Genetic Linkage
5.3 Experimental Results
The complete experimental results are listed in Tables 5.3, 5.4, 5.5, 5.6 and 5.7. According to the definition in the special session, PSO-RDL successfully solved problems 1, 2, 4, 5, 6, 7 and 12 in the experimental results. Moreover, comparable results are achieved in solving problems 3, 8, 11, 13 and 14. Unfortunately, PSO-RDL failed to solve problems 9, 10 and 15-25. Table 5.8, 5.9 , 5.10, 5.11 and 5.12 show the experimental results compared with other evolutionary algorithms proposed in the special session. Table 5.13 gives the
number of successfully solved problems. From these comparisons, it can be observed that PSO-RDL has a good performance for most problems. Table C.1 shows the solution found by PSO-RDL for each function in this benchmark. Figures 5.1, 5.2, 5.3 and 5.4 show how the dynamic linkage discovery technique changes the linkage configuration during the optimization process. Detailed discussion on the experimental results is presented in the next section.
5.4 Discussion
From the experimental results listed in Table 5.3, it can be considered that the proposed algorithm is able to provide good results for the benchmark. The first five functions are unimodal functions. Function 1 is shifted sphere function, Function 2 is shifted Schwefel’s problem 1.2, and Function 3 is shifted rotated high condition elliptic function. These three functions have different condition numbers which make Function 3 much harder than Functions 1 and 2. Function 4 is shifted Schwefel’s problem 1.2 with noise in fitness.
Function 5 is Schwefel’s problem 2.6 with global optimum on bounds. From the results, we can observe that PSO-RDL reaches the predefined error tolerance level for Functions 1, 2, 4, and 5. For Function 3, PSO-RDL achieves an error of 1e-4 but does not meet the 1e-6 criterion. It may be caused by the multiplicator 106 in the objective function which greatly amplifies the error. In summary, PSO-RDL provides a sufficiently good performance for the unimodal functions in this benchmark.
Functions 6-14 are multimodal problems. Function 6 is shifted Rosenbrock’s function, a problem with a very narrow valley from the local optimum to the global optimum, and solved by PSO-RDL. Function 7 is shifted rotated Griewank’s function without bounds, and this function makes the search easily away from the global optimum. Fortunately, PSO-RDL solved it twice in 25 trials and can achieve a comparable result in average for this function. Function 8 is shifted rotated Ackley’s function with global optimum on bounds, which has a very narrow global basin and half of the dimensions of this basin are on the boundaries. Hence, the search algorithm cannot easily find the global basin when the recombination operator is used. The PSO-RDL failed on this problem in all 25
FES12345 1st(Min)5.6975E-022.3397E+014.8244E+051.6839E+021.1410E+00 7th2.2013E+001.5546E+021.3215E+068.4541E+021.1389E+01 13th(Median)1.2539E+012.9970E+022.0604E+062.4966E+033.4230E+01 1E+0319th1.0878E+025.8835E+023.6532E+064.9921E+031.3766E+03 25th(Max)6.4912E+027.7404E+038.6580E+071.8695E+049.4677E+03 mean1.1711E+029.8902E+027.4499E+064.3445E+031.7836E+03 Std2.0425E+021.7074E+031.7630E+075.2837E+033.2111E+03 1st(Min)0.0000E+001.8605E-101.9327E+046.3928E+000.0000E+00 7th5.6843E-145.1810E-094.9197E+041.1922E+020.0000E+00 13th(Median)5.6843E-144.0501E-081.0152E+054.2557E+021.2733E-10 1E+0419th1.1369E-139.8918E-081.7427E+059.0310E+022.0135E-07 25th(Max)1.0800E-125.6686E-076.7518E+051.5376E+044.5570E-04 mean1.7053E-137.9402E-081.5610E+051.4138E+032.3191E-05 Std2.5527E-131.2049E-071.7142E+053.1880E+039.2921E-05 1st(Min)0.0000E+005.6843E-143.8107E-041.4211E-120.0000E+00 7th0.0000E+005.6843E-144.6282E-045.5718E-100.0000E+00 13th(Median)0.0000E+001.1369E-134.7232E-044.3663E-090.0000E+00 1E+0519th5.6843E-141.7053E-134.7236E-041.8184E-083.6380E-12 25th(Max)5.6843E-147.9581E-131.2360E+003.3329E-062.6100E-06 mean2.5011E-141.7735E-139.6848E-022.4686E-072.0943E-07 Std2.8798E-142.0318E-133.3375E-017.0712E-077.2248E-07 Table5.3:BestfunctionerrorvaluesachievedwhenFES=1e+3,1e+4,and1e+5forfunctions1-5.Thepredefinederroris1e-6forthese fivefunctions.Thesefunctionsareallunimodalproblems,andPSO-RDLsuccessfullysolvedfunctions1,2,4,and5.Comparableresults forfunction3wereobtained.
