International Conference on Computing, Communication and Automation (ICCCA2015)
Log-Logistic SOMA with Quadratic Approximation Crossover Dipti Singh
Seema Agrawal
Department of Applied Sciences Gautam Buddha University Greater Noida, India
[email protected] Abstract—Though population based algorithms performs well to solve many global optimization problems, many attempts has been made in literature to improve the efficiency of these algorithms. One possible way is to hybridized them with the features of other deterministic or population based techniques. This Paper presents a Log-LogisticSelf organizing migrating algorithm with quadratic approximation crossover (LLSOMAQI). This algorithm is an extension of algorithms SOMAQI,in which Self Organizing Migrating Algorithm (SOMA) has been hybridized with quadratic approximation (QA) crossover operator and SOMAM, which is hybridization of SOMA and Log-Logistic (LL)mutation. LLSOMAQI has been tested on 15 benchmark unconstrained test problems and an analysis has been made between the three algorithms. LLSOMAQI, its originator SOMA and PSO to show the efficiency of this algorithm over other population based algorithms. Keywords—Self organizing migrating algorithm; Quadratic approximation crossover operator; Log-logistic mutation operator; Particle swarm optimization; Global Optimization.
I.
INTRODUCTION
Optimization is the process of finding the best alternative among the given set of solutions. Optimization problems arise in almost every sphere of human activities. To solve these optimization problems there are many population based techniques such as genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization (ACO), differential evolution (DE), self organizing migrating algorithm (SOMA) etc. These algorithms have proved their efficiency in solving many real word global optimization problems arising in various fields. SOMA is relatively a new member to the family of population based algorithms proposed by Zelinka and Lampinen in 2000 [1]. The main feature of SOMA is that no new solutions are created during the search. Instead, only the positions of the solutions are changed during a generation, called a migration loop. Hybridization is a growing area of interest in research. Hybrid strategy is regarded as an effective method that combines the advantages of two or more algorithms in efficient manner. Only few attempts have been made in past to hybridize SOMA with the features of other existing algorithms. Hybridized SOMA was first introduced by Deep and Dipti in 2007, in which the features of Binary coded GA and Real coded SOMA has been merged [2]. In this approach crossover and mutation operators are binary coded and selection operator has
ISBN:978-1-4799-8890-7/15/$31.00 ©2015 IEEE
Department of Mathematics S.S.V.P.G. College Hapur, India
[email protected] been modified by real coded SOMA. The main feature that distinguishes SOMA with other algorithms is that it works with very less population size. To solve 100 dimensional problems it requires only 10 population size. But due to lack of diversity preserving mechanism, it may converge to local optimal solution. To maintain the diversity of the search space a new variant of SOMA (SOMAQI), has been developed by Dipti et al, in which quadratic interpolation has been combined with SOMA[3]. In successive attempts later Dipti and Seema hybridized SOMA with non uniform mutation [4] and with Log-logistic mutation operator [5]for solving the unconstrained optimization problems. In all above mentioned approaches SOMA has been combined with either crossover or mutation operator. In this paper SOMA has been combined with both, quadratic approximation crossover and log-logistic mutation. An analysis has been made between the presented algorithm LLSOMAQI, SOMA and PSO (to show its efficiency over other algorithms).For this, a set of 15 benchmark test problems has been selected. The paper is organized as follows. In section II, SOMA is described. In section III, the proposed Algorithm LLSOMAQI is presented. In section IV, the numerical results are discussed. Finally, the paper concludes with Section V drawing the conclusions of the present study. II.
SELF ORGANIZING MIGRATING ALGORITHM
Self Organizing migrating Algorithm is a population based stochastic search technique which is based on the social behavior of a group of individuals [6], [7]. Like other evolutionary algorithms it also works on the natural phenomenon. The only difference in its working principle is that it follows cooperative-competitive behavior. Rather than competing with each other, individual with highest fitness value known as leader guide the other individuals known as active, to move in its direction. Active individual may be taken only one or all individuals other than leader. Each iteration or generation is called migration and in each migration loop active individual travels a certain distance towards the leader in n steps of defined length. This path is perturbed randomly by PRT parameter. PRT vector is created before an individual proceeds towards leader. It is defined in the range (0, 1). The movement of an individual is given as follows:
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International Conference on Computing, Communication and Automation (ICCCA2015)
,
+( Where tє 0, , ,
,
, ,
.t.
, . .
