Range Constraints and Solving Integer Linear Programming

Polynomial formulation for range constraints

Let $f$ be a polynomial expression of binary variables. A range constraint has the form $l\leq f\leq u$ with $l<u$. Our goal is to design a polynomial expression that takes the minimum value 0 if and only if the range constraint is satisfied.

The key idea is to introduce an auxiliary integer variable $a$ that takes values in the range $[l,u]$. Consider the following expression:

\[\begin{aligned} g &= (f-a)^2 \end{aligned}\]

This expression $g$ takes the minimum value 0 exactly when $f=a$. Since $a$ can take any integer value in $[l,u]$, the expression $g$ achieves 0 if and only if $f$ itself takes an integer value within the same range.

Using this auxiliary-variable technique, PyQBPP implements range constraints via the constrain() function.

Solving Integer Linear Programming

An instance of integer linear programming consists of an objective function and multiple linear constraints. For example, the following integer linear program has two variables, one objective, and two constraints:

\[\begin{aligned} \text{Maximize: } & & & 5x + 4y \\ \text{Subject to: } & && 2x + 3y \le 24 \\ & & & 7x + 5y \le 54 \end{aligned}\]

The optimal solution of this problem is $x=4$, $y=5$, with the objective value $40$.

The following PyQBPP program finds this optimal solution using the Easy Solver:

import pyqbpp as qbpp

x = qbpp.var("x", between=(0, 10))
y = qbpp.var("y", between=(0, 10))
f = 5 * x + 4 * y
c1 = qbpp.constrain(2 * x + 3 * y, between=(0, 24))
c2 = qbpp.constrain(7 * x + 5 * y, between=(0, 54))
g = -f + 100 * (c1 + c2)
g.simplify_as_binary()

solver = qbpp.EasySolver(g)
sol = solver.search(time_limit=1.0)

print(f"x = {sol(x)}, y = {sol(y)}")
print(f"f = {sol(f)}")
print(f"c1 = {sol(c1)}, c2 = {sol(c2)}")
print(f"2x+3y = {sol(c1.body)}, 7x+5y = {sol(c2.body)}")

In this program,

• f represents the objective function,
• c1 and c2 represent the range constraints created using constrain(), and
• g combines them into a single optimization expression.

Since the goal is maximization, the objective is negated as -f. The constraints c1 and c2 are penalized with a weight of 100 to ensure they are satisfied with high priority.

An Easy Solver instance is created for g, and a search is performed with a time limit of 1.0 seconds passed as a parameter to search(). After obtaining the optimal solution sol, the program prints the values of x, y, f, c1, c2, and the constraint body expressions.

The program outputs:

x = 4, y = 5
f = 40
c1 = 0, c2 = 0
2x+3y = 23, 7x+5y = 53

Here,

• c1 is the penalty for the constraint 0 <= 2x + 3y <= 24, and
• c1.body represents the linear expression 2x + 3y.

We can confirm that the solver correctly finds the optimal solution.

Writing with native constraints cons()

In the program above, the constraints c1 and c2 were added to the objective as weighted penalty expressions. PyQBPP lets you go one step further and explicitly declare them as constraints by creating them with qbpp.cons():

import pyqbpp as qbpp

x = qbpp.var("x", between=(0, 10))
y = qbpp.var("y", between=(0, 10))
f = 5 * x + 4 * y
g = -f + 100 * (qbpp.cons(2 * x + 3 * y, between=(0, 24)) +
                qbpp.cons(7 * x + 5 * y, between=(0, 54)))
g.simplify_as_binary()

solver = qbpp.EasySolver(g)
sol = solver.search(time_limit=1.0)

print(f"x = {sol(x)}, y = {sol(y)}")
print(f"f = {sol(f)}")
print(f"violated constraints = {g.cons(sol)}")

The only change is using qbpp.cons() instead of constrain() (the arguments are written the same way: between=(l, u) for a range constraint and equal=n for an equality constraint). The parts created with qbpp.cons() are treated specially as constraints, and the bundled solvers search efficiently for solutions that satisfy the declared constraints. g.cons(sol) returns the number of constraints violated by the solution sol (0 means all constraints are satisfied). The program outputs:

x = 4, y = 5
f = 40
violated constraints = 0

Constraints declared with cons() are treated as hard constraints (constraints that must be satisfied) by the Exhaustive Solver, and by the MIP solvers (Gurobi, etc.) when ilp=True is specified. See Native Constraints for details.