Dogfight Search: A Swarm-Based Optimization Algorithm for Complex Engineering Optimization and Mountainous Terrain Path Planning

Yujing Sun; Jie Cai; Xingguo Xu; Yuansheng Gao; Lei Zhang; Kaichen Ouyang; Zhanyu Liu

cs.AIMar 30, 2026

Dogfight Search: A Swarm-Based Optimization Algorithm for Complex Engineering Optimization and Mountainous Terrain Path Planning

Authors: Yujing Sun, Jie Cai, Xingguo Xu, Yuansheng Gao, Lei Zhang, Kaichen Ouyang, Zhanyu Liu

Organizations: College of Science, Liaoning Technical University, Fuxin, 123000, China · School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China · College of Computer Science and Technology, Zhejiang University, Hangzhou, 310027, China · Department of Physics, University of Science and Technology of China, Hefei, 230026, China

Abstract

Dogfight is a tactical behavior of cooperation between fighters. Inspired by this, this paper proposes a novel metaphor-free metaheuristic algorithm called Dogfight Search (DoS). Unlike traditional algorithms, DoS draws algorithmic framework from the inspiration, but its search mechanism is constructed based on the displacement integration equations in kinematics. Through experimental validation on CEC2017 and CEC2022 benchmark test functions, 10 real-world constrained optimization problems and mountainous terrain path planning tasks, DoS significantly outperforms 7 advanced competitors in overall performance and ranks first in the Friedman ranking. Furthermore, this paper compares the performance of DoS with 3 SOTA algorithms on the CEC2017 and CEC2022 benchmark test functions. The results show that DoS continues to maintain its lead, demonstrating strong competitiveness. The source code of DoS is available at https://ww2.mathworks.cn/matlabcentral/fileexchange/183519-dogfight-search.

Figures & tables

Algorithm	Engineering Contribution
Dual-Pheromone Crossover Ant Colony Optimization (DPX-ACO) ( Wang et al., 2025b )	The proposed method significantly reduces the overall production costs in large-scale cotton blending optimization. Comprehensive experimental analyses further demonstrate its effectiveness, with several benchmark scenarios achieving cost reductions exceeding 30%.
Improved Discrete Particle Swarm Optimization (IDPSO) ( Wu et al., 2025 )	Provided superior scheduling solutions for automated guided vehicle manufacturing processes by considering dynamic changes, operator skills, and equipment maintenance, leading to a notable reduction in task completion time.
Artificial Rabbits Optimization (ARO) ( Kuo et al., 2025 )	Optimized control parameters of biped robots, enabling fast adaptation to external disturbances and maintaining balance in complex environments such as disaster response scenarios.
Improved Ant Colony Optimization (IACO) ( Wangying and Naiming, 2025 )	Applied to helicopter route planning for forest fire prevention, reducing total flight distance by 4.1% and improving emergency response efficiency by 7.3% in dynamic environments.

Table 1: Contributions of metaheuristic algorithms to real-world engineering problems.

Figure 1: The classification of metaheuristic algorithms.

Panel of Figure 2 — Figure 2: The inspiration diagram of DoS search strategies.

Figure 3: The iterative flowchart of DoS.

Figure 4: Visualization of the free flight strategy.

Figure 5: Visualization of the maneuver locking strategy.

Figure 6: Visualization of the maneuver evasion strategy.

No.	Type	Function	$F^{*}$
F1	Unimodal	Shifted and Rotated Bent Cigar Function	100
F3		Shifted and Rotated Zakharov Function	300
F4	Multimodal	Shifted and Rotated Rosenbrock’s Function	400
F5		Shifted and Rotated Rastrigin’s Function	500
F6		Shifted and Rotated Expanded Scaffer’s F6 Function	600
F7		Shifted and Rotated Lunacek Bi-Rastrigin Function	700

Table 2: CEC2017 Benchmark Test Functions.

No.	Type	Function	$F^{*}$
C1	Unimodal	Shifted and full Rotated Zakharov Function	300
C2	Basic	Shifted and full Rotated Rosenbrock’s Function	400
C3		Shifted and full Rotated Expanded Schaffer’s F6 Function	600
C4		Shifted and full Rotated Non-Continuous Rastrigin’s Function	800
C5		Shifted and full Rotated Levy Function	900
C6	Hybrid	Hybrid Function 1 (N = 3)	1800

Table 3: CEC2022 Benchmark Test Functions.

	CEC2017	CEC2022
Selected Dimensions ( $D$ )	30, 50, 100	10, 20
Number of Evaluations ( $E$ )	$D×10000$	200000 ( $D=10$ ), 1000000 ( $D=20$ )
Number of Independent Runs	51 times	30 times

Table 4: Standard Test Conditions.