, , , ,
III.
r (0,1) is uniformly distributed random number , u and l are the upper and lower bounds of the decision variable, / and λ is a random number T = ( following log-logistic distribution and is given by: / / 1 Where h (0,1) is a uniformly distributed random number, b> 0 is a scale parameter and α is termed as mutation index as it controls the strength of mutation. More information can be obtained from [8]. Step11:if new point is better than active replace active with the new one; Step12: if termination criterion is satisfied stop else go to step2; Step13: report the best individual as the optimal solution;
. (1)
.
PROPOSED HYBRID ALGORITHM
SOMA is an emergent search technique with unique feature of using very less population size. In the working of SOMA, individuals migrate from one position to other, no new solution are created during the search. So there are more chances of loosing diversity after certain migrations. To maintain the diversity of the search space earlier SOMA has been hybridized with QA crossover alone known as SOMAQI [3], in this approach mutation has not been used and later SOMA is hybridized with LL mutation alone without crossover called as SOMA-M [5]. The proposed approach is an extension of both the algorithms in which SOMA is merged with QA crossover and LL mutation both. The computational steps of LLSOMAQI are given as follows:
IV.
NUMERICAL RESULTS
To validate the performance of a proposed approach, an analysis has been made on the basis of results. In this section the results obtained by LLSOMAQI on a set of 15 unconstrained test problems, provided in table I, has been discussed. All the problems are of minimization and have the minimum value as 0. To evaluate the performance of this algorithm 30 trials of each problem is carried out, each time with different seed of random numbers. A run is considered to be a success within 1% accuracy of the known global optimal solution. The stopping criterion is either a run is a success or a fixed number of migrations (10,000) are performed. The comparative performance ofLLSOMAQI, SOMA and PSO are measured in terms of success rate, average number of function evaluations (ANFE), and function mean best. Trials for the 15 problems are performed for dimension n=30, 50 and 100. The value of parameters after fine tuning related to LLSOMAQI are as follows: population size is taken as 10, PRT as 0.1, 0.3 and 0.9 respectively, step size as 0.31, path length as 3.0 and total number of migrations allowed for one run are taken as 10,000. Table II, III and IV shows the number of successful runs of a total of 30 runs (success rate), ANFE and meanobjective function value corresponding to LLSOMAQI, PSO and SOMAfor dimension 30, 50 and 100 respectively.Results presented in table II, III and IV clearly shows that LLSOMAQI attains best mean function value at good success rate with lesser function evaluations as compared to SOMA and PSO in almost all problems. Since results are clearly expressible, no further discussion is required about the performance of LLSOMAQI. The important thing to mention here is that LLSOMAQI does not require any change in population size with the increment in dimension. For solving 30, 50 and 100 dimensional problems only 10 population size has been taken. A graphical analysis has also been made with the help of a Performance Index (PI) taken from [3],[4],[5] in the following manner.
Step 1: generate initial population; Step 2: evaluate all individuals in the population; Step 3: generate PRT vector for all individuals; Step 4: sort all of them; Step 5: select the best fitness individual as leader and worst as active; Step 6: for active individual new positions are created using equation (1). Then the best position is selected and replaces the active individual by the new one; Step7: create new point by QA crossover operator using equation (2) as follows: A new trial point of minima = ( , , , , ….., ) is given as (2) Where f(R1), f(R2) and f(R3) are the objective function values at randomly selected three distinct pointR1(with best fitness value) , R2 and R3 respectively. Step8:if new point is better than active replace active with the new one; Step 9: select the best fitness individual as leader and worst as active; Step10: create new point by mutation operator using equation (3) as follows: randomly selects one solutionxij and sets its value according to the following rule: (3)
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International Conference on Computing, Communication and Automation (ICCCA2015) ∑
= Total number of runs of ith problem = Mean objective function value obtained by an algorithm ofith problem = Minimum of Mean objective function value obtained by all algorithms of ith problem = Average number of function evaluations of successful runs used by an algorithm in obtaining the solution of ith problem = Minimum of average number of successful runs used all algorithms in obtaining the solution of ith problem = Total number of problems analyzed
(4) ,
Where
, 0,
0 0
,
0
0, 0 Where = Number of successful runs ofith problem TableI. Benchmark Problems
Benchmark Problems with initialized range Problem
S.No.
