Non-Extreme Individual Minima for Improved Pareto Front Sampling Efficiency and Decision-Making

TL;DR

A computationally efficient algorithm is provided to determine non-extreme individual minima that allow for a more practical restriction of the Pareto front sampling within a refined utopia-nadir hyperbox, provide a meaningful basis for image space normalization, and can enhance decision-making techniques, such as knee-point methods, by focusing on regions with acceptable trade-offs.

Abstract

In multi-objective optimization, the set of optimal trade-offs -- the Pareto front -- often contains regions that are extremely steep or flat. The Pareto optimal points in these regions are typically of limited interest for decision-making, as the marginal rate of substitution is extreme: a marginal improvement in one objective necessitates a significant deterioration in at least one other objective. These unfavorable trade-offs frequently occur near the individual minima, where single objectives attain their minimum values without considering the remaining criteria. To address this, we propose the concept of \emph{non-extreme individual minima} that relies on the notion of $L$-practical proper efficiency. These points can serve as a less sensitive replacement for \emph{standard} individual minima in subsequent related methods. Specifically, they allow for a more practical restriction of the Pareto front sampling within a refined utopia-nadir hyperbox, provide a meaningful basis for image space normalization, and can enhance decision-making techniques, such as knee-point methods, by focusing on regions with acceptable trade-offs. We provide a computationally efficient algorithm to determine these non-extreme individual minima by solving at most $2n_J$ standard weighted-sum scalarizations, where $n_J$ is the number of objectives. To ensure robustness across varying objective scales, the method incorporates an integrated image space normalization strategy. Numerical examples, specifically a convex academic case and a non-convex real-world application, demonstrate that the method successfully excludes practically irrelevant regions in the image space.