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

Ousmane Tom Bechir; Adán José-García; Zaineb Chelly Garcia; Vincent Sobanski; Clarisse Dhaenens

cs.NEMar 25, 2026

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

Authors: Ousmane Tom Bechir, Adán José-García, Zaineb Chelly Garcia, Vincent Sobanski, Clarisse Dhaenens

Organizations: Univ. Lille, CNRS, Inria, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France · Univ. Lille, Inserm, CHU Lille, U1286 – INFINITE, Lille, France · Institut Universitaire de France (IUF), Paris, France

Abstract

Several real-world optimization problems involve mixed-variable search spaces, where continuous, ordinal, and categorical decision variables coexist. However, most population-based metaheuristic algorithms are designed for either continuous or discrete optimization problems and do not naturally handle heterogeneous variable types. In this paper, we propose an adaptation of the Firefly Algorithm for mixed-variable optimization problems (FAmv). The proposed method relies on a modified distance-based attractiveness mechanism that integrates continuous and discrete components within a unified formulation. This mixed-distance approach enables a more appropriate modeling of heterogeneous search spaces while maintaining a balance between exploration and exploitation. The proposed method is evaluated on the CEC2013 mixed-variable benchmark, which includes unimodal, multimodal, and composition functions. The results show that FAmv achieves competitive, and often superior, performance compared with state-of-the-art mixed-variable optimization algorithms. In addition, experiments on engineering design problems further highlight the robustness and practical applicability of the proposed approach. These results indicate that incorporating appropriate distance formulations into the Firefly Algorithm provides an effective strategy for solving complex mixed-variable optimization problems.

Figures & tables

Category	Function ID	Function Name	Continuous	Separable	Symmetric	Optimum
Unimodal	F1	Sphere Function	✔	✔	✔	-1400
	F2	Rotated High Conditioned Elliptic Function	✔	✕	✔	-1300
	F3	Rotated Bent Cigar Function	✔	✕	✕	-1200
	F4	Rotated Discus Function	✔	✕	✕	-1100
	F5	Different Powers Function	✔	✔	✔	-1000
Multimodal	F6	Rotated Rosenbrocks Function	✔	✕	✕	-900

Table 1: The 28 benchmark functions from CEC 2013 and their main characteristics. The presence or absence of a property is indicated by ✔or ✕, respectively. Continuous denotes functions defined over continuous domains; separable indicates that the objective can be decomposed into independent contributions per variable; symmetric refers to invariance under permutation or sign changes of variables.

Variable	Description	Domain
$d_{s}$	Shell thickness (discrete variable)	multiple of $0.0625$
$d_{h}$	Head thickness (discrete variable)	multiple of $0.0625$
$r$	Inner radius of the pressure vessel	$10\leq r\leq 200$
$L$	Length of the cylindrical section	$10\leq L\leq 200$
Constraints
$g_{1}(x)$	$-d_{s}+0.0193r\leq 0$

Table 2: Mathematical formulation of the Pressure Vessel design problem.

Constraints
Variable	Description	Domain
$x_{1}$	Weld thickness	continuous
$x_{2}$	Weld length	continuous
$x_{3}$	Beam height	continuous
$x_{4}$	Beam thickness	continuous
$g_{1}(x)$	$\tau(x)-\tau_{\max}\leq 0$

Table 3: Mathematical formulation of the Welded Beam design problem.

Constraints
Variable	Description	Domain
$d$	Wire diameter	continuous, $d\geq 0.2$
$D$	Mean coil diameter	continuous, $D\leq 3.0$
$N$	Number of active coils	discrete
$g_{1}(x)$	$\dfrac{8C_{f}P_{\max}D}{\pi d^{3}}-S\leq 0$
$g_{2}(x)$	$\delta_{\max}+1.05(N+2)d-L_{\text{free}}\leq 0$

Table 4: Mathematical formulation of the Coil Spring design (CSD) problem.

Figure 1: Overall performance of MVO-based methods on CEC benchmark functions: baseline algorithms (left) and FA-based variants (right). The x-axis shows the number of results statistically similar to the best, and the y-axis shows the number of best results. Algorithms in the top-right achieve the best performance. See Tables 5 and 7 for details.