FES678910 1st(Min)3.7957E+011.2672E+032.0477E+011.6208E+011.8937E+01 7th1.2497E+031.2672E+032.0599E+012.8014E+014.0586E+01 13th(Median)1.0455E+041.2678E+032.0703E+013.0172E+014.9847E+01 1E+0319th2.4273E+051.2865E+032.0751E+014.3307E+015.9570E+01 25th(Max)2.6047E+061.9419E+032.0991E+016.5808E+017.0826E+01 mean3.9211E+051.3463E+032.0708E+013.4992E+015.0378E+01 Std7.7244E+051.6289E+021.1955E-011.3668E+011.3604E+01 1st(Min)5.5079E-095.1871E-012.0008E+013.9798E+001.5919E+01 7th3.5402E-029.1688E-012.0022E+011.5919E+012.0894E+01 13th(Median)1.6479E+009.5936E-012.0070E+012.3879E+013.5818E+01 1E+0419th1.1236E+029.9956E-012.0093E+012.9849E+015.4722E+01 25th(Max)5.8179E+021.3723E+002.0282E+016.5667E+016.9647E+01 mean8.7012E+019.7299E-012.0084E+012.3933E+013.8564E+01 Std1.6549E+021.8345E-017.7340E-021.2693E+011.7976E+01 1st(Min)1.2784E-101.6993E-102.0000E+011.9899E+001.5919E+01 7th9.0580E-101.4780E-022.0000E+017.9597E+002.0894E+01 13th(Median)2.9976E-094.9257E-022.0000E+011.0945E+013.5818E+01 1E+0519th2.6304E-071.0577E-012.0000E+011.6914E+015.4722E+01 25th(Max)3.9866E+001.5520E-012.0001E+013.6813E+016.9647E+01 mean9.5678E-015.7316E-022.0000E+011.2497E+013.8564E+01 Std1.7377E+004.6600E-022.1685E-048.1748E+001.7976E+01 Table5.4:BestfunctionerrorvaluesachievedwhenFES=1e+3,1e+4,and1e+5forfunctions6-10.Thepredefinederroris1e-2for thesefivefunctions.Thefunctionsareallmultimodalproblems,andPSO-RDLsuccessfullysolvedfunction6andgavecomparableresults onfunctions7and8.However,worseresultswereobtainedonfunctions9and10duetothelargenumberoflocaloptima.
FES1112131415 1st(Min)5.7283E+001.5379E+021.5402E+003.4972E+002.2397E+02 7th6.8054E+009.7877E+022.8680E+003.9300E+003.1785E+02 13th(Median)8.2136E+006.8587E+033.3114E+004.1025E+004.1720E+02 1E+0319th9.3260E+001.7770E+044.0523E+004.1570E+004.9406E+02 25th(Max)9.8271E+003.8007E+044.9076E+004.4537E+007.5028E+02 mean8.0589E+001.0718E+043.3269E+004.0285E+004.2130E+02 Std1.2377E+001.1550E+048.4003E-012.3182E-011.3274E+02 1st(Min)2.8733E+009.1041E-053.7507E-013.2027E+001.2408E+02 7th4.8607E+004.1626E-017.4933E-013.6311E+002.6996E+02 13th(Median)5.4240E+006.9468E+001.1691E+003.8062E+003.5151E+02 1E+0419th6.6473E+002.7252E+011.6247E+004.0219E+004.5882E+02 25th(Max)9.0392E+001.6936E+038.9953E-014.4112E+006.0597E+02 mean5.7585E+001.5910E+021.1972E+003.7981E+003.5718E+02 Std1.5200E+004.4750E+025.7742E-013.2971E-011.3133E+02 1st(Min)2.8287E+001.0967E-093.3823E-013.1359E+001.0065E+02 7th4.8524E+004.9112E-096.1622E-013.5812E+001.3786E+02 13th(Median)5.4229E+007.5948E-087.4910E-013.8060E+002.0857E+02 1E+0519th5.9537E+004.4376E-061.1119E+004.0137E+004.2923E+02 25th(Max)9.0392E+001.6936E+031.9851E+004.4108E+006.0596E+02 mean5.5754E+001.3125E+028.8732E-013.7796E+002.7113E+02 Std1.4161E+004.5006E+024.0601E-013.4364E-011.5881E+02 Table5.5:BestfunctionerrorvaluesachievedwhenFES=1e+3,1e+4,and1e+5forfunctions11-14.Thepredefinederroris1e-2for thesefourfunctions.Thefunctionsareallmultimodalproblems,andfunctions13and14areextendedfunctions.PSO-RDLsuccessfully solvedfunction12andobtainedcomparableresultsonfunctions11,13,and14.
