A Differential Evolution with Two Mutation Strategies and a Selection Based on an Improved Constraint-Handling Technique for Bilevel Programming Problems

Table 9

Best results obtained by the proposed algorithm for 5 and 10 dimensions in second set of nonlinear BLPPs.

Problem	Dimension	(𝑥^∗,𝑦^∗)	𝐹(𝑥^∗,𝑦^∗)	𝑓(𝑥^∗,𝑦^∗)
SMD1	5	(−1.33𝑒−3,−4.22𝑒−4,−2.38𝑒−9,−2.38𝑒−9,−4.22𝑒−4)	1.94𝑒−6	1.77𝑒−6
	10	(9.56𝑒−5,1.87𝑒−3,4.81𝑒−4,2.35𝑒−3,−2.89𝑒−3,−7.45𝑒−9,−7.45𝑒−9,−7.45𝑒−9,2.35𝑒−3,−2.89𝑒−3)	1.76𝑒−5	3.76𝑒−6
SMD2	5	(−5.28𝑒−4,7.12𝑒−4,−6.45𝑒−9,−6.45𝑒−9,1.00𝑒+0)	7.86𝑒−7	2.79𝑒−7
	10	(0,0,0,0,0,0,0,0,1,1)	0	0
SMD3	5	(−4.04𝑒−4,−9.92𝑒−4,−5.54𝑒−6,−5.54𝑒−6,5.26𝑒−5)	1.15𝑒−6	1.67𝑒−7
	10	(9.29𝑒−4,2.24𝑒−3,−1.43𝑒−3,−1.60𝑒−3,−2.22𝑒−4,−5.06𝑒−3,−5.06𝑒−3,−5.06𝑒−3,2.56𝑒−4,2.29𝑒−4)	1.33𝑒−5	8.03𝑒−6
SMD4	5	(−1.25𝑒−4,3.17𝑒−5,0,0,0)	1.57𝑒−8	1.67𝑒−8
	10	(0,0,0,0,0,0,0,0,0,0)	0	0
SMD5	5	(5.83𝑒−4,−1.33𝑒−3,1.00𝑒+0,1.00𝑒−0,−2.66𝑒−3)	3.58𝑒−7	2.09𝑒−6
	10	(4.53𝑒−4,1.91𝑒−3,1.75𝑒−4,2.55𝑒−3,−3.64𝑒−3,1.00𝑒+0,1.00𝑒+0,1.00𝑒−0,−2.63𝑒−4,−1.22𝑒−2)	4.94𝑒−6	2.26𝑒−5
SMD6	5	(−1.33𝑒−3,−4.22𝑒−4,0,0,0)	1.77𝑒−6	1.94𝑒−6
	10	(9.56𝑒−5,1.87𝑒−3,4.81𝑒−4,2.35𝑒−3,−2.89𝑒−3,−7.45𝑒−9,−7.45𝑒−9,−7.45𝑒−9,2.35𝑒−3,−2.89𝑒−3)	1.76𝑒−5	3.76𝑒−6
SMD7	5	(3.88𝑒−4,1.98𝑒−3,7.69𝑒−9,7.69𝑒−9,1.00𝑒+0)	4.01𝑒−6	2.45𝑒−10
	10	(0,0,0,0,0,0,0,0,1,1)	0	0
SMD8	5	(−2.12𝑒−7,−8.26𝑒−3,1.00𝑒+0,1.00𝑒+0,−2.02𝑒−1)	6.91𝑒−5	2.12𝑒−7
	10	(−6.49𝑒−6,6.99𝑒−8,−1.87𝑒−6,7.29𝑒−4,3.28𝑒−3,1.00𝑒−0,1.00𝑒−0,1.00𝑒−0,9.00𝑒−2,1.49𝑒−1)	2.69𝑒−5	8.42𝑒−6
SMD9	5	(−4.09𝑒−4,−3.62𝑒−4,2.69𝑒−8,2.69𝑒−8,−3.62𝑒−4)	2.98𝑒−7	1.67𝑒−7
	10	(0,0,0,0,0,3.13𝑒−8,3.13𝑒−8,3.13𝑒−8,1.90𝑒−8,2.17𝑒−8)	0	0
SMD10	5	(1.00𝑒+0,1.00𝑒+0,1,1,7.85𝑒−1)	4.00𝑒−0	3.00𝑒+0
	10	(5.00𝑒−1,5.00𝑒−1,5.00𝑒−1,5.00𝑒−1,5.00𝑒−1,7.07𝑒−1,7.07𝑒−1,7.07𝑒−1,4.64𝑒−1,4.64𝑒−1)	1.275𝑒+1	5.76𝑒+0
SMD11	5	(−1.77𝑒−3,4.02𝑒−3,−1.33𝑒−6,−1.33𝑒−6,3.69𝑒−1)	−1.00𝑒+0	1.00𝑒+0
	10	(2.23𝑒−3,1.18𝑒−3,1.97𝑒−3,5.74𝑒−3,5.74𝑒−3,−4.69𝑒−7,−4.69𝑒−7,−4.69𝑒−7,4.94𝑒−1,4.98𝑒−1)	−1.00𝑒+0	1.00𝑒+0
SMD12	5	(1.00𝑒+0,1.00𝑒+0,1.00𝑒−0,1.00𝑒−0,3.10𝑒−5)	3.00𝑒−0	4.00𝑒+0
	10	(5.00𝑒−1,5.00𝑒−1,5.00𝑒−1,5.00𝑒−1,5.00𝑒−1,7.07𝑒−1,7.07𝑒−1,7.07𝑒−1,−2.04𝑒−1,−2.04𝑒−1)	1.216𝑒+1	6.76𝑒+0