"""
A top-level linear programming interface.
.. 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_highs import _linprog_highs
from ._linprog_ip import _linprog_ip
from ._linprog_simplex import _linprog_simplex
from ._linprog_rs import _linprog_rs
from ._linprog_doc import (_linprog_highs_doc, _linprog_ip_doc, # noqa: F401
_linprog_rs_doc, _linprog_simplex_doc,
_linprog_highs_ipm_doc, _linprog_highs_ds_doc)
from ._linprog_util import (
_parse_linprog, _presolve, _get_Abc, _LPProblem, _autoscale,
_postsolve, _check_result, _display_summary)
from copy import deepcopy
__all__ = ['linprog', 'linprog_verbose_callback', 'linprog_terse_callback']
__docformat__ = "restructuredtext en"
LINPROG_METHODS = [
'simplex', 'revised simplex', 'interior-point', 'highs', 'highs-ds', 'highs-ipm'
]
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: f"{x: 12.4f}"})
if status:
print('--------- Simplex Early Exit -------\n')
print(f'The simplex method exited early with status {status:d}')
print(message)
elif complete:
print('--------- Simplex Complete --------\n')
print(f'Iterations required: {nit}')
else:
print(f'--------- Iteration {nit:d} ---------\n')
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(f"{nit: <5d} ", end="")
print(x)
def linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None,
bounds=(0, None), method='highs', callback=None,
options=None, x0=None, integrality=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.
Alternatively, 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``. Other bounds can be
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.
If a single tuple ``(min, max)`` is provided, then ``min`` and ``max``
will serve as bounds for all decision variables.
Use ``None`` to indicate that there is no bound. For instance, the
default bound ``(0, None)`` means that all decision variables are
non-negative, and the pair ``(None, None)`` means no bounds at all,
i.e. all variables are allowed to be any real.
method : str, optional
The algorithm used to solve the standard form problem.
The following are supported.
- :ref:`'highs' <optimize.linprog-highs>` (default)
- :ref:`'highs-ds' <optimize.linprog-highs-ds>`
- :ref:`'highs-ipm' <optimize.linprog-highs-ipm>`
- :ref:`'interior-point' <optimize.linprog-interior-point>` (legacy)
- :ref:`'revised simplex' <optimize.linprog-revised_simplex>` (legacy)
- :ref:`'simplex' <optimize.linprog-simplex>` (legacy)
The legacy methods are deprecated and will be removed in SciPy 1.11.0.
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.
Callback functions are not currently supported by the HiGHS methods.
options : dict, optional
A dictionary of solver options. All methods accept the following
options:
maxiter : int
Maximum number of iterations to perform.
Default: see method-specific documentation.
disp : bool
Set to ``True`` to print convergence messages.
Default: ``False``.
presolve : bool
Set to ``False`` to disable automatic presolve.
Default: ``True``.
All methods except the HiGHS solvers also accept:
tol : float
A tolerance which determines when a residual is "close enough" to
zero to be considered exactly zero.
autoscale : bool
Set to ``True`` to automatically perform equilibration.
Consider using this option if the numerical values in the
constraints are separated by several orders of magnitude.
Default: ``False``.
rr : bool
Set to ``False`` to disable automatic redundancy removal.
Default: ``True``.
rr_method : string
Method used to identify and remove redundant rows from the
equality constraint matrix after presolve. For problems with
dense input, the available methods for redundancy removal are:
``SVD``:
Repeatedly performs singular value decomposition on
the matrix, detecting redundant rows based on nonzeros
in the left singular vectors that correspond with
zero singular values. May be fast when the matrix is
nearly full rank.
``pivot``:
Uses the algorithm presented in [5]_ to identify
redundant rows.
``ID``:
Uses a randomized interpolative decomposition.
Identifies columns of the matrix transpose not used in
a full-rank interpolative decomposition of the matrix.
``None``:
Uses ``svd`` if the matrix is nearly full rank, that is,
the difference between the matrix rank and the number
of rows is less than five. If not, uses ``pivot``. The
behavior of this default is subject to change without
prior notice.
Default: None.
For problems with sparse input, this option is ignored, and the
pivot-based algorithm presented in [5]_ is used.
For method-specific options, see
:func:`show_options('linprog') <show_options>`.
x0 : 1-D array, optional
Guess values of the decision variables, which will be refined by
the optimization algorithm. This argument is currently used only by the
:ref:`'revised simplex' <optimize.linprog-revised_simplex>` method,
and can only be used if `x0` represents a basic feasible solution.
integrality : 1-D array or int, optional
Indicates the type of integrality constraint on each decision variable.