Name 1
Ackley
2
Cosine Mixture
3
Exponential
4
Griewank
5
Levy and Montalvo-1
20 ∑
0.1
1
0.02
8
Rosenbrock
9
Schewefel-3
10
Dejong’s function with noise
11
Step function
12
Sphere
13
Axis parallel hyper ellipsoid
14
Ellipsoidal
15
Brown3
[-1, 1]
[-600, 600]
0.5
[-10, 10] 1 4000
,
√
∑
10 1
Rastrigin
20
[-1, 1]
1
Levy and Montalvo-1
7
2
5
- 0.1∑
1 6
1
Range [-30,30]
1
1
1
[-5, 5]
10
1 [-5.12, 5.12]
0.1
3
1
1
3
1
1
2
[-10, 10]
10
10
[-30, 30]
[-1.28,1.28]
2
[-100, 100] 100
1
[-5.12, 5.12] [-5.12, 5.12]
|
|
|
1
0,1
1 2
148
|
[-n, n] [-1, 4]
International Conference on Computing, Communication and Automation (ICCCA2015)
TableII. Success Rate, ANFE& Mean objective function valueof LLSOMAQI, PSO&SOMA for dim. 30
Prob. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
PSO 21 25 30 14 09 27 0 0 11 27 0 30 22 10 28
No. of successful runs Out of 30 SOMA LLSOMAQI 03 30 18 30 30 30 03 30 30 30 23 23 0 30 02 0 30 30 30 30 29 30 30 30 30 30 29 30 30 30
Average no. of function evaluations of successful runs PSO SOMA LLSOMAQI 142777 47818 1514 141465 20023 1116 126190 13702 918 143084 43534 1518 106784 15471 12610 131592 22045 27783 200020 36010 1554 200020 180010 200010 129625 35712 320 136301 36372 1678 200020 25285 16678 125305 22924 1312 129417 26786 1396 94340 31748 46171 129973 40526 526
Mean objective function value of successful runs PSO SOMA LLSOMAQI 0.00975 0.00905 0.000530 0.00937 0.00817 0.000453 0.00810 0.000357 0.00940 0.00960 0.00758 0.000824 0.00892 0.00708 0.000687 0.00866 0.00884 0.00665 22.866 17.32 0.000623 5.38 0.0330 25.8861 0.00967 0.00832 0.000185 0.00954 0.00811 0.000513 0.0385 0.00703 0.000715 0.00992 0.00815 0.000375 0.00950 0.00807 0.000518 0.00956 0.00867 0.000965 0.00938 0.00814 0.000503
Table III. Success Rate, ANFE& Mean objective function value of LLSOMAQI, PSO&SOMA for dim. 50
Prob. No.
No. of successful runs Out of 30 PSO SOMA LLSOMAQI
Average no. of function evaluations of successful runs PSO SOMA LLSOMAQI
149
Mean objective function value of successful runs PSO SOMA LLSOMAQI
International Conference on Computing, Communication and Automation (IC CCCA2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
06 05 29 09 02 22 0 0 01 17 08 30 13 06 18
0 0 30 07 28 19 0 0 30 30 30 30 30 29 30
30 30 30 30 30 11 30 0 30 30 30 30 30 30 30
180730 189732 176762 172335 143400 165766 200020 200020 157040 168018 191830 15537 163240 125224 168554
90010 90010 27044 72521 27558 46238 90010 180010 78148 70822 64604 44712 57088 70444 75885
1810 1224 817 2040 12708 57753 1680 200010 273 2816 33078 1283 1396 95840 274
0.00995 0.00965 0.00970 0.00967 0.00951 0.00912 76.179 35.175 0.00943 0.00983 0.00981 0.00960 0.00980 0.00976 0.00978
3.413 0.5119 0.00940 0.00895 0.00785 0.00901 35.06 276.092 0.00995 0.00882 0.00868 0.00884 0.00820 0.00859 0.00845
0.000626 0.000538 0.000810 0.000648 0.000942 0.00638 0.000643 46.8746 0.000271 0.000468 0.000687 0.000375 0.000662 0.000992 0.000168
Table IV. Success Rate, ANFE E & Mean objective function value of LLSOMAQI, PSO&SO OMA for dim. 100
Prob. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
No. of successful runs Out of 30 PSO SOMA LLSOMAQI 0 0 30 0 0 30 0 30 30 0 05 30 0 28 29 0 05 05 0 0 30 0 0 0 0 0 30 0 08 30 0 12 30 30 30 30 0 30 30 03 0 26 0 01 30
Average no. of function evaluations of successful runs PSO SOMA LLSOMAQI 200020 180010 2496 200020 180010 1619 200020 84144 1607 200020 153290 2662 200020 80829 16349 200020 150842 145622 200020 180010 1920 200020 180010 200010 200020 180010 252 200020 180010 2613 200020 172555 62413 199600 129308 1849 200020 141010 2526 173520 180010 176758 200020 178372 248
Performance Index (PI)
1 0 , , 1 are , the weights assigned to percentage of successs, mean objective function value and ANFEof successful runns, respectively.In case (i), the mean objective function value and a average no. of function evaluations of successful runs are given equal weights. In case (ii), equal weights are assigned to the numbers of successful runs and meanobjective function value of successful runs. In case (iii), equal weighhts are assigned to mean objective function value and aveerage number of successful runs.
