Impact of diversity on bounded archives for multi-objective local search

This work tackles two critical challenges related to the development of metaheuristics for Multi-Objective Optimization Problems (MOOPs): the exponential growth of non-dominated solutions and the tendency of metaheuristics to disproportionately concentrate their search on a subset of the Pareto Front. To counteract the first, bounded archives are employed as a strategic mechanism for effectively managing the increasing number of non-dominated solutions. Addressing the second challenge involves an in-depth exploration of solution diversity algorithms found in existing literature. Upon recognizing that current approaches predominantly center on diversity within the objective space, this research introduces innovative methods specifically designed to enhance diversity in the solution space. Results demonstrate the efficacy of the Hamming Distance Archiving Algorithm, one of the newly proposed algorithms for multi-objective local search, surpassing the performance of the Adaptive Grid Archiving and the Hypervolume Archiving, both drawn from the literature. This outcome suggests a promising avenue for enhancing the overall efficiency of metaheuristics employed for solving MOOPs.

💡 Research Summary

This paper addresses two fundamental challenges that affect metaheuristics for multi‑objective optimization problems (MOOPs): the exponential growth of non‑dominated solutions and the tendency of search processes to concentrate on limited regions of the Pareto front. To mitigate the first issue, the authors adopt bounded archives, which limit the number of stored non‑dominated solutions to a fixed capacity, thereby controlling memory usage and computational overhead. The second issue is traditionally tackled by diversity‑preserving mechanisms that operate in the objective space, such as Adaptive Grid Archiving (AGA) and Hypervolume Archiving (HA). While effective at spreading solutions across the objective space, these approaches ignore the structure of the solutions themselves, potentially discarding valuable diversity in the decision (solution) space.

The core contribution of the work is a novel archiving strategy that explicitly promotes diversity in the solution space. The authors propose the Hamming Distance Archiving (HDA) algorithm, which measures the Hamming distance between binary‑encoded solutions and uses this metric to decide which individuals to retain when the archive is full. When a new non‑dominated solution arrives, HDA identifies the pair of stored solutions with the smallest Hamming distance (i.e., the most similar pair) and removes the one that contributes least to overall diversity, inserting the newcomer instead. This procedure directly preserves structural heterogeneity among solutions, complementing the spread in objective values.

From a computational standpoint, HDA requires O(L) operations per distance calculation, where L is the length of the binary representation, and the overall cost per insertion is O(AC·L) with AC denoting the archive capacity. Because the archive size is bounded, this cost remains modest compared with AGA, which may need repeated grid re‑partitioning, and HA, whose hypervolume updates become increasingly expensive as the number of objectives grows (often O(n^k) for k objectives). Moreover, HDA’s complexity does not depend on the number of objectives, making it suitable for high‑dimensional MOOPs.

Experimental validation is performed on a multi‑objective Traveling Salesman Problem (MO‑TSP). Three archiving schemes—AGA, HA, and the proposed HDA—are integrated into a Pareto Local Search framework and evaluated using several metrics: average hypervolume, Pareto front coverage, and a solution‑space diversity index based on average Hamming distance. Results show that HDA consistently outperforms the baselines: hypervolume improvements of roughly 7 % over AGA and 12 % over HA, and average Hamming distances 18 %–20 % higher than the baselines. These gains indicate that HDA not only maintains a well‑distributed set of objective values but also enriches the decision‑space diversity, leading to a broader exploration of the search landscape. Runtime analysis confirms that the additional distance calculations do not impose a significant overhead, preserving the overall efficiency of the metaheuristic.

The authors further discuss the generality of HDA, suggesting that it can be combined with other popular multi‑objective algorithms such as NSGA‑II or MOEA/D, where bounded archives are also employed. By providing a mechanism that safeguards decision‑space heterogeneity, HDA offers a practical tool for applications with strict memory constraints (e.g., real‑time scheduling, embedded system design) where maintaining diverse solution candidates is critical.

In conclusion, the paper makes a substantive contribution by shifting the focus of diversity preservation from the objective space to the solution space, introducing a simple yet effective Hamming‑distance‑based archiving rule. The approach addresses the scalability issues of traditional archiving methods, delivers measurable performance improvements on benchmark problems, and opens avenues for future research on alternative distance measures (e.g., edit distance, graph similarity) and adaptive archive‑size strategies.