``0`` : Continuous variable; no integrality constraint.
``1`` : Integer variable; decision variable must be an integer
within `bounds`.
``2`` : Semi-continuous variable; decision variable must be within
`bounds` or take value ``0``.
``3`` : Semi-integer variable; decision variable must be an integer
within `bounds` or take value ``0``.
By default, all variables are continuous.
For mixed integrality constraints, supply an array of shape ``c.shape``.
To infer a constraint on each decision variable from shorter inputs,
the argument will be broadcast to ``c.shape`` using `numpy.broadcast_to`.
This argument is currently used only by the
:ref:`'highs' <optimize.linprog-highs>` method and is ignored otherwise.
Returns
-------
res : OptimizeResult
A :class:`scipy.optimize.OptimizeResult` consisting of the fields
below. Note that the return types of the fields may depend on whether
the optimization was successful, therefore it is recommended to check
`OptimizeResult.status` before relying on the other fields:
x : 1-D array
The values of the decision variables that minimizes the
objective function while satisfying the constraints.
fun : float
The optimal value of the objective function ``c @ x``.
slack : 1-D array
The (nominally positive) values of the slack variables,
``b_ub - A_ub @ x``.
con : 1-D array
The (nominally zero) residuals of the equality constraints,
``b_eq - A_eq @ x``.
success : bool
``True`` when the algorithm succeeds in finding an optimal
solution.
status : int
An integer representing the exit status of the algorithm.
``0`` : Optimization terminated successfully.
``1`` : Iteration limit reached.
``2`` : Problem appears to be infeasible.
``3`` : Problem appears to be unbounded.
``4`` : Numerical difficulties encountered.
nit : int
The total number of iterations performed in all phases.
message : str
A string descriptor of the exit status of the algorithm.
See Also
--------
show_options : Additional options accepted by the solvers.
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter.
:ref:`'highs-ds' <optimize.linprog-highs-ds>`, and
:ref:`'highs-ipm' <optimize.linprog-highs-ipm>` are interfaces to the
HiGHS simplex and interior-point method solvers [13]_, respectively.
:ref:`'highs' <optimize.linprog-highs>` (default) chooses between
the two automatically. These are the fastest linear
programming solvers in SciPy, especially for large, sparse problems;
which of these two is faster is problem-dependent.
The other solvers are legacy methods and will be removed when `callback` is
supported by the HiGHS methods.
Method :ref:`'highs-ds' <optimize.linprog-highs-ds>`, is a wrapper of the C++ high
performance dual revised simplex implementation (HSOL) [13]_, [14]_.
Method :ref:`'highs-ipm' <optimize.linprog-highs-ipm>` is a wrapper of a C++
implementation of an **i**\ nterior-\ **p**\ oint **m**\ ethod [13]_; it
features a crossover routine, so it is as accurate as a simplex solver.
Method :ref:`'highs' <optimize.linprog-highs>` chooses between the two
automatically.
For new code involving `linprog`, we recommend explicitly choosing one of
these three method values.
.. versionadded:: 1.6.0
Method :ref:`'interior-point' <optimize.linprog-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 :ref:`'revised simplex' <optimize.linprog-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 :ref:`'simplex' <optimize.linprog-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 :ref:`'interior-point' <optimize.linprog-interior-point>`,
:ref:`'revised simplex' <optimize.linprog-revised_simplex>`, or
:ref:`'simplex' <optimize.linprog-simplex>`,
a presolve procedure based on [8]_ attempts
to identify trivial infeasibilities, trivial unboundedness, and potential
problem simplifications. Specifically, it checks for:
- rows of zeros in ``A_eq`` or ``A_ub``, representing trivial constraints;
- columns of zeros in ``A_eq`` `and` ``A_ub``, representing unconstrained
variables;
- column singletons in ``A_eq``, representing fixed variables; and
- column singletons in ``A_ub``, representing simple bounds.
If presolve reveals that the problem is unbounded (e.g. an unconstrained
and unbounded variable has negative cost) or infeasible (e.g., a row of
zeros in ``A_eq`` corresponds with a nonzero in ``b_eq``), the solver
terminates with the appropriate status code. Note that presolve terminates
as soon as any sign of unboundedness is detected; consequently, a problem
may be reported as unbounded when in reality the problem is infeasible
(but infeasibility has not been detected yet). Therefore, if it is
important to know whether the problem is actually infeasible, solve the
problem again with option ``presolve=False``.
