What is a Pareto set and front in multi-objective optimization?

In multi-objective optimization, the Pareto set and Pareto front are fundamental concepts.

What is a Pareto set?

The Pareto set consists of all the possible solutions that are not dominated by any other solution in the search space. It represents the set of optimal solutions with respect to multiple conflicting objectives. In other words, a solution $x$ belongs to the Pareto set if there is no other solution $x'$ in the search space that is better than $x$ in all objectives and strictly better in at least one objective.

What is a Pareto front?

The Pareto front is the set of objective vectors corresponding to the solutions in the Pareto set defined by a particular constraint. It represents the trade-offs between different objectives, where improving one objective comes at the expense of worsening another. It visually represents the non-dominated solutions in the objective space.

Each point on the Pareto front represents a unique trade-off between conflicting objectives, and the set of all such points forms the Pareto front.

Let’s take a simple example to understand these concepts in greater detail.

Example

We'll consider a simple example with two objective functions $f_1(x)$ and $f_2(x)$defined as:

Where $x$ is a vector of decision variables and these equations are subject to the constraint $x_1 + x_2 \leq 3$. We want to minimize both $f_1(x)$ and $f_2(x)$.

Let’s visualize the Pareto set and Pareto front:

Making the Pareto set

This consists of all non-dominated feasible solutions. We can find this by solving the optimization problem and identifying the non-dominated solutions. In this case, due to the simplicity of the problem, we can deduce that the Pareto set lies along the boundary of the feasible region defined by the constraint $x_1 + x_2 \leq 3$. It includes all points on the line segment where $x_1 + x_2 = 3$, as these are the non-dominated feasible solutions.

Making the Pareto front

This is the set of objective vectors corresponding to the solutions in the Pareto set. We can obtain this by evaluating the objective functions for the solutions in the Pareto set. This consists of objective vectors corresponding to points on the Pareto set.

For each point on the line segment, we can compute the objective vector $(f_1(x),f_2(x))$.

The Pareto set lies on the line segment connecting the points $(0, 3)$ and $(3, 0)$, and the Pareto front consists of objective vectors corresponding to these points. Each point on the Pareto front represents a trade-off between minimizing $f_1(x)$ and minimizing $f_2(x)$.

Conclusion

To sum up, what we have learned, the Pareto set contains all non-dominated feasible solutions, while the Pareto front represents the objective vectors associated with these non-dominated solutions, showcasing the trade-offs between conflicting objectives. These concepts are crucial in multi-objective optimization for identifying and understanding the set of optimal solutions.