FES1617181920 1st(Min)1.3947E+021.4536E+028.0049E+028.0270E+028.2370E+02 7th1.8825E+022.1723E+021.0163E+039.9133E+029.7490E+02 13th(Median)2.2334E+022.5107E+021.0489E+031.0180E+031.0230E+03 1E+0319th2.5106E+022.8939E+021.1313E+031.0836E+031.0524E+03 25th(Max)9.9560E+027.1139E+021.1991E+031.1178E+031.1251E+03 mean2.5766E+022.7683E+021.0489E+031.0102E+039.9493E+02 Std1.7382E+021.1460E+021.0500E+028.7227E+018.6921E+01 1st(Min)1.2782E+021.1722E+028.0000E+026.9578E+028.0000E+02 7th1.5093E+021.5412E+029.8800E+029.7805E+028.3695E+02 13th(Median)1.7354E+021.8304E+021.0367E+031.0134E+031.0090E+03 1E+0419th2.1744E+022.2036E+021.1182E+031.0517E+031.0416E+03 25th(Max)9.8807E+025.9989E+021.1956E+031.1084E+031.1005E+03 mean2.2028E+022.0832E+021.0248E+039.9023E+029.6112E+02 Std1.7388E+021.0039E+021.2107E+021.0272E+021.0517E+02 1st(Min)1.2782E+021.1866E+028.0000E+026.9526E+028.0000E+02 7th1.5093E+021.6793E+029.8800E+029.7674E+028.2436E+02 13th(Median)1.7342E+022.1170E+021.0362E+031.0021E+031.0090E+03 1E+0519th2.1635E+022.3655E+021.1182E+031.0480E+031.0413E+03 25th(Max)9.8632E+026.1468E+021.1905E+031.1084E+031.1002E+03 mean2.1958E+022.2222E+021.0227E+039.8499E+029.5905E+02 Std1.7350E+021.0019E+021.1883E+021.0189E+021.0567E+02 Table5.6:BestfunctionerrorvaluesachievedwhenFES=1e+3,1e+4,and1e+5forfunctions16-20.Thepredefinederroris1e-2for Function16and1e-1fortheotherfourfunctions.
FES2122232425 1st(Min)5.0026E+027.8706E+025.5401E+028.1573E+021.7342E+03 7th1.0561E+039.1841E+021.1419E+039.7928E+021.7491E+03 13th(Median)1.2217E+039.5652E+021.2438E+031.0243E+031.7682E+03 1E+0319th1.2713E+039.9860E+021.2850E+031.2637E+031.7993E+03 25th(Max)1.2814E+031.0633E+031.3264E+031.3325E+031.9582E+03 mean1.0805E+039.4973E+021.1132E+031.0796E+031.7934E+03 Std2.7233E+027.8269E+012.7359E+021.6030E+026.5746E+01 1st(Min)3.0000E+027.5236E+024.2517E+025.0000E+021.7164E+03 7th1.0450E+038.2582E+021.1417E+035.0456E+021.7290E+03 13th(Median)1.1582E+039.1275E+021.2197E+039.6491E+021.7401E+03 1E+0419th1.1899E+039.3397E+021.2594E+031.1142E+031.7478E+03 25th(Max)1.2658E+031.0209E+031.3195E+031.2883E+031.8727E+03 mean9.9795E+028.9147E+021.0874E+038.8951E+021.7452E+03 Std3.3016E+027.0707E+012.9207E+023.0372E+022.9465E+01 1st(Min)3.0000E+027.5022E+024.2517E+022.0000E+021.7372E+03 7th1.0450E+038.0458E+021.1417E+035.0000E+021.7495E+03 13th(Median)1.1582E+039.1047E+021.1881E+035.0462E+021.7554E+03 1E+0519th1.1878E+039.3324E+021.2578E+031.0213E+031.7647E+03 25th(Max)1.2647E+031.0189E+031.2846E+031.2995E+031.8087E+03 mean9.9429E+028.8666E+021.0776E+037.2004E+021.7588E+03 Std3.2749E+027.1176E+012.8681E+023.9639E+021.5416E+01 Table5.7:BestfunctionerrorvaluesachievedwhenFES=1e+3,1e+4,and1e+5forfunctions21-25.Thepredefinederroris1e-1for thesefourfunctions.
