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

TL;DR

This work addresses selecting representative Pareto knee solutions and exploring locally changing regions of the Pareto front in multi-objective optimization. It introduces the sensitivity knee (snee) approach, which formalizes knee detection by minimizing the maximal rate of change across objectives using Pareto sensitivity, and extends it to constrained problems via KKT-based derivatives. It also defines and computes most-changing Pareto sub-fronts around a given Pareto solution using ellipsoidal neighborhoods derived from the Jacobian of the weighted-sum mapping, along with a MCM metric to quantify front variation. Numerical results on synthetic unconstrained and constrained problems illustrate that snee locates knee points consistent with the verbal knee criterion and reveals structured near-front variations useful for decision-makers exploring local trade-offs.

Abstract

When dealing with a multi-objective optimization problem, obtaining a comprehensive representation of the Pareto front can be computationally expensive. Furthermore, identifying the most representative Pareto solutions can be difficult and sometimes ambiguous. A popular selection are the so-called Pareto knee solutions, 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 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 the most-changing Pareto sub-fronts around a Pareto solution, where the points are distributed along directions of maximum change, which could be of interest in a decision-making process if one is willing to explore solutions around a current one. Our approach is still restricted to scalarized methods, in particular to the weighted-sum or epsilon-constrained methods, and require the computation or approximations of first- and second-order derivatives. We include numerical results from synthetic problems that illustrate the benefits of our approach.

Figures (13)

• Figure 1: Pareto neighborhoods for problem ZLT1 in the parameter, objective, and decision spaces. The upper, middle, and lower plots were obtained by sampling weights from $\Lambda$ at equidistant points within $\mathcal{B}_r (\lambda_c)$, $\mathcal{E}_{\alpha}(\lambda_c)$, and $\mathcal{E}_{\beta}(\lambda_c)$, respectively.
• Figure 2: Pareto neighborhoods for problem GRV1 (for $\bar{n}$ in Table \ref{['tab:test_prob']} equal to 2) in the parameter, objective, and decision spaces. The upper, middle, and lower plots were obtained by sampling weights from $\Lambda$ at equidistant points within $\mathcal{B}_r (\lambda_c)$, $\mathcal{E}_{\alpha}(\lambda_c)$, and $\mathcal{E}_{\beta}(\lambda_c)$, respectively.
• Figure 3: Pareto neighborhoods for problem VFM1 in the parameter, objective, and decision spaces. The upper, middle, and lower plots were obtained by sampling weights from $\Lambda$ at equidistant points within $\mathcal{B}_r (\lambda_c)$, $\mathcal{E}_{\alpha}(\lambda_c)$, and $\mathcal{E}_{\beta}(\lambda_c)$, respectively.
• Figure 4: Knee solution for problem GRV2 (for $\bar{n}$ in Table \ref{['tab:test_prob']} equal to 2). The upper plots show the parameter, objective, and decision spaces when applying the NM algorithm. The lower plots show the values of the $\mathop{\mathrm{MCF}}\nolimits$ and MCM over the iterations (left: NM, right: DIRECT).
• Figure 5: Knee solution for problem ZLT1. The upper plots show the parameter, objective, and decision spaces when applying the NM algorithm. The lower plots show the values of the $\mathop{\mathrm{MCF}}\nolimits$ and MCM over the iterations (left: NM, right: DIRECT).

Theorems & Definitions (6)

• Definition 1: Pareto dominance
• Definition 2: Pareto minimizer
• Definition 3: Ideal and nadir points
• Proposition 2.1
• Remark 3.1
• Remark 4.1