If neither infeasibility nor unboundedness are detected in a single pass
of the presolve, bounds are tightened where possible and fixed
variables are removed from the problem. Then, linearly dependent rows
of the ``A_eq`` matrix are removed, (unless they represent an
infeasibility) to avoid numerical difficulties in the primary solve
routine. Note that rows that are nearly linearly dependent (within a
prescribed tolerance) may also be removed, which can change the optimal
solution in rare cases. If this is a concern, eliminate redundancy from
your problem formulation and run with option ``rr=False`` or
``presolve=False``.
Several potential improvements can be made here: additional presolve
checks outlined in [8]_ should be implemented, the presolve routine should
be run multiple times (until no further simplifications can be made), and
more of the efficiency improvements from [5]_ should be implemented in the
redundancy removal routines.
After presolve, the problem is transformed to standard form by converting
the (tightened) simple bounds to upper bound constraints, introducing
non-negative slack variables for inequality constraints, and expressing
unbounded variables as the difference between two non-negative variables.
Optionally, the problem is automatically scaled via equilibration [12]_.
The selected algorithm solves the standard form problem, and a
postprocessing routine converts the result to a solution to the original
problem.
References
----------
.. [1] Dantzig, George B., Linear programming and extensions. Rand
Corporation Research Study Princeton Univ. Press, Princeton, NJ,
1963
.. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to
Mathematical Programming", McGraw-Hill, Chapter 4.
.. [3] Bland, Robert G. New finite pivoting rules for the simplex method.
Mathematics of Operations Research (2), 1977: pp. 103-107.
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
.. [5] Andersen, Erling D. "Finding all linearly dependent rows in
large-scale linear programming." Optimization Methods and Software
6.3 (1995): 219-227.
.. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear
Programming based on Newton's Method." Unpublished Course Notes,
March 2004. Available 2/25/2017 at
https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf
.. [7] Fourer, Robert. "Solving Linear Programs by Interior-Point Methods."
Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at
.. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear
programming." Mathematical Programming 71.2 (1995): 221-245.
.. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear
programming." Athena Scientific 1 (1997): 997.
.. [10] Andersen, Erling D., et al. Implementation of interior point
methods for large scale linear programming. HEC/Universite de
Geneve, 1996.
.. [11] Bartels, Richard H. "A stabilization of the simplex method."
Journal in Numerische Mathematik 16.5 (1971): 414-434.
.. [12] Tomlin, J. A. "On scaling linear programming problems."
Mathematical Programming Study 4 (1975): 146-166.
.. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J.
"HiGHS - high performance software for linear optimization."
https://highs.dev/
.. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised
simplex method." Mathematical Programming Computation, 10 (1),
119-142, 2018. DOI: 10.1007/s12532-017-0130-5
Examples
--------
Consider the following problem:
.. math::
\min_{x_0, x_1} \ -x_0 + 4x_1 & \\
\mbox{such that} \ -3x_0 + x_1 & \leq 6,\\
-x_0 - 2x_1 & \geq -4,\\
x_1 & \geq -3.
The problem is not presented in the form accepted by `linprog`. This is
easily remedied by converting the "greater than" inequality
constraint to a "less than" inequality constraint by
multiplying both sides by a factor of :math:`-1`. Note also that the last
constraint is really the simple bound :math:`-3 \leq x_1 \leq \infty`.
Finally, since there are no bounds on :math:`x_0`, we must explicitly
specify the bounds :math:`-\infty \leq x_0 \leq \infty`, as the
default is for variables to be non-negative. After collecting coeffecients
into arrays and tuples, the input for this problem is:
>>> from scipy.optimize import linprog
>>> c = [-1, 4]
>>> A = [[-3, 1], [1, 2]]
>>> b = [6, 4]
>>> x0_bounds = (None, None)
>>> x1_bounds = (-3, None)
>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds])
>>> res.fun
-22.0
>>> res.x
array([10., -3.])
>>> res.message
'Optimization terminated successfully. (HiGHS Status 7: Optimal)'
The marginals (AKA dual values / shadow prices / Lagrange multipliers)
and residuals (slacks) are also available.
