A Swarm Optimization Algorithm for Multimodal Functions and Its Application in Multicircle Detection

Table 4

Performance comparison among multimodal optimization algorithms for the test functions 𝑓₅−𝑓₈. The standard unit of the column ET is seconds (numbers in parentheses are standard deviations). Bold faced letters represent best results.

Function	Algorithm	NO	DO	FE	ET
𝑓5	Deterministic crowding	62.4 (14.3)	4.72×10^−3(4.59×10^−3)	1,760,199 (254,341)	14.62 (2.83)
	Probabilistic crowding	84.7 (5.48)	1.50×10^−3(9.38×10^−4)	2,631,627 (443,522)	34.39 (5.20)
	Sequential fitness sharing	76.3 (7.08)	3.51×10^−3(1.66×10^−3)	2,726,394 (562,723)	36.55 (7.13)
	Clearing procedure	93.6 (2.31)	2.78×10^−3(1.20×10^−3)	2,107,962 (462,622)	28.61 (6.47)
	CBN	87.9 (7.78)	4.33×10^−3(2.82×10^−3)	2,835,119 (638,195)	37.05 (8.23)
	SCGA	97.4 (4.80)	1.34×10^−3(8.72×10^−4)	2,518,301 (643,129)	30.27 (7.04)
	AEGA	99.4 (1.39)	6.77×10^−4(3.18×10^−4)	978,435 (71,135)	10.56 (4.81)
	Clonal selection algorithm	90.6 (9.95)	3.15×10^−3(1.47×10^−3)	5,075,208 (194,376)	58.02 (2.19)
	AiNet	93.8 (7.8)	2.11×10^−3(3.2×10^−3)	3,342,864 (549,452)	51.65 (6.91)
	CAB	100 (2)	2.22×𝟏𝟎^−4(3.1×𝟏𝟎^−5)	680,211 (12,547)	7.33 (1.84)
𝑓6	Deterministic crowding	9.37 (1.91)	3.26×10^−3(5.34×10^−4)	832,546 (75,413)	4.58 (0.57)
	Probabilistic crowding	15.17 (2.43)	2.87×10^−3(5.98×10^−4)	1,823,774 (265,387)	12.92 (2.01)
	Sequential fitness sharing	15.29 (2.14)	1.42×10^−3(5.29×10^−4)	1,767,562 (528,317)	14.12 (3.51)
	Clearing procedure	18 (0)	1.19×10^−3(6.05×10^−4)	1,875,729 (265,173)	11.20 (2.69)
	CBN	14.84 (2.70)	4.39×10^−3(2.86×10^−3)	2,049,225 (465,098)	18.26 (4.41)
	SCGA	4.83 (0.38)	1.58×10^−3(4.12×10^−4)	2,261,469 (315,727)	13.71 (1.84)
	AEGA	18 (0)	3.34×10^−4(1.27×10^−4)	656,639 (84,213)	3.12 (1.12)
	Clonal selection algorithm	18 (0)	3.42×10^−3(1.58×10^−3)	4,989,856 (618,759)	33.85 (5.36)
	AiNet	18 (0)	2.11×10^−3(3.31×10^−3)	3,012,435 (332,561)	26.32 (2.54)
	CAB	18 (0)	𝟏.𝟎𝟐×𝟏𝟎^−𝟒(𝟒.𝟐𝟕×𝟏𝟎^−𝟓)	431,412 (21,034)	2.21 (0.51)
𝑓7	Deterministic crowding	52.6 (8.86)	3.71×10^−3(1.54×10^−3)	2,386,960 (221,982)	19.10 (2.26)
	Probabilistic crowding	79.2 (4.94)	3.48×10^−3(3.79×10^−3)	3,861,904 (457,862)	43.53 (4.38)
	Sequential fitness sharing	63.0 (5.49)	4.76×10^−3(3.55×10^−3)	3,619,057 (565,392)	42.98 (6.35)
	Clearing procedure	79.4 (4.31)	2.95×10^−3(1.64×10^−3)	3,746,325 (594,758)	45.42 (7.64)
	CBN	71.3 (9.26)	3.29×10^−3(4.11×10^−3)	4,155,209 (465,613)	48.23 (5.42)
	SCGA	94.9 (8.18)	2.63×10^−3(1.81×10^−3)	3,629,461 (373,382)	47.84 (0.21)
	AEGA	98 (2)	1.31×10^−3(8.76×10^−4)	1,723,342 (121,043)	12,54 (1.31)
	Clonal selection algorithm	89.2 (5.44)	3.02×10^−3(1.63×10^−3)	5,423,739 (231,004)	47.84 (6.09)
	AiNet	92.7 (3.21)	2.79×10^−3(3.19×10^−4)	4,329,783 (167,932)	41.64 (2.65)
	CAB	100 (1)	𝟑.𝟑𝟐×𝟏𝟎^−𝟒(𝟓.𝟐𝟓×𝟏𝟎^−𝟓)	953,832 (9,345)	8.82 (1.51)
𝑓8	Deterministic crowding	44.2 (7.93)	4.45×10^−3(3.63×10^−3)	2,843,452 (353,529)	23.14 (3.85)
	Probabilistic crowding	70.1 (8.36)	2.52×10^−3(1.47×10^−3)	4,325,469 (574,368)	49.51 (6.72)
	Sequential fitness sharing	58.2 (9.48)	4.14×10^−3(3.31×10^−3)	4,416,150 (642,415)	54.43 (12.6)
	Clearing procedure	67.5 (10.11)	2.31×10^−3(1.43×10^−3)	4,172,462 (413,537)	52.39 (7.21)
	CBN	53.1 (7.58)	4.36×10^−3(3.53×10^−3)	4,711,925 (584,396)	61.07 (8.14)
	SCGA	87.3 (9.61)	3.15×10^−3(2.07×10^−3)	3,964,491 (432,117)	53.87 (8.46)
	AEGA	90.6 (1.65)	2.55×10^−3(9.55×10^−4)	2,213,754 (412,538)	16.21 (3.19)
	Clonal selection algorithm	74.4 (7.32)	3.52×10^−3(2.19×10^−3)	5,835,452 (498,033)	74.26 (5.47)
	AiNet	83.2 (6.23)	3.11×10^−3(2.41×10^−4)	4,123,342 (213,864)	60.38 (5.21)
	CAB	97 (2)	𝟏.𝟓𝟒×𝟏𝟎^−𝟑(𝟒.𝟓𝟏×𝟏𝟎^−𝟒)	1,121,523 (51,732)	12.21 (2.66)