Parameters	Value
Population size ( $N$ )	50
Leader to follower ratio ( $k_{1}$ )	0.3
Probability factor of free flight strategy ( $k_{2}$ )	-2.5
Time ratio of free flight strategy ( $k_{3}$ )	0.2
Population random search ratio coefficient of regular fighters ( $k_{4}$ )	0.05
Flight speed ratio factor ( $k_{5}$ )	0.5

Table 5: DoS parameter settings.

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F1	Mean	100	3573.6777	4778.6883	3929809.8	3879.4607	6.758E+09	2853437.8	1093664.2
	Std	1.33E-14	4520.0466	6066.3438	10582599	5128.8228	3.892E+09	6260055.6	158463.82
	Kruskal	26	125.80392	132.64706	276.4902	126.92157	383	265.13725	300
	WSRT		1.3E-18(+)	1.3E-18(+)	1.3E-18(+)	1.3E-18(+)	1.3E-18(+)	1.3E-18(+)	1.3E-18(+)
F3	Mean	300	300.0103	593.21876	40345.956	308.37	42274.951	4450.1667	308.03883
	Std	8.51E-14	0.0063952	1553.372	8704.7464	22.671363	8942.5492	1807.7862	1.6451758

Table 6: Experimental results of DoS and advanced competitors on 30D CEC2017 benchmark test functions (Unimodal and Multimodal).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F11	Mean	1183.613	1219.0968	1253.0216	1185.991	1229.6233	2225.483	1290.5779	1188.8889
	Std	36.046174	42.029705	47.928244	26.45045	48.440969	1019.3045	52.346656	60.84981
	Kruskal	109.21569	183.86275	244.66667	116.52941	202.82353	381.21569	292.52941	105.1569
	WSRT		1.8E-05(+)	2.2E-11(+)	4.5E-01( $\approx$ )	9.3E-07(+)	3.5E-18(+)	8.2E-16(+)	3.7E-01( $\approx$ )
F12	Mean	8085.755	2137054.1	53285.144	1571978.5	1294660.8	185093497	9254480.1	1990462.5
	Std	6121.55	1627807.7	34066.696	1114836	1098320.4	197397007	6337323.7	1011953

Table 7: Experimental results of DoS and advanced competitors on 30D CEC2017 benchmark test functions (Hybrid).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F21	Mean	2338.601	2386.2256	2428.9095	2457.4887	2488.7295	2501.4059	2531.6737	2662.2154
	Std	8.448033	26.486128	36.15707	43.151154	43.680388	31.96612	51.749341	53.280363
	Kruskal	28.76471	88.078431	146.98039	193.92157	239.66667	263.54902	294.90196	380.13725
	WSRT		5.6E-16(+)	3.3E-18(+)	6.3E-17(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)
F22	Mean	2401.722	4562.4384	4530.3552	2469.8868	5455.501	7084.6276	5545.7902	7440.9618
	Std	505.0165	1694.7972	1942.0851	927.84542	2223.7592	1818.7774	2648.7982	1131.3822

Table 8: Experimental results of DoS and advanced competitors on 30D CEC2017 benchmark test functions (Composition).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F1	Mean	100	7458.7766	4205.0167	80718.056	5228.0557	2.015E+10	26034150	4965931
	Std	8.27E-10	7197.5812	6252.8935	377927.33	4731.6526	6.059E+09	26262110	568339.28
	Kruskal	26	155.92157	121.31373	195.60784	141.15686	383	329.88235	283.11765
	WSRT		3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)
F3	Mean	300	300.01759	9531.6933	155854.74	891.77674	96977.11	23051.704	471.10537
	Std	2.36E-13	0.0086403	22769.732	15777.301	827.11499	16078.457	4098.5324	102.90146

Table 9: Experimental results of DoS and advanced competitors on 50D CEC2017 benchmark test functions (Unimodal and Multimodal).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F11	Mean	1279.913	1314.3262	1372.6352	1336.712	1307.114	6080.2766	1458.8419	1366.6447
	Std	47.390331	50.612961	80.017588	66.267192	50.907215	2354.975	91.302341	32.06185
	Kruskal	86.37255	135.2549	215.33333	169.23529	124.19608	383	299.05882	223.54902
	WSRT		2.3E-03(+)	8.4E-10(+)	3.2E-05(+)	1.4E-02(+)	3.3E-18(+)	1.2E-16(+)	6.3E-14(+)
F12	Mean	6940.079	9491633.5	531961.24	7188695.6	4039899.8	4.889E+09	45078699	10512110
	Std	3988.266	4551846.5	390543.95	3675557.8	2179042.5	4.201E+09	28315049	5898223.7

Table 10: Experimental results of DoS and advanced competitors on 50D CEC2017 benchmark test functions (Hybrid).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F21	Mean	2404.251	2478.1009	2560.141	2666.5647	2740.9989	2766.9726	2809.8841	3106.3958
	Std	14.8067	40.419561	41.070358	74.808223	83.488362	52.063935	102.38993	112.45899
	Kruskal	27.13725	81.411765	131.2549	197.96078	251.56863	274.35294	291	381.31373
	WSRT		9.4E-17(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)
F22	Mean	6284.981	8371.8925	8733.8068	13007.794	10401.36	13942.92	11350.239	10992.011
	Std	892.27191	731.5659	1341.8168	4826.0349	1014.7198	1235.6665	1299.1237	961.13311

