Problems with Overlapping Ring Topology
4.1 Cyclic Trap
The cyclic trap is a problem with overlapping ring topology with one overlapping bit, and
Figure 4.1: Illustration of structure of ring topology. All subproblems are connected with each other and construct a ring structure. Each subproblem shares one bit with neighbor subproblems. Each circle represents a bit in the chromosome. The orange circles are over-lapping bits that are shared with two subproblems. The blue circles are non-overover-lapping bits.
trap in Section 3.1. 11111 is the optimum with fitness 1, and 00000 is a suboptimum with fitness 0.8. The fitness of this problem can be expressed as follows:
fm,kcyclic(⃗x) =
GHC makes chromosomes in the population climb to suboptima. Because of the charac-teristic of deceptive trap function, a bit shared with 00000 and 11111 would prefer to be 1 in GHC. After GHC, there are 4 kinds of suboptima: 11111, 00000, 10000 or 00001, and
10001 in the population. The proportion of the optimum 11111, Popt, can be calculated as:
The first two terms in equation are as usual, and the remaining term results from improve-ment of other subproblems. Due to overlapping bits, it can change 01110 to 01111 and increase the possibility of climbing to the optimum. From this result, we can estimate each suboptimum in the same method. The following equations provide estimation of each suboptimum.
To verify our estimation, we record the patterns after GHC. Table 4.1 is the average pro-portion of each pattern in 10000 deceptive trap function.
pattern 00000 10000 00001 10001 11111
theory 0.823975 0.054932 0.054932 0.003662 0.062500 experiment 0.830636 0.051480 0.051482 0.003386 0.063018 Table 4.1: The proportion of different patterns after GHC on cyclic trap function.
Figure 4.2 shows 4 types of situations in the restricted mixing. The black areas repre-sent all bits with 1, and the white areas are all bits with 0. The red frames are the bits which are changed in the restricted mixing, and they are the patterns which possibly be mixed with the whole population in the back mixing. The goal of DSMGA-II is to eliminate
sub-Figure 4.2: Illustration of the restricted mixing cases on cyclic trap function. Note that this figure is only a partial capture of the whole chromosomes. The circles are a set of non-overlapping bits in the same subproblems. The edges are a set of overlapping bits in the same subproblems. Thus, a subproblem is composed of a circle and all edges next to it.
only situation that can let the whole population converge and reach the optimum because it is edge independent improvement. In situation 1, the flipped pattern can be accepted owing to fitness increasing and lead to deceptive improvement, despite that 11111 is the optimum. In situation 2 and 3, flipped pattern can be accepted but cannot lead to pop-ulation convergence because they are edge dependent improvement. Each chromosome mixed with flipped pattern in the back mixing can not be improved if the patterns in the adjacent subproblems are not both optimal. At last, situation 4 is the only situation that can result in population convergence with improvement no matter what the adjacent patterns are.
Similar to non-overlapping problems, Poptand Pno−optare the proportions of the opti-mum and non-optiopti-mum respectively. Roughly, the NFE model provided in Chapter 3 can handle cyclic trap as NCC and Nplateausare zero. We only have to take situation 1, 2 and
3 into consideration. Thus, the NFE model for cyclic trap is written as
However, the overlapping structure may also lead to incorrect model building. Al-though linkage information between non-overlapping bits is still strong, the overlapping structure makes the linkage information between non-overlapping and overlapping bits weaker. This phenomenon increases the probability of mixing failure and leads to NFE increasing. Like experiment in Section 3.2, experiment shows that the probability of build-ing incorrect model on cyclic trap function is around 0.003.
Owing to the overlapping structure, it needs three conditions for the Nhelpless term to occur. It is similar with the Equation 3.7 in NCC. Note that if two sides in situation 4 have equal fitness for some specific parameter setting, DSMGA-II gets stuck. Our model does not handle this phenomenon. Thus, the fitness of 00000 must be adjusted carefully. In situation 1, the fitness of 0000111110000 (one optimum) must be lower than the fitness of 0000000000000 (no optimum); otherwise, cyclic trap function is similar to concatenated trap in Section 3.1. In other hand, DSMGA-II need longer size of mask for the restricted mixing if the fitness of pattern 00001111111110000 is not greater than the fitness of pattern
40 80 160 320 640 1280
Figure 4.3: NFEs with various problem sizes. k = 5 for cyclic trap.