Unimodal functions (F1–F5)
Function	Statistic	PSOmv	DEmv	GA	FA	FAmv H	FAmv ${}^{*}_{\mathrm{H}}$	FAmv G	FAmv ${}^{*}_{\mathrm{G}}$
F1	Mean	1.19e+04	5.76e+02	1.93e+03	1.71e+05	4.81e+02	4.47e+02 *	5.16e+02	4.79e+02
	STD	3.03e+03	2.13e+02	4.95e+02	1.92e+04	5.60e+01	8.43e+01 *	5.69e+01	7.92e+01
F2	Mean	3.87e+08	3.31e+08	1.25e+08	7.10e+09	2.43e+07 *	2.92e+07	3.08e+07	3.46e+07
	STD	1.26e+08	4.92e+07	3.04e+07	1.98e+09	5.69e+06 *	8.08e+06	8.63e+06	8.99e+06
F3	Mean	1.49e+11	4.77e+09	2.39e+10	1.60e+21	1.63e+09 *	2.07e+09	2.25e+09	4.14e+09

Table 5: Performance of the mixed-variable optimization methods evaluated on the 28 CEC benchmark functions. Performance is evaluated in terms of the absolute error (AE). The best value (lowest average) is indicated with (*), whereas statistically similar results are highlighted in bold .

Figure 2: Performance in terms of the absolute error (the lower the better) scored by the seven mixed-variable optimization methods on the CEC benchmark for the unimodal functions .

Figure 3: Performance in terms of the absolute error (the lower the better) scored by the seven mixed-variable optimization methods on the CEC benchmark for the multimodal functions .

Figure 4: Performance in terms of the absolute error (the lower the better) scored by the seven MVO methods on composite functions in the CEC benchmark.

Function	PSOmv	DEmv	GA	FAmv H	FAmv G
BEAM	0.85 $\pm$ 0.30	0.51 $\pm$ 0.18	1.11 $\pm$ 0.64	0.49 * $\pm$ 0.18	0.51 $\pm$ 0.27
CSD	1.13 $\pm$ 0.05	1.08 * $\pm$ 0.03	1.74 $\pm$ 0.81	1.18 $\pm$ 0.05	1.18 $\pm$ 0.06
Vessel	6363.68 $\pm$ 1978.77	4331.97 $\pm$ 2040.48	2724.67 $\pm$ 2055.74	4283.55 $\pm$ 2164.54	1903.08 * $\pm$ 2090.71

Table 6: Performance of MVO methods on the three engineering design problems, measured by the absolute error (AE). The best value (lowest) is indicated with (*), and statistically similar results are highlighted in bold .

Panel of Figure 5 — Figure 5: Performance of MVO methods on the three engineering design problems (BEAM, CSD, and Vessel), measured by absolute error (top) and convergence curves (bottom). Lower values indicate better performance.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: The influence of the control parameter $k$ on the probability function $p_{\alpha}$ presented in Section 3.2

Figure 7: Convergence curves of the MVO methods on unimodal functions in the CEC benchmark.

Figure 8: Convergence curves of the MVO methods on multimodal functions in the CEC benchmark.

Figure 9: Convergence curves of the MVO methods on composite functions in the CEC benchmark.

Func.	Statistic	FA	FAmv G	FAmv ${}^{\alpha}_{\mathrm{G}}$	FAmv ${}^{\alpha\gamma}_{\mathrm{G}}$	FAmv ${}^{\gamma}_{\mathrm{G}}$	FAmv ${}^{D}_{\mathrm{G}}$	FAmv H	FAmv ${}^{\alpha}_{\mathrm{H}}$	FAmv ${}^{\alpha\gamma}_{\mathrm{H}}$	FAmv ${}^{\gamma}_{\mathrm{H}}$
F1	Mean	1.71e+05	5.16e+02	4.35e+02 *	4.79e+02	5.53e+02	6.42e+02	4.81e+02	4.47e+02	4.90e+02	5.31e+02
F1	STD	1.92e+04	5.69e+01	7.08e+01 *	7.92e+01	6.57e+01	7.28e+01	5.60e+01	8.43e+01	7.80e+01	6.28e+01
F2	Mean	7.10e+09	3.08e+07	3.77e+07	3.46e+07	3.09e+07	4.89e+07	2.43e+07	2.92e+07	2.55e+07	2.30e+07 *
F2	STD	1.98e+09	8.63e+06	1.25e+07	8.99e+06	6.51e+06	1.86e+07	5.69e+06	8.08e+06	7.25e+06	4.64e+06 *
F3	Mean	1.60e+21	2.25e+09	5.10e+09	4.14e+09	2.84e+09	3.68e+09	1.63e+09 *	2.07e+09	2.09e+09	2.12e+09
F3	STD	3.08e+21	1.23e+09	5.22e+09	4.55e+09	1.07e+09	1.66e+09	5.47e+08 *	8.97e+08	9.82e+08	1.00e+09

