Christian Hafner, Cui Xudong, Jasmin Smajic, and Ruediger Vahldieck, "Efficient procedures for the optimization of defects in photonic crystal structures," J. Opt. Soc. Am. A 24, 1177-1188 (2007)
Seven different stochastic binary optimizers—based on the concepts of genetic algorithms and evolutionary strategies—are developed, applied to determine defect locations in several photonic crystal structures that serve as test cases, and compared by extensive statistical analysis. In addition to the stochastic optimizers, a quasi-deterministic optimizer based on an algorithm inspired by hill-climbing algorithms was implemented. The test cases include the prominent 90° photonic crystal waveguide bend and a photonic crystal power divider. The analysis of the results shows that many different photonic crystal structures with high transmission may be found for any operating frequency. All of the eight optimizers outperform standard codes—because they maintain an incomplete fitness table—and find the global optima with a high probability even when the number of fitness evaluations is much smaller than the number of potential solutions contained in the discrete search space. Based on the incomplete fitness table, an algorithm to estimate bit-fitness values is presented. The bit-fitness values are then used to improve the performance of some algorithms. The four best algorithms—an extended microgenetic algorithm, two mutation-based algorithms, and the quasi-deterministic algorithm inspired by hill-climbing algorithms—are considered to be of high value for the optimization of defects in photonic crystals and for similar binary optimization problems.
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Population Size for the Eight Different Runs for the Seven Stochastic Optimizers (STAT, MGA0, MGA1, MGA2, MUT0, MUT1, MUT2) and for the HC Algorithm RHC1a
Stochastic
RHC1
1
4
1
2
5
2
3
7
4
4
10
7
5
14
11
6
19
16
7
25
22
8
32
29
is used in the following tables.
Table 2
Probabilities of Finding Global Optimum in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizersa
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
33
36
39
43
22
46
49
64
2
34
38
40
46
29
50
52
62
3
34
39
42
46
30
55
56
58
4
34
39
43
45
31
57
55
45
5
36
41
44
49
31
53
49
45
6
36
42
45
50
32
44
40
45
7
35
40
45
50
32
34
33
43
8
36
39
43
50
32
29
29
42
av
35
39
43
47
30
46
46
51
indicates the population size according to Table 1; av is the average of the values in the corresponding column.
Table 3
Average Relative Fitness (Fitness Found by the Algorithm or Fitness of the Global Optimum) in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizersa
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
99
99
99
99
96
99
99
100
2
99
99
99
99
98
99
99
100
3
99
99
99
99
99
99
99
100
4
99
99
99
99
99
99
99
99
5
99
99
99
99
99
99
99
99
6
99
99
99
100
99
99
99
99
7
99
99
99
100
99
99
99
99
8
99
99
99
100
99
99
99
99
av
99
99
99
99
98
99
99
100
indicates the population size according to Table 1.
Table 4
Average Number of Fitness Calls When an IFT Is Used and the Algorithm Is Stopped As Soon As It Finds the Global Optimum, Averaged over All Test Case 100, 200, and 400 Fitness Evaluations, for All Eight Optimizersa
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
184
180
175
167
172
163
158
111
2
183
178
173
163
191
157
151
118
3
185
175
171
164
192
149
146
131
4
186
176
171
166
190
144
141
166
5
186
173
167
159
190
126
119
168
6
187
173
167
157
191
110
102
169
7
193
176
165
158
191
99
91
174
8
194
178
168
160
191
89
83
179
av
187
176
170
162
188
130
124
152
indicates the population size according to Table 1.