00000000000000000. The size of mask for the successful restricted mixing can be as large as 3k. In this case, we only focus on mask size with at most 2k long. Assume that the fitness of 00000 is fsubopt, the condition can be expressed as:
3
4fsubopt+ 1.0 +3
4fsubopt< 3fsubopt
4fsubopt< 3
4fsubopt+ 2.0 + 3 4fsubopt
⇒ 2
3 < fsubopt< 0.8
Therefore, we choose fsubopt = 0.72 as example for convenient in this experiment. Fig-ure 4.3 provides an empirical verification for our analysis.
Also, the complexity of NFEs for the mixing on cyclic trap is also derived:
Nmixing = O(m(k + n)) = (2km ln m) (4.9)
Nf e = NGHC+ Nmixing = O(k2km ln m) (4.10)
4.2 0-1 Trap
Cyclic trap mentioned above is an overlapping problem without conflict [3]. In this sec-tion, we investigate cyclic trap with conflict by changing a half of subproblems into 0-trap function, and we call this problem 0-1 trap. An overlapping problem with conflict means that at least two subproblems share overlapping bits, and the local optimal patterns in the adjacent subproblems conflict on those overlapping bits, causing the global optimum can-not consist of all optimal patterns to all subproblems. For instance, regarding to the 0-1 trap problem, the overlapping bits are shared between 0-trap function and 1-trap function.
Their optima are 00000 and 11111, respectively. In order to make the optimum unique for the whole problem, the fitness of suboptimum in 0-trap is 0.9 which is greater than the fitness of suboptimum in 1-trap. Thus, the global optimum for 0-1 trap is a chromosome with all 1s. The landscapes of 0-trap and 1-trap are shown in Figure 4.4.
To clarify our statement, in the following context, the local optimal pattern means the pattern of the optimum in the subproblems, and the global optimal fragment is the fragment in the subproblem when the chromosome is the global optimal solution. For example, the local optimal patterns are 00000 in 0-trap function and 11111 in 1-trap function. However, the global optimal fragments in these two function are both 11111.
To investigate the difficulty difference between cyclic trap and 0-1 trap, we vary the ratio of number of 0-trap to number of 1-trap. Figure 4.5 shows the NFEs of optimizing cyclic trap and different ratio of 0-1 trap. We find that the ratio of 1 : 1 makes the problem most difficult. As our first attempt, we focus our investigation on this problem. The fitness
0 1 2 3 4 5
Figure 4.4: The landscape of 0-trap function and 1-trap function
48 96 192 384 768 1536 104
106
Problem Size
NFE
Different ratio of 0−trap and 1−trap 1:1 0−1 trap
1:2 0−1 trap 1:3 0−1 trap cyclic trap
Figure 4.5: Comparison of cyclic trap and 0-1 trap with various ratios. We change the ratio of number of 0-trap to number of 1-trap. For example, 1 : 1 represents the problem is composed of one 0-trap and one 1-trap iteratively. NFEs of 0-1 trap with ratio of 1 to 1 is the greatest in our experiment.
of 0-1 trap function can be expressed as follows:
Similarly, we start our analysis by investigating the proportion of the local optimal pat-terns in different subproblems. Since the optima in 1-trap and 0-trap are different, we di-vide the discussion into two parts. The fitness of suboptimum in 0-trap, 11111, is greater than the fitness of suboptimum in 1-trap, 00000, so the overlapping bit prefers 1 when both subproblem is at each suboptimum. Thus, there are only two kinds of patterns after GHC:11111 and 00000. Like Section 4.1, the optimum can be generated by initialization and GHC. The proportions of the optimum 00000, P00000,0-trap, can be expressed as:
P00000,0-trap = 1
The first term is from initial supply, and the last two terms are from chromosomes with one Hamming distance. However, the overlapping bits may be changed due to conflict.