>>> res.ineqlin
residual: [ 3.900e+01 0.000e+00]
marginals: [-0.000e+00 -1.000e+00]
For example, because the marginal associated with the second inequality
constraint is -1, we expect the optimal value of the objective function
to decrease by ``eps`` if we add a small amount ``eps`` to the right hand
side of the second inequality constraint:
>>> eps = 0.05
>>> b[1] += eps
>>> linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun
-22.05
Also, because the residual on the first inequality constraint is 39, we
can decrease the right hand side of the first constraint by 39 without
affecting the optimal solution.
>>> b = [6, 4] # reset to original values
>>> b[0] -= 39
>>> linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun
-22.0
"""
meth = method.lower()
methods = {"highs", "highs-ds", "highs-ipm",
"simplex", "revised simplex", "interior-point"}
if meth not in methods:
raise ValueError(f"Unknown solver '{method}'")
if x0 is not None and meth != "revised simplex":
warning_message = "x0 is used only when method is 'revised simplex'. "
warn(warning_message, OptimizeWarning, stacklevel=2)
if np.any(integrality) and not meth == "highs":
integrality = None
warning_message = ("Only `method='highs'` supports integer "
"constraints. Ignoring `integrality`.")
warn(warning_message, OptimizeWarning, stacklevel=2)
elif np.any(integrality):
integrality = np.broadcast_to(integrality, np.shape(c))
else:
integrality = None
lp = _LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0, integrality)
lp, solver_options = _parse_linprog(lp, options, meth)
tol = solver_options.get('tol', 1e-9)
# Give unmodified problem to HiGHS
if meth.startswith('highs'):
if callback is not None:
raise NotImplementedError("HiGHS solvers do not support the "
"callback interface.")
highs_solvers = {'highs-ipm': 'ipm', 'highs-ds': 'simplex',
'highs': None}
sol = _linprog_highs(lp, solver=highs_solvers[meth],
**solver_options)
sol['status'], sol['message'] = (
_check_result(sol['x'], sol['fun'], sol['status'], sol['slack'],
sol['con'], lp.bounds, tol, sol['message'],
integrality))
sol['success'] = sol['status'] == 0
return OptimizeResult(sol)
warn(f"`method='{meth}'` is deprecated and will be removed in SciPy "
"1.11.0. Please use one of the HiGHS solvers (e.g. "
"`method='highs'`) in new code.", DeprecationWarning, stacklevel=2)
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.
rr_method = solver_options.pop('rr_method', None) # need to pop these;
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):
(lp, c0, x, undo, complete, status, message) = _presolve(lp, rr,
rr_method,
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)
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, integrality)
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)top level linear prog interface
versioadded
functions autosummary
LINPROG_METHODS = [‘simplex’, ‘revised simplex’, ‘]
def linprog_verbose_callback(res):
f”{x: 12.4f}”}
.. 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 ,
solving linear programming problem using Simplex method, interior-point method, and HiGHS method.
meth = method.lower()
methods = {“highs”, “highs-ds”}
lp, solver_options = _parse_linprog(lp, options, meth)
Each solver (_linprog_simplex, _linprog_ip, _linprog_rs)
dive into _linprog_simplex.py to see how Lagrangian is used in simplex method
give a simplex tableau
pivot_col
pivot_row
apply_pivot
solve_simplex
linprog_simplex, below is the actual _linprog_simplex.py
"""Simplex method for linear programming
The *simplex* method 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
Warnings
--------
The simplex method may encounter numerical difficulties when pivot
values are close to the specified tolerance. If encountered try
remove any redundant constraints, change the pivot strategy to Bland's
rule or increase the tolerance value.
Alternatively, more robust methods maybe be used. See
:ref:`'interior-point' <optimize.linprog-interior-point>` and
:ref:`'revised simplex' <optimize.linprog-revised_simplex>`.
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.
"""
import numpy as np
from warnings import warn
from ._optimize import OptimizeResult, OptimizeWarning, _check_unknown_options
from ._linprog_util import _postsolve
def _pivot_col(T, tol=1e-9, bland=False):
"""
Given a linear programming simplex tableau, determine the column
of the variable to enter the basis.
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
tol : float
Elements in the objective row larger than -tol will not be considered
for pivoting. Nominally this value is zero, but numerical issues
cause a tolerance about zero to be necessary.
bland : bool
If True, use Bland's rule for selection of the column (select the
first column with a negative coefficient in the objective row,
regardless of magnitude).
