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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)
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
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.
.. [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
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