Table5.8:Resultscomparedwithotherevolutionaryalgorithmsforfunctions1-5 PSO-RDLmean2.5011E-141.7735E-139.6848E-022.4686E-072.0943E-07 std.2.8798E-142.0318E-133.3375E-017.0712E-077.2248E-07 PSO[50]mean0.0000E+000.0000E+002.2903E+050.0000E+002.6278E+03 std.0.0000E+000.0000E+004.0637E+050.0000E+004.8124E+02 DMS-PSO[51]mean0.0000E+001.2960E-137.0064E-091.8851E-031.1383E-06 std.0.0000E+001.5612E-132.6589E-091.8932E-032.1828E-06 SPC-PNX[52]mean8.8967E-099.6317E-091.0806E+059.3788E-099.1535E-09 std.9.3915E-103.2989E-108.7160E+046.3274E-106.3186E-10 DE[53]mean0.0000E+000.0000E+001.9400E-069.0900E-150.0000E+00 std.0.0000E+000.0000E+004.6300E-063.1500E-140.0000E+00 SaDE[54]mean0.0000E+001.0459E-131.6720E-051.4182E-051.2300E-02 std.0.0000E+005.1124E-133.1196E-057.0912E-051.4600E-02 restartCMA-ES[55]mean5.2000E-094.7000E-095.6000E-095.0200E-096.5800E-09 std.1.9400E-091.5600E-091.9300E-091.7100E-092.1700E-09
Table5.9:Resultscomparedwithotherevolutionaryalgorithmsforfunctions6-10 PSO-RDLmean9.5678E-015.7316E-022.0000E+011.2497E+013.8564E+01 std.1.7377E+004.6600E-022.1685E-048.1748E+001.7976E+01 PSO[50]mean1.5529E+011.7184E-012.0303E+012.7859E+001.8582E+01 std.3.0962E+019.9578E-028.1576E-021.5732E+007.7603E+00 DMS-PSO[51]mean6.8925E-084.5189E-022.0000E+010.0000E+003.6217E+00 std.3.1904E-073.2611E-025.5382E-090.0000E+008.5509E-01 SPC-PNX[52]mean1.8909E+018.2610E-022.0991E+014.0196E+007.3044E+00 std.3.9977E+016.2418E-025.7946E-022.2703E+005.2116E+00 DE[53]mean1.5900E-011.4600E-012.0400E+019.5500E-011.2500E+01 std.7.9700E-011.3800E-017.5800E-029.7300E-017.9600E+00 SaDE[54]mean1.1987E-081.9900E-022.0000E+010.0000E+004.9685E+00 std.1.9282E-081.0700E-025.3901E-080.0000E+001.6918E+00 restartCMA-ES[55]mean4.8700E-093.3100E-092.0000E+012.3900E-017.9600E-02 std.1.6600E-092.0200E-093.8900E-034.3400E-012.7500E-01
Table5.10:Resultscomparedwithotherevolutionaryalgorithmsforfunctions11-15 PSO-RDLmean5.5754E+001.3125E+028.8732E-013.7796E+002.7113E+02 std.1.4161E+004.5006E+024.0601E-013.4364E-011.5881E+02 PSO[50]mean4.1633E+003.3058E+026.2393E-013.0386E+00 std.1.3470E+005.0884E+022.2529E-013.8894E-01 DMS-PSO[51]mean4.6229E+002.4007E+003.6865E-012.3601E+004.8539E+00 std.5.8400E-014.3602E+005.6411E-023.3750E-011.3415E+01 SPC-PNX[52]mean1.9098E+002.5951E+028.3793E-013.0456E+002.5376E+02 std.1.1598E+004.8933E+022.6913E-014.3662E-011.5052E+02 DE[53]mean8.4700E-013.1700E+019.7700E-013.4500E+002.5900E+02 std.1.4000E+001.4200E+024.6700E-014.4000E-011.8300E+02 SaDE[54]mean4.8909E+004.5011E-072.2020E-012.9153E+003.2000E+01 std.6.6190E-018.5062E-074.1100E-022.0630E-011.1076E+02 restartCMA-ES[55]mean9.3400E-012.9300E+016.9600E-013.0100E+002.2800E+02 std.9.0000E-011.4200E+021.5000E-013.4900E-016.8000E+01