Returns
-------
status: bool
True if a suitable pivot column was found, otherwise False.
A return of False indicates that the linear programming simplex
algorithm is complete.
col: int
The index of the column of the pivot element.
If status is False, col will be returned as nan.
"""
ma = np.ma.masked_where(T[-1, :-1] >= -tol, T[-1, :-1], copy=False)
if ma.count() == 0:
return False, np.nan
if bland:
# ma.mask is sometimes 0d
return True, np.nonzero(np.logical_not(np.atleast_1d(ma.mask)))[0][0]
return True, np.ma.nonzero(ma == ma.min())[0][0]
def _pivot_row(T, basis, pivcol, phase, tol=1e-9, bland=False):
"""
Given a linear programming simplex tableau, determine the row for the
pivot operation.
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a Problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
basis : array
A list of the current basic variables.
pivcol : int
The index of the pivot column.
phase : int
The phase of the simplex algorithm (1 or 2).
tol : float
Elements in the pivot column smaller than tol will not be considered
for pivoting. Nominally this value is zero, but numerical issues
cause a tolerance about zero to be necessary.
bland : bool
If True, use Bland's rule for selection of the row (if more than one
row can be used, choose the one with the lowest variable index).
Returns
-------
status: bool
True if a suitable pivot row was found, otherwise False. A return
of False indicates that the linear programming problem is unbounded.
row: int
The index of the row of the pivot element. If status is False, row
will be returned as nan.
"""
if phase == 1:
k = 2
else:
k = 1
ma = np.ma.masked_where(T[:-k, pivcol] <= tol, T[:-k, pivcol], copy=False)
if ma.count() == 0:
return False, np.nan
mb = np.ma.masked_where(T[:-k, pivcol] <= tol, T[:-k, -1], copy=False)
q = mb / ma
min_rows = np.ma.nonzero(q == q.min())[0]
if bland:
return True, min_rows[np.argmin(np.take(basis, min_rows))]
return True, min_rows[0]
def _apply_pivot(T, basis, pivrow, pivcol, tol=1e-9):
"""
Pivot the simplex tableau inplace on the element given by (pivrow, pivol).
The entering variable corresponds to the column given by pivcol forcing
the variable basis[pivrow] to leave the basis.
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
basis : 1-D array
An array of the indices of the basic variables, such that basis[i]
contains the column corresponding to the basic variable for row i.
Basis is modified in place by _apply_pivot.
pivrow : int
Row index of the pivot.
pivcol : int
Column index of the pivot.
"""
basis[pivrow] = pivcol
pivval = T[pivrow, pivcol]
T[pivrow] = T[pivrow] / pivval
for irow in range(T.shape[0]):
if irow != pivrow:
T[irow] = T[irow] - T[pivrow] * T[irow, pivcol]
# The selected pivot should never lead to a pivot value less than the tol.
if np.isclose(pivval, tol, atol=0, rtol=1e4):
message = (
f"The pivot operation produces a pivot value of:{pivval: .1e}, "
"which is only slightly greater than the specified "
f"tolerance{tol: .1e}. This may lead to issues regarding the "
"numerical stability of the simplex method. "
"Removing redundant constraints, changing the pivot strategy "
"via Bland's rule or increasing the tolerance may "
"help reduce the issue.")
warn(message, OptimizeWarning, stacklevel=5)
def _solve_simplex(T, n, basis, callback, postsolve_args,
maxiter=1000, tol=1e-9, phase=2, bland=False, nit0=0,
):
"""
Solve a linear programming problem in "standard form" using the Simplex
Method. Linear Programming is intended to solve the following problem form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Parameters
----------
T : 2-D array
A 2-D array representing the simplex tableau, T, corresponding to the
linear programming problem. It should have the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0]]
for a Phase 2 problem, or the form:
[[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]],
[A[1, 0], A[1, 1], ..., A[1, n_total], b[1]],
.
.
.
[A[m, 0], A[m, 1], ..., A[m, n_total], b[m]],
[c[0], c[1], ..., c[n_total], 0],
[c'[0], c'[1], ..., c'[n_total], 0]]
for a Phase 1 problem (a problem in which a basic feasible solution is
sought prior to maximizing the actual objective. ``T`` is modified in
place by ``_solve_simplex``.
n : int
The number of true variables in the problem.
basis : 1-D array
An array of the indices of the basic variables, such that basis[i]
contains the column corresponding to the basic variable for row i.
