Variable Search Stepsize for Randomized Local Search in Multi-Objective Combinatorial Optimization

TL;DR

This work tackles multi-objective combinatorial optimization (MOCOPs) by addressing the limitations of fixed-neighborhood local search. It introduces Variable Size Randomized Local Search (VS-RLS), which progressively expands and then refines the search neighbourhood to escape Pareto local optima and improve diversity, all within an archive-based Pareto framework. Across four representative MOCOPs (Knapsack, TSP, QAP, NK-Landscape), VS-RLS demonstrates competitive HV performance and robust behavior, often outperforming both fixed-stepsize local search and several MOEAs, particularly in maintaining diverse Pareto fronts. The study also provides extensive parameter analyses and ablation experiments showing the importance of the exploration phase and reasonable tuning of the exploitation threshold, with future directions including integration with population-based methods and data-driven adaptation of stepsize.

Abstract

Over the past two decades, research in evolutionary multi-objective optimization has predominantly focused on continuous domains, with comparatively limited attention given to multi-objective combinatorial optimization problems (MOCOPs). Combinatorial problems differ significantly from continuous ones in terms of problem structure and landscape. Recent studies have shown that on MOCOPs multi-objective evolutionary algorithms (MOEAs) can even be outperformed by simple randomised local search. Starting with a randomly sampled solution in search space, randomised local search iteratively draws a random solution (from an archive) to perform local variation within its neighbourhood. However, in most existing methods, the local variation relies on a fixed neighbourhood, which limits exploration and makes the search easy to get trapped in local optima. In this paper, we present a simple yet effective local search method, called variable stepsize randomized local search (VS-RLS), which adjusts the stepsize during the search. VS-RLS transitions gradually from a broad, exploratory search in the early phases to a more focused, fine-grained search as the search progresses. We demonstrate the effectiveness and generalizability of VS-RLS through extensive evaluations against local search and MOEAs methods on diverse MOCOPs.

Figures (6)

• Figure 1: An illustration of the failure of fixed stepsize randomized local search on a bi-objective minimization toy example (adapted from III1). Each vertex represents a solution, and the numbers in the upper-right corner indicate its objective values. An edge connects two solutions whose decision variables differ by one bit. The blue point represents the starting location of the search, while the dashed rectangle indicates the search region when the stepsize is fixed to one (i.e., 1-bit neighbourhood). When performing randomized local search with a stepsize of one, one of the four neighbouring solutions around the blue point is randomly selected as an offspring. The red point represents a solution that is accepted into the archive according to the dominance relation. As can be seen, once the search reaches the local optimum at $(6,6)$ (red point), all neighbouring solutions within the fixed stepsize are dominated by it, and the algorithm is therefore unable to move forward.
• Figure 2: An illustration of the proposed VS-RLS on the bi-objective toy example in Fig. \ref{['F1']}. The dashed rectangles in different colours indicate the search regions corresponding to different stepsize, where the yellow and pink rectangles represent the search regions with stepsize two and three, respectively. Specifically, in the first subfigure, assume that the algorithm is trapped at a local optimum $(6,6)$ when the stepsize is equal to one. VS-RLS first randomly samples one solution from the search region (black dashed rectangle) corresponding to stepsize of one. Since the sampled solution is not acceptable for the archive (i.e., being dominated by at least one solution in the archive), the stepsize is increased to two (yellow dashed rectangle) and another solution is randomly sampled. Although multiple acceptable solutions exist in the search region with stepsize two, such as $(3,8)$, $(4,7)$, and $(6,5)$, suppose that none of these acceptable solutions is sampled. The stepsize is therefore further increased to three, as shown in the second subfigure. This process illustrates that, even if no acceptable solution is sampled under small stepsize, expanding the stepsize allows the algorithm to sample solutions from a broader region and thus escape from the local optimum. At the same time, since the stepsize is increased progressively from small to large values, VS-RLS preserves the ability to sample acceptable solutions near the current solution. Once an acceptable solution is sampled, the current iteration is terminated. For example, in the third subfigure, suppose that the solution $(2,9)$ is sampled when the stepsize is equal to three. Since $(2,9)$ is non-dominated with respect to $(6,6)$, it is added to the archive. The algorithm then starts a new iteration by randomly selecting a solution from the archive as the parent. In this case, if $(2,9)$ is selected as the parent, its stepsize of one region may be more likely to contain acceptable solutions than that of $(6,6)$, allowing the search to move from a less promising region to a more promising one.
• Figure 3: The final archive sets are the results of the seven algorithms on 4 different settings of the first experiment from a single run. This particular run is associated with the result closest to the average HV value.
• Figure 4: The final archive sets are the results of the seven algorithms on 4 different settings of the second experiment from a single run. This particular run is associated with the result closest to the average HV value.
• Figure 5: The final archive sets are the results of the seven algorithms on 4 different settings of the third experiment from a single run. This particular run is associated with the result closest to the average HV value.