Example on using the scalarization methods for scalarizing and minimizing a problem which is based on discrete data

In this example, we will go through the following two topics: 1. How to define a scalarization method for scalarizing discrete data representing a multiobjective optimization problem; 2. How to find a solution to the scalarized problem.

We will start by defining simple 2-dimensional data representing a set of Pareto optimal solutions.

import numpy as np
import matplotlib.pyplot as plt

f1 = np.linspace(1, 100, 50)
f2 = f1[::-1]**2

plt.scatter(f1, f2)
plt.title("Pareto front")
plt.xlabel("f1")
plt.ylabel("f2")
plt.show()

Let us pretend the points represent the Pareto front for a problem with two objectives to be minimized. We can easily determine the ideal and nadir points as follows:

pfront = np.stack((f1, f2)).T

ideal = np.min(pfront, axis=0)
nadir = np.max(pfront, axis=0)

print(f"Ideal point: {ideal}")
print(f"Nadir point: {nadir}")

plt.scatter(f1, f2, label="Pareto front")
plt.scatter(ideal[0], ideal[1], label="ideal")
plt.scatter(nadir[0], nadir[1], label="nadir")
plt.title("Pareto front")
plt.xlabel("f1")
plt.ylabel("f2")
plt.legend()
plt.show()

Ideal point: [1. 1.]
Nadir point: [  100. 10000.]

Next, suppose we would like to find a solution close to the point (80, 2500), let us define that point as a reference point.

z = np.array([80, 2500])

Clearly, z is not on the Pareto front. We can find a closest solution by scalarizing the problem using an achievement scalarizing function (ASF) and minimizing the related achievement scalarizing optimization problem. We will do that next.

from desdeo_tools.scalarization.ASF import PointMethodASF
from desdeo_tools.scalarization.Scalarizer import DiscreteScalarizer
from desdeo_tools.solver.ScalarSolver import DiscreteMinimizer

# define the achievement scalarizing function
asf = PointMethodASF(nadir, ideal)
# the scalarizer
dscalarizer = DiscreteScalarizer(asf, scalarizer_args={"reference_point": z})
# the solver (minimizer)
dminimizer = DiscreteMinimizer(dscalarizer)

solution_i = dminimizer.minimize(pfront)

print(f"Index of the objective vector minimizing the ASF problem: {solution_i}")

Index of the objective vector minimizing the ASF problem: 32

When a scalar problem is minimized using a DiscreteMinimizer, the result will be the index of the objective vector in the supplied vector argument minimizing the DiscreteScalarizer defined in DiscreteMinimizer. This is done because it is assumed that the corresponding decision variables are also kept in a vector somewhere, and the variables are ordered in a manner where the ith element in vectors corresponds to the ith variables in the vector storing the variables.

Anyway, let us plot the solution:

plt.scatter(f1, f2, label="Pareto front")
plt.scatter(ideal[0], ideal[1], label="ideal")
plt.scatter(nadir[0], nadir[1], label="nadir")
plt.scatter(pfront[solution_i][0], pfront[solution_i][1], label="Preferred solution")
plt.scatter(z[0], z[1], label="Reference point")
plt.title("Pareto front")
plt.xlabel("f1")
plt.ylabel("f2")
plt.legend()
plt.show()

Suppose now that there is the following constraint to our problem: values of f1 should be less than 50 or more than 77. We can easily deal with this situation as well, and we will conclude our example here.

# define the constraint function, it should return either True of False for each
# objective vector defined in its argument.
def con(fs):
    fs = np.atleast_2d(fs)

    return np.logical_or(fs[:, 0] < 50, fs[:, 0] > 77)

dminimizer_con = DiscreteMinimizer(dscalarizer, con)

solution_con = dminimizer_con.minimize(pfront)

mask = con(pfront)
plt.scatter(f1[mask], f2[mask], label="Pareto front with constraint")
plt.scatter(ideal[0], ideal[1], label="ideal")
plt.scatter(nadir[0], nadir[1], label="nadir")
plt.scatter(pfront[solution_i][0], pfront[solution_i][1], label="Preferred solution wihtout constraint")
plt.scatter(pfront[solution_con][0], pfront[solution_con][1], label="Preferred solution with constraint", color="black")
plt.scatter(z[0], z[1], label="Reference point")
plt.title("Pareto front")
plt.xlabel("f1")
plt.ylabel("f2")
plt.legend()
plt.show()