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Problem Formulation

In the PYOMO language, like any other algebraic modeling languages (AMLs), the optimization problem consists of three main components: decision variables, constraints, and an objective function. This section briefly introduces these components and explains the modalities supported by the platform.

Model

As an object-oriented language, PYOMO organizes the optimization problem into a single object of type ConcreteModel. It contains all the other components as Python attributes. The model is declared in the following way:

import pyomo.environ as pyo

model = pyo.ConcreteModel()

Decision Variables

The variables component is the “unknown part” of the model. The solver aims to assign values that constitute the best possible solution. Following is a description of the possible arguments for variable declaration in PYOMO.

In many optimization problems, the variables are indexed. The index set is provided as the first argument to the PYOMO var object.

index_set = [0, 1, 2, 3]

model.x = pyo.Var(index_set)

Next, state the variable domain. It may be a binary variable, an integer variable, a real variable, etc. The platform currently supports the binary and integer domains:

• Binary:

model.x = pyo.Var(index_set, domain=pyo.Binary)

• Integer:

model.x = pyo.Var(index_set, domain=pyo.NonNegativeIntegers, bounds=(0, 7))

If you know the variable value, fix it now to save qubits:

model.x[0].fix(3)

See PYOMO variables.

Constraints

Constraints enrich the descriptive power of optimization problems. Different scenarios, rules, regulations, and variable relations may be phrased as conditions on decision variables that must be satisfied. In PYOMO, specify the constraints using equality or inequality expressions. There are several ways to integrate them inside the existing PYOMO model:

  • Using the expr argument:
model = pyo.ConcreteModel()
model.x = pyo.Var(index_set, domain=pyo.Binary)
model.amount_constraint = pyo.Constraint(expr=sum(model.x[i] for i in model.x) == 3)
  • Using a rule argument and a separate Python function:
def amount_rule(model):
    return sum(model.x[i] for i in model.x) == 3


model.amount = pyo.Constraint(rule=amount_rule)
  • Using a Python constraint decorator:
@model.Constraint()
def amount_rule(model):
    return sum(model.x[i] for i in model.x) == 3

Index the constraints similarly to the variables.

def size_rule(model, i):
    return model.x[i] <= model.x[i + 1]


model.size_rule = pyo.Constraint(index_set[:-1], rule=size_rule)

See PYOMO constraints.

Objective Function

The objective function encodes the essence of the best solution. It is a function of the decision variables, which return a real value that the optimization solver tries to minimize or maximize.

It is incorporated into the PYOMO model similarly to the constraints: with an expr argument, with a rule argument, or with a decorator. An additional declaration argument is sense, which is set to minimize or maximize.

model.cost = pyo.Objective(expr=sum(model.x[i] for i in index_set), sense=pyo.maximize)

See PYOMO objective functions.

Importing

You can import all PYOMO components from the pyomo.environ sub-package:

import pyomo.environ as pyo

Model = pyo.ConcreteModel()
model.variable = pyo.Var()
model.constraint = pyo.Constraint()

Complete Example

from typing import Union, List
import numpy as np
import pyomo.core as pyo
import networkx as nx


def mis(graph: Union[nx.Graph, List[List[int]]]) -> pyo.ConcreteModel:
    if isinstance(graph, list):
        graph = nx.convert_matrix.from_numpy_matrix(np.array(graph))
    model = pyo.ConcreteModel()
    model.Nodes = pyo.Set(initialize=list(graph.nodes))
    model.Arcs = pyo.Set(initialize=list(graph.edges))

    model.x = pyo.Var(model.Nodes, domain=pyo.Binary)

    @model.Constraint(model.Arcs)
    def independent_rule(model, node1, node2):
        return model.x[node1] + model.x[node2] <= 1

    model.cost = pyo.Objective(expr=sum(list(model.x.values())), sense=pyo.maximize)

    return model

This example combines all components under a single Python function. It accepts user-defined inputs, and returns a complete PYOMO model with the assigned parameters. The function is saved in a file of the same name. The graph may be either an instance of the networkx.Graph class or the adjacency matrix in the form of List[List[int]].