Local Optimal Sets and Bounded Archiving on Multi-objective NK-Landscapes with Correlated Objectives

The properties of local optimal solutions in multi-objective combinatorial optimization problems are crucial for the effectiveness of local search algorithms, particularly when these algorithms are based on Pareto dominance. Such local search algorithms typically return a set of mutually nondominated Pareto local optimal (PLO) solutions, that is, a PLO-set. This paper investigates two aspects of PLO-sets by means of experiments with Pareto local search (PLS). First, we examine the impact of several problem characteristics on the properties of PLO-sets for multi-objective NK-landscapes with correlated objectives. In particular, we report that either increasing the number of objectives or decreasing the correlation between objectives leads to an exponential increment on the size of PLO-sets, whereas the variable correlation has only a minor effect. Second, we study the running time and the quality reached when using bounding archiving methods to limit the size of the archive handled by PLS, and thus, the maximum size of the PLO-set found. We argue that there is a clear relationship between the running time of PLS and the difficulty of a problem instance.

💡 Research Summary

This paper investigates two fundamental aspects of Pareto local optimal (PLO) sets in multi‑objective combinatorial optimization, using Pareto Local Search (PLS) on correlated multi‑objective NK‑landscapes (ρ‑MNK). First, the authors examine how problem characteristics—namely the number of objectives (m), the epistatic degree (k), and the correlation coefficient between objectives (ρ)—affect the size and structure of PLO‑sets. By exhaustively enumerating the solution space for instances with n ∈ {8, 16}, k ∈ {1, 2, 4, 8}, m ∈ {2, 3, 5}, and ρ ∈ {−0.7, −0.2, 0.0, 0.2, 0.7}, they generate 91 distinct problem configurations and run each 25 times with different random seeds, yielding 20 475 PLS executions. The results show that the cardinality of maximal PLO‑sets grows exponentially with the number of objectives and with decreasing correlation (more conflict between objectives). The epistatic degree k has a much smaller impact, and the variance across random seeds is low, indicating that for a given instance the PLO‑set size is fairly stable.

Second, the paper evaluates the effect of bounding the archive size during PLS. Two state‑of‑the‑art archiving strategies are considered: a hypervolume‑based archiver (HV_A) and a multi‑level grid archiver (MGA). Both limit the archive to a maximum of µ ∈ {10, 20, 40, 80} solutions; when a new nondominated solution would exceed µ, the archiver discards one existing solution according to its selection rule. The quality of the resulting PLO‑sets is measured by (i) hypervolume relative difference (hvr) and (ii) multiplicative epsilon (ε), both lower values indicating a set closer to the true Pareto front. As µ increases, both quality metrics improve, but the improvement plateaus beyond µ ≈ 80. When the number of objectives is high and ρ is low, the quality degradation caused by a small µ is especially pronounced. HV_A tends to produce slightly better hypervolume values than MGA, while both achieve comparable ε values, suggesting that the choice between them should be guided by the decision maker’s preferred quality indicator.

The third experimental focus is the “length” of PLS, i.e., the number of iterations required to terminate. With an unbounded archive, PLS often requires many thousands of iterations, reflecting the exponential growth of the PLO‑set. Bounding the archive dramatically reduces the number of iterations, with µ = 10 yielding the shortest runs. However, shorter runs correlate with lower solution quality, confirming a trade‑off between computational effort and approximation accuracy. Moreover, longer PLS runs are associated with larger PLO‑sets, implying that instances with many local Pareto basins cause the algorithm to stall more frequently, thereby serving as a proxy for problem difficulty.

Overall, the study delivers several actionable insights: (1) the number of objectives and inter‑objective correlation dominate the size of PLO‑sets; (2) bounded archiving can control the exponential blow‑up of the archive and runtime, but at the cost of solution quality, which can be mitigated by increasing µ; (3) HV_A offers a modest advantage in hypervolume preservation, while MGA is competitive in ε; (4) the PLS length can be used as an empirical difficulty indicator for multi‑objective NK‑landscapes. These findings guide the design of future multi‑objective local search algorithms, suggesting that adaptive archive‑size policies or hybrid strategies that switch between unbounded and bounded modes could balance exploration depth with computational feasibility, especially for high‑dimensional, highly conflicting objective spaces.