Mean objective o function value of successfull runs PSO SOMA LLSOMAQI 20.016 9 9.06 0.000810 36.364 2.336 0.000606 0.999 0.00810 0.000513 0.0552 0.00940 0.000687 5.023 0.00758 0.000953 4.327 0.00926 0.00811 345.606 118.156 0.000648 195.04 1284.4 97.1142 23.244 1.078 0.000242 * 0.0488 0.000468 * 0.00969 0.000860 0.00999 0.00771 0.000739 * 0.00921 0.000416 0.00945 6.862 0.00994 * 0.00984 8.09E-05
PSO
SOM MA
LLSOMAQI
1 0.8 0.6 0.4 0.2 0 0
0.2
0.4 0.6 0 Weights
0.8
1
Figure 1.1 PI for combination of SOMA, PSO &LLSOMAQI for caase1
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International Conference on Computing, Communication and Automation (IC CCCA2015)
Performance Index (PI)
PSO
SOMA
[2]
LLSOMA AQI
1 0.8 0.6 0.4 0.2 0
[3]
[4]
0
0.2
0.4 0.6 weights
0.8 8
1
[5]
Figure 1.2 PI for combination of SOMA, PSO &LLSOMAQI for case 2
Performance Index (PI)
PSO
SOMA
[6]
LLSOMA AQI
[7]
1 0.8 0.6 0.4 0.2 0
[8]
0
0.2
0.4
0.6
0.8 8
1
Weights Figure 1.3 PI for combination of SOMA, PSO &LLSOMAQI for case 3
The graphs corresponding to each of casse (i), (ii) and (iii) for dim. 30 are shown in Figs. 1.1, 1.2 and 1.3. 1 The horizontal axis represents the weight w and the verticcal axis represents the performance index PI.From Fig. 1.1, 1.22 and 1.3 it is clear that LLSOMAQI has the highest PI as com mpare to PSO and SOMA. This PI has shown only the compaarative analysis of these three algorithms. It does not show thhat LLSOMAQI is globally best among all population based alggorithms.
V.
CONCLUSION
In the present study we have proposed LLSOMAQI, the hybridization of SOMA with QA crossoveer operator and LL mutationoperator to maintain the diversity of the population and increase the search capability. The efficiency of the proposed algorithm has been tested on 15 unconstrained benchmark problems. Results clearly show that LLSOMAQI f an analysis performs better than SOMA and PSO. In future can be made on SOMA with varying populattion size. REFERENCES [1]
I.Zelinka, and J. Lampinen, “SOMA- Self Organizing Migrating Algorithm”, in proceedings of the 6th International MendelConference on Soft Computing, pp. 177-187, Brno, Czech, Republic (2000).
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K. Deep, and Dipti, “A New Hyybrid Self Organizing MigratingGenetic Algorithm for Function Optimization”, O IEEE Congress on Evolutionary Computation, pp 2796-2803 2 (2007). Dipti Singh, Seema Agarwal, andNidhi Singh, “A Novel Variant of Self-Organizing Migrating Algorithm A for Global Optimization”, Advances in Intelligent System ms and Computing, Springer, Vol. 258, pp. 225-233 (2014). Dipti Singh, andSeema Agraw wal, “A Novel Hybrid Self Organizing Migrating Algorithm With Mutation for Global Optimization”, International Journal of Soft Computing C and Engineering, Vol.3, pp. 101-106 (2014). Dipti Singh, and SeemaAgraw wal,“Hybridization of self organizing migrating algorithm with mutation for global optimization”, In proceedings of the internationaal conference on mathematical sciences (ICMS), Elsevier, pp. 605-609 (2014). ( I. Zelinka, “SOMA- Self Orgganizing Migrating Algorithm, in New Optimization Techniques in Enngineering”, G. C. Onwubolu and B.V. Babu, Eds. Berlin, Germany: Sppringer (2004). C.O. Godfrey and B. V. Bab abu, “New optimization techniques in engineering”, Springer-Verlag Berlin, Germany Heidelberg,pp. 167215(2004). K. Deep, Shashi and V. K. Katiyar, K “A New Real Coded Genetic Algorithm Operator: Log Loggistic Mutation”,In proceedings of the international conference on soft s computing for problem solving, Advances in intelligent and Soft S Computing Vol. 130, pp. 193200(2012).