Appendix

Table 7: Performance of the FA-based variants evaluated on the 28 CEC benchmark functions. Performance is evaluated in terms of the absolute error (AE). The best value (lowest average) is indicated with (*), whereas statistically similar results are highlighted in bold .

Figure 10: Performance scored by the FA-based variants on the CEC benchmark for the unimodal functions .

Figure 11: Performance scored by the FA-based variants on the CEC benchmark for the multimodal functions .

Figure 12: Performance scored by the FA-based variants on the CEC benchmark for the composition functions .

Figure 13: Convergence curves of the FA-based variants on unimodal functions in the CEC benchmark.

Figure 14: Convergence curves of the FA-based variants on multimodal functions in the CEC benchmark.

Figure 15: Convergence curves of the FA-based variants on composition functions in the CEC benchmark.

Metaheuristic Optimization

Metaheuristic optimization develops flexible search strategies for finding strong solutions to difficult problems when exact methods are too costly.

See all 115 papers

Decision Variable Analysis-Guided Differentiated Fuzzy Search for Large-Scale Multi-Objective Optimization
Jul 18, 2026Boxi Xiao, Hui Bai, Jinhua Zheng +2Multi-Objective Evolutionary Optimization Multi-Objective Optimization
Key Laboratory of Hunan Province for Internet of Things and Information Security, Xiangtan University, Xiangtan 411105, China, and also with the Key Laboratory of Intelligent Computing and Information Processing, Ministry of Education, Xiangtan University, Xiangtan 411105, China
Mixed-Categorical Black-Box Optimization via Information-Geometric Bilevel Decomposition
Jun 11, 2026Marc Ong, Shinichi Shirakawa, Youhei AkimotoBlack-Box Optimization Bilevel Optimization
University of Tsukuba, Tsukuba, Japan · Yokohama National University, Yokohama, Japan · RIKEN Center for Advanced Intelligence Project, Tokyo, Japan
Constrained Hybrid Metaheuristic: A Universal Framework for Continuous Optimisation
Mar 18, 2026Piotr A. Kowalski, Szymon Kucharczyk, Jacek MańdziukMetaheuristic Optimization Black-Box Optimization
Faculty of Physics and Applied Computer Science, AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Cracow, Poland. · Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland. · AGH Doctoral School, AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Cracow, Poland. +2

Jul 18, 2026J/K move · Enter open · S save

Decision Variable Analysis-Guided Differentiated Fuzzy Search for Large-Scale Multi-Objective Optimization

Boxi Xiao, Hui Bai, Jinhua Zheng, Yu Li+1

Key Laboratory of Hunan Province for Internet of Things and Information Security, Xiangtan University, Xiangtan 411105, China, and also with the Key Laboratory of Intelligent Computing and Information Processing, Ministry of Education, Xiangtan University, Xiangtan 411105, China

Multi-Objective Evolutionary Optimization Multi-Objective Optimization

Open paper AskPDF arXiv

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.

Abstract

Metaheuristic Optimization

Explore similar work

Decision Variable Analysis-Guided Differentiated Fuzzy Search for Large-Scale Multi-Objective Optimization

Mixed-Categorical Black-Box Optimization via Information-Geometric Bilevel Decomposition

Constrained Hybrid Metaheuristic: A Universal Framework for Continuous Optimisation