Table 11: Experimental results of DoS and advanced competitors on 50D CEC2017 benchmark test functions (Composition).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F1	Mean	100	11434.827	9153.6919	2544239.6	15030.121	8.79E+10	553528597	28429335
	Std	1.79E-09	12022.747	10985.079	1057815.3	26942.256	1.178E+10	478815373	2733487.8
	Kruskal	26	132.80392	123.88235	230	127.31373	383	332	281
	WSRT		3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)
F3	Mean	300.0001	316.54711	60838.978	489769.43	16618.625	247187.64	125616.3	15583.486
	Std	0.000108	13.01766	63938.281	35277.33	4668.4778	19100.352	14829.806	3253.5886

Table 12: Experimental results of DoS and advanced competitors on 100D CEC2017 benchmark test functions (Unimodal and Multimodal).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F11	Mean	2093.826	2336.136	2499.4601	45253.726	2364.2269	63454.389	5624.7345	2666.6838
	Std	260.32616	223.32864	365.46358	6151.0608	203.67499	11307.545	1001.9915	185.3469
	Kruskal	60.82353	112.07843	150.45098	335.92157	119.47059	379.07843	281	197.17647
	WSRT		1.2E-05(+)	6.1E-08(+)	3.3E-18(+)	1.1E-06(+)	3.3E-18(+)	3.3E-18(+)	1.6E-16(+)
F12	Mean	70696.23	27497571	1321869	26204306	10543347	2.452E+10	378055056	46529684
	Std	46332.36	12742127	689098.2	9472511.5	4533071.7	1.194E+10	159585023	6876212.1

Table 13: Experimental results of DoS and advanced competitors on 100D CEC2017 benchmark test functions (Hybrid).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
F21	Mean	2687.31	2817.1405	3089.7996	3195.7371	3549.1034	3773.2339	3891.7374	4531.5168
	Std	42.61848	85.264587	104.49185	176.04385	192.28252	111.68982	239.68755	165.59949
	Kruskal	30.2549	75.627451	142.72549	166	240.68627	289.29412	309.23529	382.17647
	WSRT		2.8E-13(+)	3.3E-18(+)	5.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)	3.3E-18(+)
F22	Mean	13378.01	15909.747	17807.914	33116.244	19590.348	30978.01	23117.352	20489.071
	Std	1682.1851	1481.4432	1560.3427	539.5097	1692.7854	1388.0878	2169.571	1328.2754

Table 14: Experimental results of DoS and advanced competitors on 100D CEC2017 benchmark test functions (Composition).

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
C1	Mean	300	300	300	300.33526	300	2591.1845	300.02927	300.0381
	Std	0	5.047E-08	6.922E-14	0.5229235	1.301E-13	2181.0563	0.0450559	0.0115081
	Kruskal	17	105.5	55.066667	186.33333	64.433333	225.5	145.4	164.76667
	WSRT		1.2E-12(+)	1.1E-11(+)	1.2E-12(+)	3.2E-12(+)	1.2E-12(+)	1.2E-12(+)	1.2E-12(+)
C2	Mean	403.1328	405.99319	403.92253	407.34966	413.59611	428.71844	418.64804	409.12183
	Std	1.030256	3.5072486	3.7235414	3.2614581	23.134137	37.315088	28.497419	17.201281

Table 15: Experimental results of DoS and advanced competitors on 10D CEC2022 benchmark test functions.

No.	Metrics	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
C1	Mean	300	300	300	306.03495	300	11803.089	307.16622	300.39313
	Std	3.66E-14	8.449E-09	1.84E-12	6.8698472	1.855E-13	4703.1851	8.1940942	0.1019005
	Kruskal	15.5	105.5	68.983333	179.23333	52.016667	225.5	177.26667	140
	WSRT		1.3E-11(+)	1.3E-11(+)	1.3E-11(+)	1.2E-11(+)	1.3E-11(+)	1.3E-11(+)	1.3E-11(+)
C2	Mean	409.2096	441.76036	436.91201	453.40179	431.14145	530.20493	452.20248	451.34328
	Std	9.3737047	16.365079	22.07657	5.766707	24.286936	60.651785	22.946428	12.37996

Table 16: Experimental results of DoS and advanced competitors on 20D CEC2022 benchmark test functions.