Table5.11:Resultscomparedwithotherevolutionaryalgorithmsforfunctions16-20 PSO-RDLmean2.1958E+022.2222E+021.0227E+039.8499E+029.5905E+02 std.1.7350E+021.0019E+021.1883E+021.0189E+021.0567E+02 DMS-PSO[51]mean9.4756E+011.1009E+027.6067E+027.1430E+028.2196E+02 std.1.0086E+014.3453E+001.8458E+022.0105E+024.5874E+01 SPC-PNX[52]mean1.0962E+021.1898E+024.3956E+023.8000E+024.4000E+02 std.9.8654E+001.0707E+012.2494E+021.8708E+022.2913E+02 DE[53]mean1.1300E+021.1500E+024.0000E+024.2000E+024.6000E+02 std.1.8000E+012.0100E+012.0400E+022.1800E+022.3800E+02 SaDE[54]mean1.0121E+021.1406E+027.1939E+027.0494E+027.1302E+02 std.6.1686E+009.9679E+002.0852E+021.9040E+022.0134E+02 restartCMA-ES[55]mean9.1300E-011.2300E+023.3200E+022.2600E+023.0000E+02 std.3.4900E+002.0900E+011.1200E+029.9300E+010.0000E+00
Table5.12:Resultscomparedwithotherevolutionaryalgorithmsforfunctions21-25 PSO-RDLmean9.9429E+028.8666E+021.0776E+037.2004E+021.7588E+03 std.3.2749E+027.1176E+012.8681E+023.9639E+021.5416E+01 DMS-PSO[51]mean5.3600E+026.9242E+027.3034E+022.2400E+023.6571E+02 std.2.1772E+021.5647E+021.6620E+028.3066E+011.5096E+02 SPC-PNX[52]mean6.8006E+027.4927E+025.7591E+022.0000E+024.0641E+02 std.2.6873E+029.3700E+018.2207E+010.0000E+002.3848E-01 DE[53]mean4.9200E+027.1800E+025.7200E+022.0000E+029.2300E+02 std.4.0000E+011.5800E+024.4800E+010.0000E+003.4000E-01 SaDE[54]mean4.6400E+027.3190E+026.6406E+022.0000E+023.7586E+02 std.1.5780E+029.1523E+011.5266E+020.0000E+003.1453E+00 restartCMA-ES[55]mean5.0000E+027.2900E+025.5900E+022.0000E+023.7400E+02 std.0.0000E+003.1800E-136.8600E+003.2400E-113.2200E+00
Table 5.13: Problems solved in different evolutionary algorithms
Method Unimodal Basic Multimodal Expanded Hybrid Composition Functions(5) Functions (7) Function (2) Function (11)
PSO-RDL 1,2,4,5 (4) 6,7,12 (3) *
PSO 1,2,4,5 (4) 6,7,12 (3) * *
SPC-PNX 1,2,4,5 (4) 6,7,11 (3) * *
Sa-DE 1,2,4 (3) 9,12 (2) 15 *
DE 1,2,3,4,5 (5) 6,9 (2) * *
DMS-PSO 1,2,3,5 (4) 6,7,9,12 (4) 15 *
LR-CMA-ES 1,2,3,4,5 (5) 6,7,12 (3) * *
changing unchanged
epoch
Figure 5.1: Fitness convergence and linkage dynamics of the Sphere function. A unimodal function which PSO-RDL solved successfully. The gray area in the figure represents the proper building blocks can improve the fitness and stay unchanged. Once the building blocks do not work well, the linkage configuration will change until the next suitable set is found.
runs. Functions 9, 10, and 11 are shifted Rastrigin’s function, shifted rotated Rastrigin’s function, and shifted rotated Weierstrass function, respectively, all of them have a huge number of local optima. The PSO-RDL has a relatively bad performance on the first two problems comparing with traditional PSO [50] and DMS-PSO [51]. Comparable results were obtained for Function 11. It may be because when the number of local optima is huge, the dissimilar individuals would likely to have similar fitness values. Although they could provide good building blocks, when different building blocks are combined to create new individuals, the offspring could have worse fitness values instead. As long as the building blocks cannot be identified correctly, the genetic operator cannot work well, either. Function 12 is Schwefel’s problem, and PSO-RDL achieves a 80% success rate.
