A Firefly Algorithm for Mixed-Variable Optimization Based on Hybrid Distance Modeling

Large-scale multi-objective optimization problems (LSMOPs) are challenging due to their high-dimensional decision spaces. Fuzzy search is an effective technique for improving search efficiency, while decision variable analysis can reveal the distinct roles of variables in promoting convergence and maintaining diversity. However, existing fuzzy search methods generally employ a uniform search granularity for all variables, overlooking the heterogeneous search requirements implied by variable roles. To address this limitation, this paper proposes a Decision variable analysis-guided Differentiated Fuzzy Search method, termed DDFS. The proposed method establishes an explicit mapping between decision-variable roles and fuzzy search granularities. Decision variable analysis is employed to identify variable roles and search sensitivities, enabling different variable groups to adopt differentiated fuzzy search behaviors during offspring generation. Furthermore, a Dual-Indicator Stage Transition Mechanism is developed to dynamically adjust fuzzy-updating intensity throughout the evolutionary process, balancing early-stage search-space compression and late-stage convergence refinement. Extensive experiments on the LSMOP and UF benchmark suites with up to 1000 decision variables show that DDFS generally achieves competitive performance against several representative large-scale multi-objective evolutionary algorithms. The results suggest that explicitly incorporating decision-variable roles into fuzzy search can help improve optimization performance in high-dimensional decision spaces.

Mixed categorical-continuous optimization arises in many practical domains, yet remains challenging. In the black-box setting, evolution strategy-based approaches have shown promise in extending the efficiency and robustness of the CMA-ES to mixed-variable spaces. However, these methods exhibit worsened performance when strong categorical-continuous interactions are present, as their underlying search distributions assume independence between categorical and continuous variables. To address this limitation, we propose a bilevel optimization framework that explicitly captures such interactions by optimizing over categorical variables in an outer loop, and over continuous variables conditioned on each categorical configuration in an inner loop. We formulate each level of the bilevel problem as a stochastic relaxation under information-geometric optimization. To mitigate the high computational cost inherent to bilevel optimization, we introduce a warm-starting strategy that accelerates the lower-level search by selecting the best among multiple cached configurations and updating the cache after each iteration. Experimental results on binary-continuous domain demonstrate that the proposed method outperforms existing state-of-the-art approaches in interaction-handling capability while also being more computationally efficient across benchmarks encompassing both previously reported and newly proposed types of interaction.

This paper presents the constrained Hybrid Metaheuristic (cHM) algorithm as a general framework for continuous optimisation. Unlike many existing metaheuristics that are tailored to specific function classes or problem domains, cHM is designed to operate across a broad spectrum of objective functions, including those with unknown, heterogeneous, or complex properties such as non-convexity, non-separability, and varying smoothness. We provide a formal description of the algorithm, highlighting its modular structure and two-phase operation, which facilitates dynamic adaptation to the problem's characteristics. A key feature of cHM is its ability to harness synergy between both candidate solutions and component metaheuristic strategies. This property allows the algorithm to apply the most appropriate search behaviour at each stage of the optimisation process, thereby improving convergence and robustness. Our extensive experimental evaluation on 28 benchmark functions demonstrates that cHM consistently matches or outperforms traditional metaheuristics in terms of solution quality and convergence speed. In addition, a practical application of the algorithm is demonstrated for a feature selection problem in the context of data classification. The results underscore its potential as a versatile and effective black-box optimiser suitable for both theoretical research and practical applications.