No.	Metrics	Mean	Best	Std	Kruskal	WSRT
F1	DoS	100	100	8.266E-10	86.676471
	LSHADE	100	100	1.591E-07	163.98039	2.4E-14(+)
	LSHADE-SPACMA	100	100	2.57E-14	27.84314	4.4E-16(-)
	AL-SHADE	100	100	1.164E-09	131.5	5.0E-09(+)
F3	DoS	300	300	2.36E-13	46.16667
	LSHADE	40946.98	300	79720.112	153.36275	1.8E-17(+)

Table 17: Experimental results of DoS and SOTA algorithms on 50D CEC2017 benchmark test functions (Unimodal and Multimodal).

No.	Metrics	Mean	Best	Std	Kruskal	WSRT
F11	DoS	1279.913	1167.419	47.39033	77.84314
	LSHADE	1335.352	1237.6025	70.316482	127.86275	2.4E-05(+)
	LSHADE-SPACMA	1317.3168	1216.7653	56.918422	112.60784	2.0E-03(+)
	AL-SHADE	1294.5099	1190.8966	59.899022	91.686275	2.7E-01( $\approx$ )
F12	DoS	6940.079	2276.173	3988.2658	95.784314
	LSHADE	14013.363	4292.871	10769.826	146.66667	8.7E-07(+)

Table 18: Experimental results of DoS and SOTA algorithms on 50D CEC2017 benchmark test functions (Hybrid).

No.	Metrics	Mean	Best	Std	Kruskal	WSRT
F21	DoS	2404.2514	2372.333	14.806696	177.70588
	LSHADE	2355.131	2334.8268	11.24392	68.76471	5.3E-18(-)
	LSHADE-SPACMA	2360.6729	2339.5417	11.606271	88.72549	1.0E-17(-)
	AL-SHADE	2356.7652	2331.429	14.092693	74.803922	3.2E-17(-)
F22	DoS	6284.9812	2300	892.27191	109.5098
	LSHADE	6459.7616	2301.4107	685.0383	124	1.9E-01( $\approx$ )

Table 19: Experimental results of DoS and SOTA algorithms on 50D CEC2017 benchmark test functions (Composition).

No.	Metrics	Mean	Best	Std	Kruskal	WSRT
C1	DoS	300	300	3.657E-14	54.3
	LSHADE	1321.9344	300	4227.5481	84.25	3.1E-05(+)
	LSHADE-SPACMA	3754.3535	300	6223.297	54.6	8.4E-01( $\approx$ )
	AL-SHADE	300	300	3.17E-14	48.85	4.3E-01( $\approx$ )
C2	DoS	409.2096	400	9.3737047	18.25
	LSHADE	448.80521	444.89547	1.062786	81.216667	4.7E-12(+)

Table 20: Experimental results of DoS and SOTA algorithms on 20D CEC2022 benchmark test functions.

Panel of Figure 7 — Figure 7: Convergence curves of objective function values.

Panel of Figure 8 — Figure 8: Convergence curves of objective function values.

Panel of Figure 9 — Figure 9: Radar chart of Mean rankings on CEC2017 benchmark test functions.

Figure 10: Bar chart of Friedman rankings on CEC2017 benchmark test functions.

Panel of Figure 11 — Figure 11: Radar chart of Mean rankings on CEC2022 benchmark test functions.

Figure 12: Bar chart of Friedman rankings on CEC2022 benchmark test functions.

Panel of Figure 13 — Figure 13: Search History, Trajectory and Average Fitness of DoS on Some CEC2017 Benchmark Test Functions (2D).

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$T_{0}$	0.0073590	0.0075530	0.0076597	0.0075053	0.0068882	0.0074479	0.0071382	0.0070808
$T_{1}$	0.3870446	0.4021236	0.3944610	0.4043148	0.3995194	0.387512	0.3841997	0.3897004
$T_{2}^{mean}$	1.1057091	1.3215604	0.8729215	1.5493374	0.9475669	1.9344475	0.6012483	4.7229055
$(T_{2}^{mean}-T_{1})/T_{0}$	97.657899	121.73134	62.464655	152.56187	79.563241	207.70089	30.406635	611.96547

Table 21: Results of computational time cost.

No.	Type	Name	$D$	$g$	$h$
R1	Industrial Chemical Processes	Optimal operation of alkylation unit	7	14	0
R2	Process Synthesis and Design	Process flow sheeting	3	3	0
R3	Mechanical Engineering	Welded beam design	4	7	0
R4	Mechanical Engineering	Pressure vessel design	4	4	0
R5	Mechanical Engineering	Side impact design of automobiles	11	10	0
R6	Mechanical Engineering	Optimal design of industrial refrigeration system	14	15	0

Table 22: Basic information of constrained optimization problems.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	2000	1999.96884	1292.95679	1999.99371	2000	1994.89058	NaN	1195.04863
$x_{2}$	0	0	59.1542907	0	0	0	NaN	43.9829715
$x_{3}$	2576.38006	2498.27354	2158.35587	3162.21346	2403.38284	3223.05907	NaN	2116.12799
$x_{4}$	0	0	85.5187486	0	0	0	NaN	82.1842824
$x_{5}$	58.1606111	57.8291168	92.1783142	60.8575728	57.4311232	61.8920633	NaN	91.0010561
$x_{6}$	1.25994095	0.92996449	13.4113046	4.33703641	0.5441601	5.18089297	NaN	11.3993706

