Integer Linear Programming¶
Introduction¶
In Integer Linear Programming (ILP), we seek to find 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 be optimized and all the constraints are linear and the decision variables are integer.
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 the 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*}
Solving with the Classiq platform¶
We go through the steps of solving the problem with the Classiq platform, using QAOA algorithm [2]. The solution is based on defining a pyomo model for the optimization problem we would like to solve.
from typing import cast
import networkx as nx
import numpy as np
import pyomo.core as pyo
from IPython.display import Markdown, display
from matplotlib import pyplot as plt
Building the Pyomo model from a graph input¶
We proceed by defining the pyomo model that will be used 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, 4)
ilp_model.pprint()
Setting Up the Classiq Problem Instance¶
In order to solve the Pyomo model defined above, we use the Classiq combinatorial optimization engine. For the quantum part of the QAOA algorithm (QAOAConfig) - define the number of repetitions (num_layers):
from classiq import construct_combinatorial_optimization_model
from classiq.applications.combinatorial_optimization import OptimizerConfig, QAOAConfig
qaoa_config = QAOAConfig(num_layers=3)
For the classical optimization part of the QAOA algorithm we define the maximum number of classical iterations (max_iteration) and the $\alpha$-parameter (alpha_cvar) for running CVaR-QAOA, an improved variation of the QAOA algorithm [3]:
optimizer_config = OptimizerConfig(max_iteration=90, alpha_cvar=0.7)
Lastly, we load the model, based on the problem and algorithm parameters, which we can use to solve the problem:
qmod = construct_combinatorial_optimization_model(
pyo_model=ilp_model,
qaoa_config=qaoa_config,
optimizer_config=optimizer_config,
)
We also set the quantum backend we want to execute on:
from classiq import set_execution_preferences
from classiq.execution import ClassiqBackendPreferences, ExecutionPreferences
backend_preferences = ExecutionPreferences(
backend_preferences=ClassiqBackendPreferences(backend_name="aer_simulator")
)
qmod = set_execution_preferences(qmod, backend_preferences)
with open("integer_linear_programming.qmod", "w") as f:
f.write(qmod)
Synthesizing the QAOA Circuit and Solving the Problem¶
We can now synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:
from classiq import show, synthesize
qprog = synthesize(qmod)
show(qprog)
We now solve the problem using the generated circuit by using the execute method:
from classiq import execute
res = execute(qprog).result()
We can check the convergence of the run:
from classiq.execution import VQESolverResult
vqe_result = res[1].value
vqe_result.convergence_graph
Optimization Results¶
We can also examine the statistics of the algorithm:
import pandas as pd
optimization_result = pd.DataFrame.from_records(res[0].value)
optimization_result.sort_values(by="cost", ascending=False).head(5)
And the histogram:
optimization_result.hist("cost", weights=optimization_result["probability"])
Let us plot the solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmax()]
best_solution
Comparison to a classical solver¶
Lastly, we can 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)
References¶
[1]: Integer Programming (Wikipedia).
[2]: Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).
[3]: Barkoutsos, Panagiotis Kl, et al. "Improving variational quantum optimization using CVaR." Quantum 4 (2020): 256.