Pareto sensitivity, most-changing sub-fronts, and knee solutions

Abstract

When dealing with a multi-objective optimization problem, obtaining a comprehensive representation of the set of Pareto optimal solutions can be computationally expensive. However, identifying the most representative solutions can be difficult and sometimes ambiguous, since what constitutes a representative solution depends on the decision maker's preferences. A popular selection are the so-called Pareto knee solutions, which correspond to nondominated points on the Pareto front where a small improvement in any objective leads to a large deterioration in at least one other objective. In this paper, using Pareto sensitivity, we show how to compute Pareto knee solutions according to their verbal (informal) definition of least maximal change. We refer to the resulting approach as the sensitivity knee (snee) approach, and we apply it to unconstrained and constrained problems. Pareto sensitivity can also be used to compute local most-changing Pareto sub-fronts around a nondominated point, where points on the sub-fronts are distributed along directions of maximum change. Our approach is still restricted to scalarized methods, in particular to the weighted-sum or epsilon-constrained methods, and requires the computation or approximations of first- and second-order derivatives. We include numerical results from synthetic problems that illustrate the benefits of our approach.