Functions 13 and 14 are extended functions, and the PSO-RDL produces comparable results in solving these two functions.
Function 15-25 are eleven composition functions. They are all built up with basic functions. They give a big challenge to any search algorithm. PSO-RDL has failed in all the experiment runs. These composition functions possess a huge number of local optima, thus made PSO-RDL performs worse due to the similar reason discussed for Functions 9 and 10. The failure of identifying building block makes the PSO-RDL also failed in the search process.
Observe the fitness convergence and linkage dynamics in Figures 5.1, 5.2, 5.3, and 5.4. The gray areas represent the time frames when a proper linkage configuration can assist the optimization process. When the current linkage groups are not suitable, i.e.
the linkage configuration cannot assist the search, the linkage group composition will start to vibrate for some iterations until the next proper set of linkage groups is found.
The phenomenon can explain the assumption that the building block’s composition is dynamically changed during the search process in the real-parameter optimization prob-lem. Thus, it is reasonable that we hand over the linkage adaptation to the mechanism of natural selection. Moreover, Figure 7 shows the function with a large number of local optima and PSO-RDL failed. It is clearly that the linkage configuration keeps changing all the time. As discussed above, this phenomenon indicates that when the function has a large number of local optima, it is hard to recognize the building blocks because totally
different individuals may have similar fitness values. In such a case, different individual may provide their own good building blocks, but worse individuals may still be created by recombining these incompatible pieces of solutions.
Focusing on the time ratio of the linkage status (changing vs. unchanged), we can observe that for Figures 5.1 and 5.2, the linkage configuration stay unchanged most of the time. Correspondingly, the proposed algorithm provide good results on these two func-tions. On the contrary, the linkage configuration keeps changing in the Figures 5.3 and 5.4. Thus, our algorithm do not work very well on these two functions, although we men-tioned that PSO-RDL can obtain comparable results on the shifted expanded Griewank’s plus Rosenbrock’s function. Because there does not exist a very efficient algorithm for this problem so far. Hence, we can conclude that when the linkage configuration changes too often, the algorithm will fail to solve the problem with a high probability.
5.5 Summary
In this chapter, we evaluate the performance of PSO-RDL by conducting the search on 25 numerical functions that were proposed in CEC2005 special session on real-parameter optimization. The description of the test functions, parameter setting and experimental results were given in the above context. We also discussed the strength and weakness of the PSO-RDL by analyzed the search result and linkage dynamics. From the results, it is considered that PSO-RDL can work well and produce a better performance than the traditional particle swarm optimizer. Further, from the observation of linkage dynamics, it is considered that dynamic linkage discovery and recombination operator do improve the performance of the particle swarm optimizer when the building blocks are successfully identified. A detailed discussion and summary of this research work will be provided in the next chapter.
changing
linkage state unchanged
epoch
Figure 5.2: Fitness convergence and linkage dynamics of the Shifted Rotated Griewank’s function. A multimodal function which PSO-RDL produced comparable results. The gray area in the figure represents the proper building blocks can improve the fitness and stay unchanged. Once the building blocks do not work well, the linkage configuration will change until the next suitable set is found.
changing
linkage state unchanged
Figure 5.3: Fitness convergence and linkage dynamics of the Shifted Expanded Griewank’s plus Rosenbrock’s function. A multimodal function which PSO-RDL produced compara-ble results. The gray area in the figure represents the proper building blocks can improve the fitness and stay unchanged. Once the building blocks do not work well, the linkage configuration will change until the next suitable set is found.
changing
unchanged
epoch
Figure 5.4: Fitness convergence and linkage dynamics of the Shifted Rastrigin’s function.
A multimodal function with large number of local optima and PSO-RDL failed to solve.
The gray area in the figure represents the proper building blocks can improve the fitness and stay unchanged. Once the building blocks do not work well, the linkage configuration will change until the next suitable set is found.