`` : Optimization terminated successfully.
``1`` : Iteration limit reached.
``2`` : Problem appears to be infeasible.
``3`` : Problem appears to be unbounded.
``4`` : Numerical difficulties encountered.
nit : int
The total number of iterations performed in all phases.
message : str
A string descriptor of the exit status of the algorithm.
See Also
--------
show_options : Additional options accepted by the solvers.
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter.
:ref:`'interior-point' <optimize.linprog-interior-point>` is the default
as it is typically the fastest and most robust method.
:ref:`'revised simplex' <optimize.linprog-revised_simplex>` is more
accurate for the problems it solves.
:ref:`'simplex' <optimize.linprog-simplex>` is the legacy method and is
included for backwards compatibility and educational purposes.
Method *interior-point* uses the primal-dual path following algorithm
as outlined in [4]_. This algorithm supports sparse constraint matrices and
is typically faster than the simplex methods, especially for large, sparse
problems. Note, however, that the solution returned may be slightly less
accurate than those of the simplex methods and will not, in general,
correspond with a vertex of the polytope defined by the constraints.
.. versionadded:: 1.0.0
Method *revised simplex* uses the revised simplex method as described in
[9]_, except that a factorization [11]_ of the basis matrix, rather than
its inverse, is efficiently maintained and used to solve the linear systems
at each iteration of the algorithm.
.. versionadded:: 1.3.0
Method *simplex* uses a traditional, full-tableau implementation of
Dantzig's simplex algorithm [1]_, [2]_ (*not* the
Nelder-Mead simplex). This algorithm is included for backwards
compatibility and educational purposes.
.. versionadded:: 0.15.0
Before applying any method, a presolve procedure based on [8]_ attempts
to identify trivial infeasibilities, trivial unboundedness, and potential
problem simplifications. Specifically, it checks for:
- rows of zeros in ``A_eq`` or ``A_ub``, representing trivial constraints;
- columns of zeros in ``A_eq`` `and` ``A_ub``, representing unconstrained
variables;
- column singletons in ``A_eq``, representing fixed variables; and
- column singletons in ``A_ub``, representing simple bounds.
If presolve reveals that the problem is unbounded (e.g. an unconstrained
and unbounded variable has negative cost) or infeasible (e.g., a row of
zeros in ``A_eq`` corresponds with a nonzero in ``b_eq``), the solver
terminates with the appropriate status code. Note that presolve terminates
as soon as any sign of unboundedness is detected; consequently, a problem
may be reported as unbounded when in reality the problem is infeasible
(but infeasibility has not been detected yet). Therefore, if 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)
lp = _LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0)
lp, solver_options = _parse_linprog(lp, options)
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.
lp_o = deepcopy(lp)
# 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)
(lp, c0, x, undo, complete, status, message) = _presolve(lp, rr, tol)
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)
if not complete:
A, b, c, c0, x0 = _get_Abc(lp, c0, 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)
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