Basis is modified in place by _solve_simplex
callback : callable, optional
If a callback function is provided, it will be called within each
iteration of the algorithm. The callback must accept a
`scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
Current solution vector
fun : float
Current value of the objective function
success : bool
True only when a phase has completed successfully. This
will be False for most iterations.
slack : 1-D array
The values of the slack variables. Each slack variable
corresponds to an inequality constraint. If the slack is zero,
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.
postsolve_args : tuple
Data needed by _postsolve to convert the solution to the standard-form
problem into the solution to the original problem.
maxiter : int
The maximum number of iterations to perform before aborting the
optimization.
tol : float
The tolerance which determines when a solution is "close enough" to
zero in Phase 1 to be considered a basic feasible solution or close
enough to positive to serve as an optimal solution.
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.
bland : bool
If True, choose pivots using Bland's rule [3]_. In problems which
fail to converge due to cycling, using Bland's rule can provide
convergence at the expense of a less optimal path about the simplex.
nit0 : int
The initial iteration number used to keep an accurate iteration total
in a two-phase problem.
Returns
-------
nit : int
The number of iterations. Used to keep an accurate iteration total
in the two-phase problem.
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 = nit0
status = 0
message = ''
complete = False
if phase == 1:
m = T.shape[1]-2
elif phase == 2:
m = T.shape[1]-1
else:
raise ValueError("Argument 'phase' to _solve_simplex must be 1 or 2")
if phase == 2:
# Check if any artificial variables are still in the basis.
# If yes, check if any coefficients from this row and a column
# corresponding to one of the non-artificial variable is non-zero.
# If found, pivot at this term. If not, start phase 2.
# Do this for all artificial variables in the basis.
# Ref: "An Introduction to Linear Programming and Game Theory"
# by Paul R. Thie, Gerard E. Keough, 3rd Ed,
# Chapter 3.7 Redundant Systems (pag 102)
for pivrow in [row for row in range(basis.size)
if basis[row] > T.shape[1] - 2]:
non_zero_row = [col for col in range(T.shape[1] - 1)
if abs(T[pivrow, col]) > tol]
if len(non_zero_row) > 0:
pivcol = non_zero_row[0]
_apply_pivot(T, basis, pivrow, pivcol, tol)
nit += 1
if len(basis[:m]) == 0:
solution = np.empty(T.shape[1] - 1, dtype=np.float64)
else:
solution = np.empty(max(T.shape[1] - 1, max(basis[:m]) + 1),
dtype=np.float64)
while not complete:
# Find the pivot column
pivcol_found, pivcol = _pivot_col(T, tol, bland)
if not pivcol_found:
pivcol = np.nan
pivrow = np.nan
status = 0
complete = True
else:
# Find the pivot row
pivrow_found, pivrow = _pivot_row(T, basis, pivcol, phase, tol, bland)
if not pivrow_found:
status = 3
complete = True
if callback is not None:
solution[:] = 0
solution[basis[:n]] = T[:n, -1]
x = solution[:m]
x, fun, slack, con = _postsolve(
x, postsolve_args
)
res = OptimizeResult({
'x': x,
'fun': fun,
'slack': slack,
'con': con,
'status': status,
'message': message,
'nit': nit,
'success': status == 0 and complete,
'phase': phase,
'complete': complete,
})
callback(res)
if not complete:
if nit >= maxiter:
# Iteration limit exceeded
status = 1
complete = True
else:
_apply_pivot(T, basis, pivrow, pivcol, tol)
nit += 1
return nit, status
def _linprog_simplex(c, c0, A, b, callback, postsolve_args,
maxiter=1000, tol=1e-9, disp=False, bland=False,
**unknown_options):
"""
Minimize a linear objective function subject to linear equality and
non-negativity constraints using the two phase simplex method.
Linear programming is intended to solve problems of the following form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
User-facing documentation is in _linprog_doc.py.
Parameters
----------
c : 1-D array
Coefficients of the linear objective function to be minimized.
c0 : float
Constant term in objective function due to fixed (and eliminated)
variables. (Purely for display.)