Same as in Table 2, When All Algorithms Are Stopped After 100 Fitness Evaluations
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
16
17
17
20
11
21
22
44
2
15
17
17
20
12
23
22
37
3
16
17
19
21
12
24
25
30
4
16
17
19
20
13
25
25
20
5
16
19
20
23
12
30
29
22
6
17
17
21
23
14
29
27
23
7
16
16
20
24
14
26
25
21
8
16
16
19
22
14
24
25
20
av
16
17
19
22
13
25
25
27
Table 7
Same as in Table 2, When All Algorithms Are Stopped After 200 Fitness Evaluations
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
29
31
35
39
21
42
46
67
2
29
35
35
43
25
47
50
65
3
30
36
38
43
25
54
54
56
4
30
36
38
41
27
59
58
40
5
32
37
39
46
27
56
53
41
6
31
39
40
48
28
47
44
41
7
32
38
42
49
28
36
34
39
8
32
36
39
48
28
30
30
39
av
31
36
38
44
26
46
46
48
Table 8
Same as in Table 2, When All Algorithms Are Stopped After 400 Fitness Evaluations
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
55
60
65
71
34
75
80
81
2
57
62
68
75
51
81
85
85
3
55
64
68
74
53
86
88
89
4
56
65
70
75
54
88
83
75
5
58
68
73
77
53
73
66
72
6
60
69
74
78
54
56
50
71
7
58
67
74
78
54
42
39
70
8
59
66
72
79
54
34
32
68
av
57
65
70
76
50
67
65
76
Table 9
Probabilities of Finding the Global Optimum in Percent for the RHC1 Algorithm, for the 14 Bit Bend Optimization Stopped after 100, 200, 400 Fitness Evaluations (Columns B1, B2, B4, Respectively) and for the Power Divider Optimization Stopped After 100, 200, 400 Fitness Evaluations (Columns D1, D2, D4, Respectively)a
B1
B2
B4
D1
D2
D4
1
7
17
27
51
81
89
2
7
17
40
43
74
86
4
39
64
86
18
35
68
7
6
16
38
16
31
59
11
7
13
27
16
30
55
16
7
14
28
16
30
53
22
6
13
27
15
27
50
29
6
10
23
14
28
47
indicates the number of individuals in the initialization step.
Table 10
Bit-Fitness Values for the Ten Bits of the HF Solution of the 90° Bend Structurea
Bit
Bit Fitness
Best
1
0.5020626
1
2
0.4509298
0
3
0.3468390
4
0.0569637
0
5
0.3744641
0
6
0.3250102
7
0.2519560
0
8
0.3823080
0
9
0.3907066
0
10
0.4766179
0
The “Best” column shows the bits of the global optimum solution. For the underlined values, the bit-fitness value is misleading; i.e., the corresponding bit of the best solution is 1, although the bit-fitness value is . Note that 80% of the bit-fitness values provides a correct estimate. Bit 4 has a very small value, indicating very high probability for a defect at position 4. According to Fig. 1, this is the cell at the input and output ports, where one also intuitively expects a defect.
Tables (10)
Table 1
Population Size for the Eight Different Runs for the Seven Stochastic Optimizers (STAT, MGA0, MGA1, MGA2, MUT0, MUT1, MUT2) and for the HC Algorithm RHC1a
Stochastic
RHC1
1
4
1
2
5
2
3
7
4
4
10
7
5
14
11
6
19
16
7
25
22
8
32
29
is used in the following tables.
Table 2
Probabilities of Finding Global Optimum in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizersa
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
33
36
39
43
22
46
49
64
2
34
38
40
46
29
50
52
62
3
34
39
42
46
30
55
56
58
4
34
39
43
45
31
57
55
45
5
36
41
44
49
31
53
49
45
6
36
42
45
50
32
44
40
45
7
35
40
45
50
32
34
33
43
8
36
39
43
50
32
29
29
42
av
35
39
43
47
30
46
46
51
indicates the population size according to Table 1; av is the average of the values in the corresponding column.
Table 3
Average Relative Fitness (Fitness Found by the Algorithm or Fitness of the Global Optimum) in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizersa
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
99
99
99
99
96
99
99
100
2
99
99
99
99
98
99
99
100
3
99
99
99
99
99
99
99
100
4
99
99
99
99
99
99
99
99
5
99
99
99
99
99
99
99
99
6
99
99
99
100
99
99
99
99
7
99
99
99
100
99
99
99
99
8
99
99
99
100
99
99
99
99
av
99
99
99
99
98
99
99
100
indicates the population size according to Table 1.
