Integer Linear Programming
Integer Linear Programming (ILP) seeks a vector of integer numbers that maximizes (or minimizes) a linear cost function under a set of linear equality or inequality constraints [1]. In other words, it is an optimization problem where the cost function to optimize and all the constraints are linear and the decision variables are integers.
Mathematical Formulation
The ILP problem can be formulated as follows: given an \(n\)-dimensional vector \(\vec{c} = (c_1, c_2, \ldots, c_n)\), an \(m \times n\) matrix \(A = (a_{ij})\) with \(i=1,\ldots,m\) and \(j=1,\ldots,n\), and an \(m\)-dimensional vector \(\vec{b} = (b_1, b_2, \ldots, b_m)\), find an \(n\)-dimensional vector \(\vec{x} = (x_1, x_2, \ldots, x_n)\) with integer entries that maximizes (or minimizes) the cost function:
\(\begin{align*}\vec{c} \cdot \vec{x} = c_1x_1 + c_2x_2 + \ldots + c_nx_n\end{align*}\)
subject to these constraints:
\(\begin{align*}A \vec{x} & \leq \vec{b} \\ x_j & \geq 0, \quad j = 1, 2, \ldots, n \\ x_j & \in \mathbb{Z}, \quad j = 1, 2, \ldots, n\end{align*}\)
This tutorial guides you through the steps of solving the problem with the Classiq platform, using QAOA [2]. The solution is based on defining a Pyomo model for the optimization problem to solve.
Building the Pyomo Model from a Graph Input
Define the Pyomo model to use on the Classiq platform, using the mathematical formulation defined above:
import numpy as np
import pyomo.core as pyo
def ilp(a: np.ndarray, b: np.ndarray, c: np.ndarray, bound: int) -> pyo.ConcreteModel:
# model constraint: a*x <= b
model = pyo.ConcreteModel()
assert b.ndim == c.ndim == 1
num_vars = len(c)
num_constraints = len(b)
assert a.shape == (num_constraints, num_vars)
model.x = pyo.Var(
# here we bound x to be from 0 to to a given bound
range(num_vars),
domain=pyo.NonNegativeIntegers,
bounds=(0, bound),
)
@model.Constraint(range(num_constraints))
def monotone_rule(model, idx):
return a[idx, :] @ list(model.x.values()) <= float(b[idx])
# model objective: max(c * x)
model.cost = pyo.Objective(expr=c @ list(model.x.values()), sense=pyo.maximize)
return model
A = np.array([[1, 1, 1], [2, 2, 2], [3, 3, 3]])
b = np.array([1, 2, 3])
c = np.array([1, 2, 3])
# Instantiate the model
ilp_model = ilp(A, b, c, 3)
ilp_model.pprint()
2 Set Declarations
monotone_rule_index : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {0, 1, 2}
x_index : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {0, 1, 2}
1 Var Declarations
x : Size=3, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : None : 3 : False : True : NonNegativeIntegers
1 : 0 : None : 3 : False : True : NonNegativeIntegers
2 : 0 : None : 3 : False : True : NonNegativeIntegers
1 Objective Declarations
cost : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : maximize : x[0] + 2*x[1] + 3*x[2]
1 Constraint Declarations
monotone_rule : Size=3, Index=monotone_rule_index, Active=True
Key : Lower : Body : Upper : Active
0 : -Inf : x[0] + x[1] + x[2] : 1.0 : True
1 : -Inf : 2*x[0] + 2*x[1] + 2*x[2] : 2.0 : True
2 : -Inf : 3*x[0] + 3*x[1] + 3*x[2] : 3.0 : True
5 Declarations: x_index x monotone_rule_index monotone_rule cost
Setting Up the Classiq Problem Instance
To solve the Pyomo model defined above, use the CombinatorialProblem quantum object. Under the hood it translates the Pyomo model to a quantum model of QAOA, with the cost Hamiltonian translated from the Pyomo model. Choose the number of layers for the QAOA ansatz using the num_layers argument. The penalty_factor is the coefficient of the constraints term in the cost Hamiltonian.