Table 23: The best results for the Optimal operation of alkylation unit.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	-4481.477	-3649.8628	620.373012	-2583.1734	-4200.1079	-4034.2287	NaN	726.378768
Std	238.213415	634.096046	165.068727	2075.36504	373.127814	332.001258	NaN	0
Success	1	1	0.48	0.6	1	0.28	0	0.04
WSRT		8.3E-09(+)	1.0E-09(+)	3.2E-09(+)	2.6E-08(+)	1.2E-08(+)	NaN	9.7E-11(+)
FMR	1.12	3.64	6	4.56	2.64	3.2	NaN	6.84
F-Rank	1	4	6	5	2	3	8	7

Table 24: Statistical results for the Optimal operation of alkylation unit.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	0.94193734	0.94193735	0.94193734	0.94193759	0.94193734	0.94212928	0.94193738	0.94201652
$x_{2}$	-2.1	-2.1	-2.1	-2.1000003	-2.1	-2.1002767	-2.1000001	-2.1001605
$x_{3}$	0.75209273	1.31123275	0.95099277	1.1686118	1.48992977	1.00614078	1.26073299	1.1339543
Best	1.0765431	1.07654309	1.07654308	1.07654415	1.07654308	1.07739151	1.07654326	1.07689302

Table 25: The best results for the Process flow sheeting.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	1.0765431	1.07654364	1.1320493	1.08980276	1.18061723	1.0804378	1.19451316	1.12838518
Std	4.532E-16	1.3065E-06	0.08258201	0.02321725	0.08672846	0.00176624	0.08255323	0.07745561
Success	1	1	1	1	1	0.6	1	1
WSRT		9.7E-11(+)	2.4E-03(+)	9.7E-11(+)	4.9E-11(+)	8.0E-11(+)	3.6E-11(+)	9.1E-11(+)
FMR	1.34	3.2	3.4	5.04	5.44	5.44	6.26	5.88
F-Rank	1	2	3	4	5	6	8	7

Table 26: Statistical results for the Process flow sheeting.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	0.20572964	0.20460826	0.20572964	0.20572963	0.20565992	0.20379636	0.20385754	0.20151701
$x_{2}$	3.25312004	3.2733965	3.25312004	3.25312028	3.25436302	3.28201771	3.28630361	3.34964561
$x_{3}$	9.03662391	9.03605126	9.0366239	9.03662391	9.03662131	9.07355424	9.0385155	9.08696903
$x_{4}$	0.20572964	0.20575609	0.20572964	0.20572964	0.20572976	0.20554814	0.20572023	0.20557583
Best	1.6952472	1.69644556	1.69524717	1.6952472	1.69531381	1.70126017	1.69723581	1.70952413

Table 27: The best results for the Welded beam design.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	1.6952472	1.7033707	1.76975031	1.73309548	1.71426008	1.70811517	1.71747601	1.82058308
Std	2.266E-16	0.01235605	0.1188277	0.04718389	0.01906936	0.00371913	0.024741	0.16032823
Success	1	1	1	0.08	1	1	1	1
WSRT		9.7E-11(+)	9.7E-11(+)	6.7E-12(+)	9.7E-11(+)	9.7E-11(+)	9.7E-11(+)	9.7E-11(+)
FMR	1	3.44	4.72	6.44	4.4	4.32	4.68	7
F-Rank	1	2	6	7	4	3	5	8

Table 28: Statistical results for the Welded beam design.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	12.7927137	12.8923688	13.018652	13.4194694	13.0437498	12.7584516	13.3917073	15.7473509
$x_{2}$	7.31679176	7.22512106	6.88413741	7.17723922	7.10333375	6.78178608	6.71662934	8.40498716
$x_{3}$	42.0984456	42.0982098	42.0984456	42.0721363	42.0984456	42.0512357	42.0984135	51.8132818
$x_{4}$	176.636596	176.639524	176.636596	176.963169	176.636596	177.425062	176.636993	84.5827817
Best	6059.7143	6059.74319	6059.71434	6062.92769	6059.71434	6070.20351	6059.71824	6410.20622

Table 29: The best results for the Pressure vessel design.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	6067.5069	6349.98002	6399.20768	6539.31568	6744.7382	6471.54327	6495.56359	6808.79589
Std	14.273598	498.958892	221.108314	264.377381	532.303828	424.078767	385.542707	161.908079
Success	1	1	1	1	1	0.96	1	1
WSRT		1.3E-05(+)	4.5E-09(+)	7.1E-10(+)	9.2E-09(+)	5.7E-09(+)	1.8E-08(+)	5.5E-10(+)
FMR	1.32	3.52	4.16	5.08	5.44	5.16	4.48	6.84
F-Rank	1	2	3	5	7	6	4	8