The last term is the estimation of non-conflict probability. In addition, the proportion of 11111 can be calculated by P11111,0-trap = 1−P00000,0-trap. In other hand, the estimation of
proportions of suboptima in 1-trap is more complicated. Due to overlapping, there are 3 kinds of patterns that can climb to 11111 after GHC. With investigation on the probability of each situation, we can derive the proportion of 11111. Accordingly, the proportions of other suboptima can be derived from following equations:
P00000,1-trap = (1− P11111,1-trap)× P00000,0-trap2
P10000,1-trap = P00001,1-trap = (1− P11111,1-trap)× P11111,0-trapP00000,0-trap
P10001,1-trap = (1− P11111,1-trap)× P11111,0-trap2
(4.16)
Also, to verify our estimation, we record the patterns after GHC. Tables 4.2 and 4.3 are the average proportion of each pattern in 10000 0-trap and 1-trap function.
pattern 00000 11111 theory 0.061719 0.938281 experiment 0.061379 0.938620
Table 4.2: The proportion of different patterns after GHC in 0-trap on 0-1 trap function.
pattern 00000 10000 00001 10001 11111
theory 0.003449 0.052438 0.052438 0.797195 0.094479 experiment 0.003755 0.057615 0.057615 0.786330 0.094685
Table 4.3: The proportion of different patterns after GHC in 1-trap on 0-1 trap function.
To investigate NFEs, we need to find each possible case of the restricted mixing and the back mixing. Figure 4.6 shows 3 major types of situations in the restricted mixing on problem of 0-1 trap. Unlike cyclic trap in Section 4.1, it is more difficult to flip to the global optimal pattern because of conflict. Without conflict, the chromosomes can be flipped into global optimal fragments when only one adjacent subproblem is optimal. However, with conflict, the chromosomes can be flipped into global optimal fragments only if it is between two global optimal fragments. In addition, the characteristic of overlapping and
Figure 4.6: Illustration of the major restricted mixing cases on 0-1 trap. Note that this figure is only a partial capture of the whole chromosomes. The circles are a set of non-overlapping bits in the same subproblems.
conflict can possibly make the overlapping bits not linked to non-overlapping bits in ILS even though they are in the same subproblem. Therefore, the phenomenon of Figure 4.7 can also occur and increase NFEs.
Based on the previous work in 2015 [18], two recurrence relations are derived to es-timate the proportion of 11111 in 0-trap and trap. The proportions of 0-trap and 1-trap in the population at the t-th mixing are denoted as R0,t and R1,t, respectively. Thus, R0,0 = P11111,0-trap and R1,0 = P11111,1-trapin the initial population. By investigating the
Figure 4.7: Illustration of the restricted mixing special case on 0-1 trap. This case is different from above-mentioned cases. It may generate new pattern which does not exist in population at initialization. Note that this figure is only a part capture of the whole chromosome. The rectangle frames each subproblem in cyclic trap. The overlapping bits are framed by multiple rectangles with conflict.
possible mixing situation in 4.6, R1,t+1and R0,t+1can be expressed as:
m Equations 4.17 and 4.18 are two recurrence relations that describe the relation between the proportions of the optimum in 0-trap and 1-trap. In Figure 4.6, we can find that situa-tion 1 decreases R0,t, and situation 2 increases R1,t. Specially, situation 3 increases both R0,t and R1,t. Thus, R0,t and R1,t are affected by each other. Based on the recurrence re-lations, we can estimate NFEs of the t-th mixing from R0,t and R1,tand can be expressed
as: Also, the phenomenon of incorrect model-building can also happen on 0-1 trap, and the linkage information among non-overlapping bits also become weaker. To provide a lower bound of NFEs, we consider a model incorrect only when the adjacent non-overlapping bits are not linked together in ILS. This phenomenon increases the probability of the failed mixing which delays the growth of R0,t and R1,t and leads to NFEs increasing. Like experiment in Section 3.2, experiment shows that the probability of building incorrect model in 0-1 trap function is around 0.034.
As R0,tand R1,tincrease, the possibility of finding the optimum also increases. Owing to the overlapping bits’ preference for 0-trap, DSMGA-II finds the optimum of 1-trap only when both adjacent 0-trap are optimal. Thus, The probability of finding the optimum of 0-1 trap at the t-th mixing can be expressed as
R
m 2
1,t (4.20)
and the probability that the optimum exists in the population is
1− (1 − (R1,tm2 ))n (4.21)
40 80 160 320 640 1280 2560 5120
Figure 4.8: NFEs with various problem sizes. k = 5 for 0-1 trap.
as
When the probability is greater than mm−1, we consider the optimum is found and NFEs can be calculated by the summation∑
tNt. Based on above analyses, NFEs of 0-1 trap can be enumerated with various problem sizes, and the results are shown in Figure 4.8.