from classiq import *
from classiq.applications.combinatorial_optimization import CombinatorialProblem
combi = CombinatorialProblem(pyo_model=ilp_model, num_layers=3, penalty_factor=10)
qmod = combi.get_model()
Synthesizing the QAOA Circuit and Solving the Problem
Synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:
qprog = combi.get_qprog()
show(qprog)
Opening: https://nightly.platform.classiq.io/circuit/2d391a9b-4f75-4b90-ba76-fe6eb0cc3dec?version=0.63.0.dev2
Set the quantum backend on which to execute:
from classiq.execution import *
execution_preferences = ExecutionPreferences(
backend_preferences=ClassiqBackendPreferences(backend_name="simulator"),
)
Solve the problem by calling the optimize method of the CombinatorialProblem object. For the classical optimization part of QAOA, define the maximum number of classical iterations (maxiter) and the \(\alpha\)-parameter (quantile) for running CVaR-QAOA, an improved variation of QAOA [3]:
optimized_params = combi.optimize(execution_preferences, maxiter=90, quantile=0.7)
Optimization Progress: 76%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████▎ | 68/90 [02:24<00:46, 2.12s/it]
Check the convergence of the run:
import matplotlib.pyplot as plt
plt.plot(combi.cost_trace)
plt.xlabel("Iterations")
plt.ylabel("Cost")
plt.title("Cost convergence")
Text(0.5, 1.0, 'Cost convergence')
Optimization Results
Examine the statistics of the algorithm. The optimization is always defined as a minimization problem, so the positive maximization objective is translated to negative minimization by the Pyomo-to-Qmod translator.
To get samples with the optimized parameters, call the sample method:
optimization_result = combi.sample(optimized_params)
optimization_result.sort_values(by="cost").head(5)
| solution | probability | cost | |
|---|---|---|---|
| 18 | {'x': [0, 0, 1], 'monotone_rule_1_slack_var': ... | 0.011719 | -3.000000e+00 |
| 0 | {'x': [0, 1, 0], 'monotone_rule_1_slack_var': ... | 0.030762 | -2.000000e+00 |
| 9 | {'x': [0, 0, 0], 'monotone_rule_1_slack_var': ... | 0.018066 | 1.527468e-150 |
| 150 | {'x': [0, 0, 1], 'monotone_rule_1_slack_var': ... | 0.001465 | 7.000000e+00 |
| 130 | {'x': [0, 1, 0], 'monotone_rule_1_slack_var': ... | 0.001953 | 8.000000e+00 |
Compare the optimized results to uniformly sampled results:
uniform_result = combi.sample_uniform()
And compare the histograms:
optimization_result["cost"].plot(
kind="hist",
bins=30,
edgecolor="black",
weights=optimization_result["probability"],
alpha=0.6,
label="optimized",
)
uniform_result["cost"].plot(
kind="hist",
bins=30,
edgecolor="black",
weights=uniform_result["probability"],
alpha=0.6,
label="uniform",
)
plt.legend()
plt.ylabel("Probability", fontsize=16)
plt.xlabel("cost", fontsize=16)
plt.tick_params(axis="both", labelsize=14)
Plot the solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]
best_solution
{'x': [0, 0, 1],
'monotone_rule_1_slack_var': [0],
'monotone_rule_2_slack_var': [0]}
Comparing to a Classical Solver
Compare to the classical solution of the problem:
from pyomo.opt import SolverFactory
solver = SolverFactory("couenne")
solver.solve(ilp_model)
classical_solution = [int(pyo.value(ilp_model.x[i])) for i in range(len(ilp_model.x))]
print("Classical solution:", classical_solution)
Classical solution: [0, 0, 1]
References
[1] Integer Programming (Wikipedia).
[2] Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. (2014). "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028.
[3] Barkoutsos, Panagiotis Kl, et al. (2020). "Improving variational quantum optimization using CVaR." Quantum 4: 256.