A : 2-D array
2-D array such that ``A @ x``, gives the values of the equality
constraints at ``x``.
b : 1-D array
1-D array of values representing the right hand side of each equality
constraint (row) in ``A``.
callback : callable, optional
If a callback function is provided, it will be called within each
iteration of the algorithm. The callback function must accept a single
`scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1-D array
Current solution vector
fun : float
Current value of the objective function
success : bool
True when an algorithm has completed successfully.
slack : 1-D array
The values of the slack variables. Each slack variable
corresponds to an inequality constraint. If the slack is zero,
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 algorithm being executed.
status : int
An integer representing the status of the optimization::
0 : Algorithm proceeding nominally
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.
postsolve_args : tuple
Data needed by _postsolve to convert the solution to the standard-form
problem into the solution to the original problem.
Options
-------
maxiter : int
The maximum number of iterations to perform.
disp : bool
If True, print exit status message to sys.stdout
tol : float
The tolerance which determines when a solution is "close enough" to
zero in Phase 1 to be considered a basic feasible solution or close
enough to positive to serve as an optimal solution.
bland : bool
If True, use Bland's anti-cycling rule [3]_ to choose pivots to
prevent cycling. If False, choose pivots which should lead to a
converged solution more quickly. The latter method is subject to
cycling (non-convergence) in rare instances.
unknown_options : dict
Optional arguments not used by this particular solver. If
`unknown_options` is non-empty a warning is issued listing all
unused options.
Returns
-------
x : 1-D array
Solution vector.
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
message : str
A string descriptor of the exit status of the optimization.
iteration : int
The number of iterations taken to solve the 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.
Notes
-----
The expected problem formulation differs between the top level ``linprog``
module and the method specific solvers. The method specific solvers expect a
problem in standard form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Whereas the top level ``linprog`` module expects a problem of form:
Minimize::
c @ x
Subject to::
A_ub @ x <= b_ub
A_eq @ x == b_eq
lb <= x <= ub
where ``lb = 0`` and ``ub = None`` unless set in ``bounds``.
The original problem contains equality, upper-bound and variable constraints
whereas the method specific solver requires equality constraints and
variable non-negativity.
``linprog`` module converts the original problem to standard form by
converting the 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.
"""
_check_unknown_options(unknown_options)
status = 0
messages = {0: "Optimization terminated successfully.",
1: "Iteration limit reached.",
2: "Optimization failed. Unable to find a feasible"
" starting point.",
3: "Optimization failed. The problem appears to be unbounded.",
4: "Optimization failed. Singular matrix encountered."}
n, m = A.shape
# All constraints must have b >= 0.
is_negative_constraint = np.less(b, 0)
A[is_negative_constraint] *= -1
b[is_negative_constraint] *= -1
# As all constraints are equality constraints the artificial variables
# will also be basic variables.
av = np.arange(n) + m
basis = av.copy()
# Format the phase one tableau by adding artificial variables and stacking
# the constraints, the objective row and pseudo-objective row.
row_constraints = np.hstack((A, np.eye(n), b[:, np.newaxis]))
row_objective = np.hstack((c, np.zeros(n), c0))
row_pseudo_objective = -row_constraints.sum(axis=0)
row_pseudo_objective[av] = 0
T = np.vstack((row_constraints, row_objective, row_pseudo_objective))
nit1, status = _solve_simplex(T, n, basis, callback=callback,
postsolve_args=postsolve_args,
maxiter=maxiter, tol=tol, phase=1,
bland=bland
)
# if pseudo objective is zero, remove the last row from the tableau and
# proceed to phase 2
nit2 = nit1
if abs(T[-1, -1]) < tol:
# Remove the pseudo-objective row from the tableau
T = T[:-1, :]
# Remove the artificial variable columns from the tableau
T = np.delete(T, av, 1)
else:
# Failure to find a feasible starting point
status = 2
messages[status] = (
"Phase 1 of the simplex method failed to find a feasible "
"solution. The pseudo-objective function evaluates to "
f"{abs(T[-1, -1]):.1e} "
f"which exceeds the required tolerance of {tol} for a solution to be "
"considered 'close enough' to zero to be a basic solution. "
"Consider increasing the tolerance to be greater than "
f"{abs(T[-1, -1]):.1e}. "
"If this tolerance is unacceptably large the problem may be "
"infeasible."
)
if status == 0:
# Phase 2
nit2, status = _solve_simplex(T, n, basis, callback=callback,
postsolve_args=postsolve_args,
maxiter=maxiter, tol=tol, phase=2,
bland=bland, nit0=nit1
)
solution = np.zeros(n + m)
solution[basis[:n]] = T[:n, -1]
x = solution[:m]
return x, status, messages[status], int(nit2)
Naixian Zhang 