Table 30: Statistical results for the Pressure vessel design.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	0.5	0.50263548	0.62281348	NaN	0.5	0.5	0.51948093	0.5
$x_{2}$	0.97984709	0.97778074	0.88730099	NaN	0.96383745	0.92325516	0.96325679	0.94018371
$x_{3}$	0.5	0.50000005	0.5	NaN	0.5	0.5	0.5	0.5
$x_{4}$	1.00244987	1.0056303	1.02435596	NaN	1.04103104	1.14925255	1.00876643	1.12618121
$x_{5}$	0.5	0.50004965	0.5	NaN	0.50000002	0.5	0.5	0.5
$x_{6}$	0.5	0.50854536	0.5	NaN	0.5000684	0.51201813	0.50000657	0.52275068

Table 31: The best results for the Side impact design of automobiles.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	20.850749	21.0935379	21.093146	NaN	21.0122346	21.1392367	21.1213781	21.2455305
Std	0.1790251	0.1333431	0.17630027	NaN	0.36347931	0.16308521	0.39526243	0.20830952
Success	1	1	1	0	1	1	1	1
WSRT		4.0E-08(+)	5.3E-06(+)	NaN	4.1E-04(+)	2.2E-07(+)	4.5E-05(+)	4.0E-08(+)
FMR	1.76	3.92	3.72	NaN	3.92	5	3.72	5.96
F-Rank	1	4	2	8	5	6	3	7

Table 32: Statistical results for the Side impact design of automobiles.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	0.001	0.001	0.001	0.001	0.001	0.00100193	0.001	0.001
$x_{2}$	0.001	0.00100134	0.001	0.001	0.001	0.0012729	0.001	0.001
$x_{3}$	0.001	0.00100008	0.001	0.001	0.001	0.00130519	0.001	0.0010957
$x_{4}$	0.001	0.01079786	0.001	0.001	0.00100005	0.00153374	0.001571	0.01001128
$x_{5}$	0.001	0.00100067	0.001	0.001	0.0010001	0.001	0.001	0.00659281
$x_{6}$	0.001	0.00100002	0.001	0.001	0.001	0.001	0.00105786	0.00191787

Table 33: The best results for the Optimal design of industrial refrigeration system.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	0.032213	0.04270679	0.0651192	0.11484867	0.0403492	0.05902045	0.05023075	1.34361959
Std	1.018E-07	0.01463627	0.02767783	0.05306657	0.01459214	0.02002695	0.01265639	1.2266915
Success	0.64	1	0.72	1	0.68	0.88	1	0.84
WSRT		1.3E-09(+)	1.3E-09(+)	1.3E-09(+)	1.2E-09(+)	1.3E-09(+)	1.3E-09(+)	1.2E-09(+)
FMR	1	3.12	5.2	6.52	3.08	4.92	4.16	8
F-Rank	1	3	6	7	2	5	4	8

Table 34: Statistical results for the Optimal design of industrial refrigeration system.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$d_{1}$	38.4139618	NaN	38.4415703	38.4160269	39.6685254	NaN	NaN	NaN
$d_{2}$	52.8586378	NaN	52.8966562	52.8614815	54.5862806	NaN	NaN	NaN
$d_{3}$	70.4726957	NaN	70.523379	70.4764868	72.775855	NaN	NaN	NaN
$d_{4}$	84.4957161	NaN	84.5564332	84.5002576	87.2547612	NaN	NaN	NaN
$\omega$	90	NaN	89.9351938	89.9951493	87.2375442	NaN	NaN	NaN
Best	16.090273	NaN	16.1018141	16.0911367	16.6320727	NaN	NaN	NaN

Table 35: The best results for the Step-cone pulley.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	16.090273	NaN	16.6411059	16.4156844	17.2273553	NaN	NaN	NaN
Std	6.405E-15	NaN	0.35976121	0.3277681	0.47618814	NaN	NaN	NaN
Success	1	0	1	0.52	0.84	0	0	0
WSRT		NaN	4.7E-10(+)	3.5E-10(+)	4.7E-10(+)	NaN	NaN	NaN
FMR	1	NaN	2.84	2.36	3.8	NaN	NaN	NaN
F-Rank	1	5	3	2	4	6	7	8

Table 36: Statistical results for the Step-cone pulley.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	5.95551185	5.98000801	6.4667959	6.26446032	6.0918833	6.16587283	7.20625039	6.37381799
$x_{2}$	5.38871638	5.41558346	5.94891526	5.70944689	5.53900748	5.55478777	5.68618996	5.76494585
$x_{3}$	5.3587E-06	6.4328E-06	8.2294E-06	5.9139E-06	6.6888E-06	5.5526E-06	7.5162E-06	7.9497E-06
$x_{4}$	2.25664104	3.26750791	7.65466293	3.01829092	3.68679675	2.69046358	10.3889963	7.00843564
Best	1616.1204	1744.98267	2193.90348	1836.03564	1813.66823	1819.17843	3761.60047	2283.20929