Table 4
Average Number of Fitness Calls When an IFT Is Used and the Algorithm Is Stopped As Soon As It Finds the Global Optimum, Averaged over All Test Case 100, 200, and 400 Fitness Evaluations, for All Eight Optimizersa
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
184
180
175
167
172
163
158
111
2
183
178
173
163
191
157
151
118
3
185
175
171
164
192
149
146
131
4
186
176
171
166
190
144
141
166
5
186
173
167
159
190
126
119
168
6
187
173
167
157
191
110
102
169
7
193
176
165
158
191
99
91
174
8
194
178
168
160
191
89
83
179
av
187
176
170
162
188
130
124
152
indicates the population size according to Table 1.
Same as in Table 2, When All Algorithms Are Stopped After 100 Fitness Evaluations
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
16
17
17
20
11
21
22
44
2
15
17
17
20
12
23
22
37
3
16
17
19
21
12
24
25
30
4
16
17
19
20
13
25
25
20
5
16
19
20
23
12
30
29
22
6
17
17
21
23
14
29
27
23
7
16
16
20
24
14
26
25
21
8
16
16
19
22
14
24
25
20
av
16
17
19
22
13
25
25
27
Table 7
Same as in Table 2, When All Algorithms Are Stopped After 200 Fitness Evaluations
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
29
31
35
39
21
42
46
67
2
29
35
35
43
25
47
50
65
3
30
36
38
43
25
54
54
56
4
30
36
38
41
27
59
58
40
5
32
37
39
46
27
56
53
41
6
31
39
40
48
28
47
44
41
7
32
38
42
49
28
36
34
39
8
32
36
39
48
28
30
30
39
av
31
36
38
44
26
46
46
48
Table 8
Same as in Table 2, When All Algorithms Are Stopped After 400 Fitness Evaluations
STAT
MGA0
MGA1
MGA2
MUT0
MUT1
MUT2
RHC1
1
55
60
65
71
34
75
80
81
2
57
62
68
75
51
81
85
85
3
55
64
68
74
53
86
88
89
4
56
65
70
75
54
88
83
75
5
58
68
73
77
53
73
66
72
6
60
69
74
78
54
56
50
71
7
58
67
74
78
54
42
39
70
8
59
66
72
79
54
34
32
68
av
57
65
70
76
50
67
65
76
Table 9
Probabilities of Finding the Global Optimum in Percent for the RHC1 Algorithm, for the 14 Bit Bend Optimization Stopped after 100, 200, 400 Fitness Evaluations (Columns B1, B2, B4, Respectively) and for the Power Divider Optimization Stopped After 100, 200, 400 Fitness Evaluations (Columns D1, D2, D4, Respectively)a
B1
B2
B4
D1
D2
D4
1
7
17
27
51
81
89
2
7
17
40
43
74
86
4
39
64
86
18
35
68
7
6
16
38
16
31
59
11
7
13
27
16
30
55
16
7
14
28
16
30
53
22
6
13
27
15
27
50
29
6
10
23
14
28
47
indicates the number of individuals in the initialization step.
Table 10
Bit-Fitness Values for the Ten Bits of the HF Solution of the 90° Bend Structurea
Bit
Bit Fitness
Best
1
0.5020626
1
2
0.4509298
0
3
0.3468390
4
0.0569637
0
5
0.3744641
0
6
0.3250102
7
0.2519560
0
8
0.3823080
0
9
0.3907066
0
10
0.4766179
0
The “Best” column shows the bits of the global optimum solution. For the underlined values, the bit-fitness value is misleading; i.e., the corresponding bit of the best solution is 1, although the bit-fitness value is . Note that 80% of the bit-fitness values provides a correct estimate. Bit 4 has a very small value, indicating very high probability for a defect at position 4. According to Fig. 1, this is the cell at the input and output ports, where one also intuitively expects a defect.