bundles / scipy 1.17.1 / scipy / optimize / _milp / milp

function

`scipy.optimize._milp:milp`

source: /scipy/optimize/_milp.py :154

Signature

   def     milp (    c    ,  * ,     integrality    =  None   ,    bounds    =  None   ,    constraints    =  None   ,    options    =  None     )

Summary

Mixed-integer linear programming

Extended Summary

Solves problems of the following form:

\min_x \ & c^T x \\
\mbox{such that} \ & b_l \leq A x \leq b_u,\\
& l \leq x \leq u, \\
& x_i \in \mathbb{Z}, i \in X_i

where $x$ is a vector of decision variables; $c$ , $b_{l}$ , $b_{u}$ , $l$ , and $u$ are vectors; $A$ is a matrix, and $X_{i}$ is the set of indices of decision variables that must be integral. (In this context, a variable that can assume only integer values is said to be "integral"; it has an "integrality" constraint.)

Alternatively, that's:

minimize

c @ x

such that

b_l <= A @ x <= b_u
l <= x <= u
Specified elements of x must be integers

By default, l = 0 and u = np.inf unless specified with bounds.

Parameters

c : 1D dense array_like

The coefficients of the linear objective function to be minimized. c is converted to a double precision array before the problem is solved.

integrality : 1D dense array_like, optional

Indicates the type of integrality constraint on each decision variable.

0Continuous variable; no integrality constraint.

1Integer variable; decision variable must be an integer within bounds.

2Semi-continuous variable; decision variable must be within bounds or take value 0.

3Semi-integer variable; decision variable must be an integer within bounds or take value 0.

By default, all variables are continuous. integrality is converted to an array of integers before the problem is solved.

bounds : scipy.optimize.Bounds, optional

Bounds on the decision variables. Lower and upper bounds are converted to double precision arrays before the problem is solved. The keep_feasible parameter of the Bounds object is ignored. If not specified, all decision variables are constrained to be non-negative.

constraints : sequence of scipy.optimize.LinearConstraint, optional

Linear constraints of the optimization problem. Arguments may be one of the following:

A single LinearConstraint object
A single tuple that can be converted to a LinearConstraint object as LinearConstraint(*constraints)
A sequence composed entirely of objects of type 1. and 2.

Before the problem is solved, all values are converted to double precision, and the matrices of constraint coefficients are converted to instances of scipy.sparse.csc_array. The keep_feasible parameter of LinearConstraint objects is ignored.

options : dict, optional

A dictionary of solver options. The following keys are recognized.

disp: disp
node_limit: node_limit
presolve: presolve
time_limit: time_limit
mip_rel_gap: mip_rel_gap

Returns

res : OptimizeResult

An instance of scipy.optimize.OptimizeResult. The object is guaranteed to have the following attributes.

status: status
success: success
message: message

The following attributes will also be present, but the values may be None, depending on the solution status.

x: x
fun: fun
mip_node_count: mip_node_count
mip_dual_bound: mip_dual_bound
mip_gap: mip_gap

Notes

milp is a wrapper of the HiGHS linear optimization software ^[1]. The algorithm is deterministic, and it typically finds the global optimum of moderately challenging mixed-integer linear programs (when it exists).

Examples

Consider the problem at https://en.wikipedia.org/wiki/Integer_programming#Example, which is expressed as a maximization problem of two variables. Since `milp` requires that the problem be expressed as a minimization problem, the objective function coefficients on the decision variables are:

import numpy as np
c = -np.array([0, 1])

✓

Note the negative sign: we maximize the original objective function by minimizing the negative of the objective function. We collect the coefficients of the constraints into arrays like:

A = np.array([[-1, 1], [3, 2], [2, 3]])
b_u = np.array([1, 12, 12])
b_l = np.full_like(b_u, -np.inf, dtype=float)

✓

Because there is no lower limit on these constraints, we have defined a variable ``b_l`` full of values representing negative infinity. This may be unfamiliar to users of `scipy.optimize.linprog`, which only accepts "less than" (or "upper bound") inequality constraints of the form ``A_ub @ x <= b_u``. By accepting both ``b_l`` and ``b_u`` of constraints ``b_l <= A_ub @ x <= b_u``, `milp` makes it easy to specify "greater than" inequality constraints, "less than" inequality constraints, and equality constraints concisely. These arrays are collected into a single `LinearConstraint` object like:

from scipy.optimize import LinearConstraint
constraints = LinearConstraint(A, b_l, b_u)

✓

The non-negativity bounds on the decision variables are enforced by default, so we do not need to provide an argument for `bounds`. Finally, the problem states that both decision variables must be integers:

integrality = np.ones_like(c)

✓

We solve the problem like:

from scipy.optimize import milp
res = milp(c=c, constraints=constraints, integrality=integrality)
res.x

✓

Note that had we solved the relaxed problem (without integrality constraints):

res = milp(c=c, constraints=constraints)  # OR:
res.x

✓

we would not have obtained the correct solution by rounding to the nearest integers. Other examples are given :ref:`in the tutorial <tutorial-optimize_milp>`.

Aliases

scipy.optimize.milp