Table 37: The best results for the Hydro-static thrust bearing design.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	1650.2441	2284.54068	2728.81527	2206.51479	2170.76997	2156.02939	5601.09312	5118.3029
Std	57.051917	369.796558	478.199976	290.012353	276.19162	314.054991	2556.66797	2376.17662
Success	1	0.96	1	1	0.96	1	0.2	0.44
WSRT		7.4E-09(+)	1.4E-09(+)	1.6E-09(+)	2.3E-09(+)	2.3E-09(+)	3.9E-10(+)	9.3E-10(+)
FMR	1	4.12	5.36	3.72	3.52	3.36	7.72	7.2
F-Rank	1	5	6	4	3	2	8	7

Table 38: Statistical results for the Hydro-static thrust bearing design.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$N_{p1}$	21.5320385	14.4365735	16.3522535	27.6467997	14.2885826	28.4713456	14.541271	24.1665764
$N_{g1}$	46.6205336	31.0391991	38.1855094	40.6005345	28.4852633	40.2217087	45.1645854	63.3848144
$N_{p2}$	21.0587073	18.453822	21.4922182	23.1810638	17.9460929	14.009929	15.2618438	14.2695344
$N_{g2}$	42.4589623	40.109072	38.4061982	49.1868321	41.1803625	35.8717594	19.7370829	31.0863413
$N_{p3}$	21.4872325	21.2135998	10.7031089	18.2407005	18.6306136	21.4574928	15.2525391	16.5003483
$N_{g3}$	45.3812778	42.3807148	23.4193648	46.304301	40.6289769	40.5799586	30.8480731	41.069902

Table 39: The best results for the Four-stage gear box.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	42.973019	62.8111705	45.9896535	60.0108344	47.3720259	75.0605268	93.1821241	51.349004
Std	7.65659425	20.7589046	0	13.570429	5.49153005	0	0	8.21682197
Success	0.84	0.4	0.04	0.92	0.2	0.04	0.04	0.24
WSRT		2.6E-08(+)	1.1E-05(+)	4.5E-07(+)	6.0E-05(+)	9.6E-11(+)	9.6E-11(+)	5.3E-06(+)
FMR	1.56	5.36	2.08	5.04	3.16	6.76	7.92	4.12
F-Rank	1	6	2	5	3	7	8	4

Table 40: Statistical results for the Four-stage gear box.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	1462.63623	686.727895	40.0266304	40	1144.00883	NaN	1077.02246	569.828992
$x_{2}$	788.942125	522.561888	1959.9987	40	1392.90003	NaN	1704.95946	1377.66413
$x_{3}$	1547.23036	945.95155	1841.77117	1959.68289	147.650041	NaN	1555.44313	1859.44004
$x_{4}$	1960	1329.53655	40.000101	46.0147221	230.504712	NaN	86.8739308	1954.81841
$x_{5}$	272.001925	1922.48364	40.0000206	300.774847	40.0000368	NaN	316.67443	1018.64235
$x_{6}$	842.392038	1324.08245	40.0576999	490.453593	41.1868254	NaN	634.290145	1727.97915

Table 41: The best results for the Wind farm layout.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	-6205.214	-5806.3767	-5913.9266	-5496.6945	-5747.3537	NaN	-5546.75	-4913.6592
Std	30.581606	125.853091	86.1246327	76.656197	135.130191	NaN	193.119997	239.132835
Success	1	1	1	1	1	0	0.88	1
WSRT		1.4E-09(+)	1.4E-09(+)	1.4E-09(+)	1.4E-09(+)	NaN	1.4E-09(+)	1.4E-09(+)
FMR	1	3.16	2.4	5.4	3.92	NaN	5.16	6.96
F-Rank	1	3	2	6	4	8	5	7

Table 42: Statistical results for the Wind farm layout.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	12.9888909	13.3089578	14.3835712	NaN	10.3624115	6.10146067	2.2170757	9.90246732
$x_{2}$	13.8690372	13.7325103	14.8694544	NaN	8.77075093	4.52803833	2.05372255	10.4415022
$x_{3}$	2.50699876	2.40820912	2.59246594	NaN	2.77900294	2.72990543	2.21872515	3.07949599
$x_{4}$	6.48131699	9.37718037	10.5927012	NaN	7.25823212	0.72235314	4.2335812	5.81442362
$x_{5}$	6.33422166	11.6377417	12.3866256	NaN	9.5216832	2.72726476	1.46826182	5.2906585
$x_{6}$	3.95329554	3.40000558	2.39321317	NaN	3.78592371	2.54084754	3.9254009	4.17795437

Table 43: The best results for mountainous terrain path planning.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	115.70175	116.664178	117.451852	NaN	116.386235	126.408888	124.175462	115.897928
Std	0.15175712	1.12785545	2.36536695	NaN	1.37315754	4.90373033	8.51917331	0.1045847
Success	1	0.96	0.92	0	1	1	1	1
WSRT		2.4E-06(+)	2.2E-07(+)	NaN	5.5E-03(+)	1.4E-09(+)	1.4E-09(+)	3.3E-07(+)
FMR	1.52	3.84	4.12	NaN	2.96	6.52	6.2	2.84
F-Rank	1	4	5	8	3	7	6	2

Table 44: Statistical results for mountainous terrain path planning.

