An Efficient Evolutionary Algorithm for Few-for-Many Optimization

TL;DR

The paper tackles Few-for-Many (F4M) optimization by promoting a compact set of solutions that jointly cover many conflicting objectives. It introduces SoM-EMOA, a $(\mu+1)$-evolutionary algorithm that optimizes a set-level coverage objective $G_{ws}$ with archive-guided offspring and a removal rule, and it pairs this with a scalable, R2-based benchmark suite to evaluate high-dimensional F4M instances. Empirical results on synthetic and real-world problems up to $m=100$ objectives show SoM-EMOA achieving superior coverage and robustness compared with state-of-the-art EMO methods and dedicated F4M solvers, with public code available at https://github.com/MOL-SZU/SoM-EMOA. The work advances practical F4M optimization by providing an effective solver and a flexible benchmark that decouples problem difficulty from naive front-coverage requirements, enabling broader application in domains with many objectives.

Abstract

Few-for-many (F4M) optimization, recently introduced as a novel paradigm in multi-objective optimization, aims to find a small set of solutions that effectively handle a large number of conflicting objectives. Unlike traditional many-objective optimization methods, which typically attempt comprehensive coverage of the Pareto front, F4M optimization emphasizes finding a small representative solution set to efficiently address high-dimensional objective spaces. Motivated by the computational complexity and practical relevance of F4M optimization, this paper proposes a new evolutionary algorithm explicitly tailored for efficiently solving F4M optimization problems. Inspired by SMS-EMOA, our proposed approach employs a $(μ+1)$-evolution strategy guided by the objective of F4M optimization. Furthermore, to facilitate rigorous performance assessment, we propose a novel benchmark test suite specifically designed for F4M optimization by leveraging the similarity between the R2 indicator and F4M formulations. Our test suite is highly flexible, allowing any existing multi-objective optimization problem to be transformed into a corresponding F4M instance via scalarization using the weighted Tchebycheff function. Comprehensive experimental evaluations on benchmarks demonstrate the superior performance of our algorithm compared to existing state-of-the-art algorithms, especially on instances involving a large number of objectives. The source code of the proposed algorithm will be released publicly. Source code is available at https://github.com/MOL-SZU/SoM-EMOA.

Figures (5)

• Figure 1: An illustration of F4M optimization with $m=100$ objectives and a cover set of size $k=5$, reproduced from lin2024few. Each polar plot represents one solution, where each angular tick corresponds to a specific objective index, and each radial tick indicates the objective value. The five solutions in (a)–(e) collectively handle different subsets of objectives in a complementary manner. Plot (f) shows the union of all five solutions, effectively handling the entire objective space.
• Figure 2: Illustration of the F4M formulation on the 2-objective DTLZ2 problem. The quarter-circle curve represents the Pareto front of DTLZ2. Ten directional scalarized objectives $F_1,\ldots,F_{10}$ are shown as arrows originating from the utopian point and pointing toward uniformly distributed directions on the front. A small set of three representative solutions located near $20^\circ$, $45^\circ$, and $70^\circ$ collectively covers these directional objectives, demonstrating how F4M approximates a continuous Pareto front using a compact solution set.
• Figure 3: The convergence curve of different algorithms on 5 existing F4M optimization benchmark problems. The number of objectives $m=50$ and the size of the solution set $k=5$.
• Figure 4: The convergence curve of different algorithms on the proposed F4M optimization benchmark problems. The number of objectives $m=50$ and the size of the solution set $k=5$.
• Figure 5: Comparison of runtime (in seconds) for different algorithms on synthetic and real-world problems. Each bar represents the average runtime of a single algorithm over all problems with $k=5$.

Theorems & Definitions (3)

• Definition 1: Dominance
• Definition 2: Pareto Optimality
• Definition 3: Few-for-Many (F4M) Optimization