A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems

Table 1

The results found by EDA-NM and compared algorithms in the corresponding references.

Number	𝐹(𝑥,𝑦)				𝑓(𝑥,𝑦)		Reference
	𝐹(𝑥^∗,𝑦^∗)	𝐹(𝑥,𝑦)	Avg.	Std.	𝑓(𝑥^∗,𝑦^∗)	𝑓(𝑥,𝑦)	𝐹(𝑥^∗,𝑦^∗)	𝑓(𝑥^∗,𝑦^∗)
1^∗ [11]	85.090909	85.090908	85.090909	2.6𝑒−7	−50.181818	−50.181818	85.0909	−50.1818
2^∗ [11]	11.0	10.999999	11.000000	2.7𝑒−7	−10.999999	−11.0	11	−11
3^∗ [11]	16.0	15.999999	16.000000	3.1𝑒−7	−4.000000	−4.000000	16	−4
4^∗ [11]	26.0	25.999999	25.999999	2.1𝑒−7	−3.2	−3.199999	25.9827	−3.1932
5 [19]	22.5	22.500001	22.500000	3.1𝑒−7	−4.499981	−1.500850	22.500610	−1.523438
6^∗ [20]	469.142857	469.142856	469.142857	1.8𝑒−7	8.857162	8.857344	469.14286	8.85714
7 [15]	0.000000	0.000001	0.000000	2.0𝑒−7	−54.999690	−54.997967	0	5
8 [21]	252.0	252.000001	252.000001	2.5𝑒−7	−8.000000	−7.999999	252	−8
9 [22]	7.500000	7.500001	7.500001	2.6𝑒−7	0.000000	0.000000	10.625	−0.5
10^∗ [23]	32.468750	32.468748	32.468749	3.2𝑒−7	63.0	62.999998	32.4688	63
11 [24]	0.000000	0.000000	0.000000	6.6𝑒−7	0.142411	1.967259	0	1.0440
12 [24]	0.000000	0.000000	0.000000	3.7𝑒−7	0.142409	2.002728	0	1.0573
13 [25]	−469.142857	−469.142856	−469.142857	1.7𝑒−7	−0.794858	−0.794922	−469.14	−0.795332
14 [26]	−1.688887	−1.688148	−1.688716	1.8𝑒−6	4.333333	4.333328	−1.6889	4.3333