Learning to Analyze the Pareto-Optimal Front

As mentioned in the previous chapters, evolutionary multi- and many-objective optimization algorithms (EMâOAs) attempt to find a set of well-converged and well-diversified solutions to approximate the true Pareto front ( \(P\!F\) ). In general, a uniform distribution of solutions across \(P\!F\) is desired. However, this cannot be guaranteed due to the stochasticity involved in EMâOAs. In a contrasting scenario, even a biased distribution of solutions across \(P\!F\) , with a higher concentration of solutions in specific parts of \(P\!F\) , may be desired by the decision maker for a subsequent multi-criterion decision-making (MCDM) task. indexMulti-criterion decision-making (MCDM) To meet such requirements, this chapter presents a machine learning (ML)-based approach, which treats a given \(P\!F\) -approximation as input and trains an ML model to capture the relationship between pseudo-weight vectorsPseudo-weight vector derived from the objective vectors in the \(P\!F\) -approximation (F in  \(\mathcal{Z}\) ), and their underlying variable vectors (X in  \(\mathcal{X}\) ). Subsequently, the trained ML model is applied to predict the solution’s X vector for any desired pseudo-weight vector. In other words, the trained ML model is used to create new non-dominated solutions in any desired region of the obtained \(P\!F\) -approximation. Such new solutions could be created to fill apparent gaps in the input \(P\!F\) -approximation toward a more uniform distribution or to enhance the concentration of solutions as desired by the decision maker. The working and usefulness of the above post-optimality analysis basis approach have been demonstrated over several problem instances. However, this approach also has the potential to be integrated within an EMâOA to arrive at the desired distribution.