Panel of Figure 14 — Figure 14: Visualization of mountainous terrain path planning.

	No fly zone 1	No fly zone 2	No fly zone 3	No fly zone 4	No fly zone 5
$x_{c}$	56.7157	32.6590	13.1987	65.9033	64.0290
$y_{c}$	18.5965	39.5000	45.5621	70.4151	44.1858
$r_{c}$	5.0676	4.7530	10.5505	4.9743	8.7564
$h_{c}$	6.4409	8.2620	3.8032	8.5927	9.5035

Table 45: Information of no fly zones.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
$x_{1}$	24.6151727	6.19575074	5.83758569	NaN	9.653058	8.35983887	1.35877043	9.10803916
$x_{2}$	24.3137897	7.18586063	6.44106836	NaN	9.27005726	7.07298245	-0.1705658	8.19400313
$x_{3}$	3.69506866	3.39562003	3.25904729	NaN	2.69756453	1.91506739	2.38150614	3.55724855
$x_{4}$	10.4177494	5.70624126	11.9844314	NaN	9.03563824	0.33891949	2.50262731	9.27551588
$x_{5}$	9.43944082	6.02840097	9.26556372	NaN	6.61331424	0.62194985	1.69667884	8.90176369
$x_{6}$	3.02437287	1.75030092	4.31175872	NaN	4.22084634	1.96825997	3.71290435	4.76670039

Table 46: The best results of mountainous terrain path planning with no fly zones.

	DoS	AOO	PSA	FFA	AVOA	ETO	SGA	WAA
Mean	118.36611	122.097895	129.442151	NaN	120.574604	138.719413	133.276447	119.678824
Std	1.2245701	8.08930431	8.54127858	NaN	7.3629909	7.81870547	12.0252177	7.52920892
Success	1	0.44	0.6	0	0.96	0.88	0.96	0.56
WSRT		3.8E-05(+)	1.8E-07(+)	NaN	9.4E-01( $\approx$ )	1.4E-09(+)	1.2E-08(+)	6.8E-01( $\approx$ )
FMR	2.2	3.88	5.08	NaN	2.6	6.48	5.32	2.44
F-Rank	1	4	5	8	3	7	6	2

Table 47: Statistical results of mountainous terrain path planning with no fly zones.

Panel of Figure 15 — Figure 15: Visualization of mountainous terrain path planning with no fly zones.

Metaheuristic Optimization

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

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Bandits attack function optimization
May 5, 2026Philippe Preux, Rémi Munos, Michal ValkoBlack-Box Optimization Global Optimization
Université de Lille, LIFL (UMR 8022 CNRS), and INRIA, France · INRIA, and LIFL, Lille, France
Drain-Vortex Optimization: A Population-Based Metaheuristic Inspired by Multi-Drain Free-Vortex Flow
May 9, 2026Mohsen Omidi, Brian VaughanBlack-Box Optimization Metaheuristic Optimization
Technological University Dublin, Grangegorman Lower, Dublin, D07 C972, Ireland
optimize_anything: A Universal API for Optimizing any Text Parameter
May 19, 2026Lakshya A Agrawal, Donghyun Lee, Shangyin Tan +11Black-Box Optimization Large Language Model-Guided Optimization
UC Berkeley USA · MIT USA

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Bandits attack function optimization

Philippe Preux, Rémi Munos, Michal Valko

Université de Lille, LIFL (UMR 8022 CNRS), and INRIA, France · INRIA, and LIFL, Lille, France

Black-Box Optimization Global Optimization

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We consider function optimization as a sequential decision making problem under budget constraint. This constraint limits the number of objective function evaluations allowed during the optimization. We consider an algorithm inspired by a continuous version of a multi-armed bandit problem which attacks this optimization problem by solving the tradeoff between exploration (initial quasi-uniform search of the domain) and exploitation (local optimization around the potentially global maxima). We introduce the so-called Simultaneous Optimistic Optimization (SOO), a deterministic algorithm that works by domain partitioning. The benefit of such approach are the guarantees on the returned solution and the numerical efficiency of the algorithm. We present this machine learning approach to optimization, and provide the empirical assessment of SOO on the CEC'2014 competition on single objective real-parameter numerical optimization test-suite.

Abstract

Metaheuristic Optimization

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Bandits attack function optimization

Drain-Vortex Optimization: A Population-Based Metaheuristic Inspired by Multi-Drain Free-Vortex Flow

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