Immigrant Population Search Algorithm for Solving Constrained Optimization Problems

Figures & data

Table

Algorithm 1.

1- Initializing

2-

3- C = C + 1

4- Migrate New Population Group

5- Eliminate Weak Groups

6- Local Search to Find Better Habitats

7- Updating Ratios

8- if , then go to 3

9- End

Table

Algorithm 2.
1- for = 1 to do
1-1- for = 1 to do
1-1-1-
1-1-2-
1-1-3- End for
1-2-
1-3- End for
2-

Table

Algorithm 3.
1- for = 1 to do
1-1- Select from by roulette wheel
1-2-
1-3-
1-4-
1-5-
1-6-
1-7- if then
1-8- if then
1-9-
1-10- End for

Table

Algorithm 4.

1- for = 1 to do

1-1-

1-2- for = 1 to do

1-2-1-

1-2-2- if then

1-2-2-1-

1-2-2-2-

1-2-2-3-

1-2-2-4- if then

1-2-2-5- if then

1-2-2-6- End if

1-2-3- End for

1-3- If is better than then

1-3-1- ,

1-3-2- End if

1-4- End for

TABLE 1 Results of Solving Sample Problems by IPSA

Problem	Goal	Optimal	Mean	Best	Worst	S.D.
g01	Min	−15.000	−14.799998	−14.999999	−12.999998	0.610257
g02	Max	0.803619	0.759755	0.803562	0.706772	0.020177
g03	Max	1.000	1.004634	1.004866	1.004206	0.000192
g04	Min	−30665.539	−30665.538639	−30665.538660	−30665.538599	0.000014
g05	Min	5126.498	5914.861264	5127.526282	6112.198192	360.711661
g06	Min	−6961.814	−6961.809276	−6961.813026	−6961.802327	0.002391
g07	Min	24.306	25.139169	24.498111	26.764343	0.581283
g08	Max	0.095825	0.095825	0.095825	0.095825	0.000000
g09	Min	680.63	680.793875	680.658206	681.069672	0.093441
g10	Min	7049.25	7704.834277	7362.194521	8355.855430	222.668683
g11	Min	0.75	0.749005	0.749000	0.749034	0.000008
g12	Max	1.000	1.000000	1.000000	1.000000	0.000000
g13	Min	0.053950	1.477502	0.135751	10.560852	1.865055

TABLE 2 Mean Solutions Obtained by IPSA and Three Other Methods

Problem	Goal	Optimal	PSO (Muñoz Zavala, Aguirre, and Villa Diharce 2005)	DE (Rocha and Fernandes 2008)	EM (Rocha and Fernandes 2008)	IPSA
g01	Min	−15.000	−14.710	−14.555	−14.590470	−14.799998
g02	Max	0.803619	0.419960	0.665	0.427705	0.759755
g03	Max	1.000	0.764813	1.000	0.999661	1.000000
g04	Min	−30665.539	−30665.539	−30665.539	−30591.897	−30665.539000
g05	Min	5126.498	5135.973	5264.270	5338.583	5914.861264
g06	Min	−6961.814	−6961.814	—	−6942.915	−6961.809276
g07	Min	24.306	32.407	24.310	53.08965	25.139169
g08	Max	0.095825	0.095825	0.095825	0.095825	0.095825
g09	Min	680.63	680.630	680.630	689.896	680.793875
g10	Min	7049.25	7205.5	7147.334	9492.735	7704.834277
g11	Min	0.75	0.749	0.901	0.749079	0.749005
g12	Max	1.000	0.998875	1.000	1.000000	1.000000
g13	Min	0.053950	0.569358	0.872	2.083421	1.477502

TABLE 3 Best Solutions Obtained by IPSA and Three Other Methods

Problem	Goal	Optimal	PSO (Muñoz Zavala, Aguirre, and Villa Diharce 2005)	DE (Rocha and Fernandes 2008)	EM (Rocha and Fernandes 2008)	IPSA
g01	Min	−15.000	−15.000	−15.000	−14.999805	−14.999999
g02	Max	0.803619	0.669158	0.472	0.444939	0.803562
g03	Max	1.000	0.993930	1.000	1.002983	1.004866
g04	Min	−30665.539	−30665.539	−30665.539	−30642.600	−30665.538660
g05	Min	5126.498	5126.484	5126.484	5135.818	5127.526282
g06	Min	−6961.814	−6161.814	−6954.434	−6953.411	−6961.813026
g07	Min	24.306	24.370153	24.306	27.46692	24.498111
g08	Max	0.095825	0.095825	0.095825	0.095825	0.095825
g09	Min	680.63	680.630	680.630	681.777	680.658206
g10	Min	7049.25	7049.381	7049.248	7187.181	7362.194521
g11	Min	0.75	0.749	0.752	0.749001	0.749000
g12	Max	1.000	1.000	1.00	1.000000	1.000000
g13	Min	0.053950	0.085655	0.385	0.063196	0.135751

TABLE 4 Worst Solutions Obtained by IPSA and Three Other Methods

Problem	Goal	Optimal	PSO (Muñoz Zavala, Aguirre, and Villa Diharce 2005)	DE (Rocha and Fernandes 2008)	EM (Rocha and Fernandes 2008)	IPSA
g01	Min	−15.000	−13.000	−11.828	−12.500515	−12.999998
g02	Max	0.803619	0.299426	0.472	0.377406	0.706772
g03	Max	1.000	0.464	1.000	0.996438	1.004206
g04	Min	−30665.539	−30665.539	−30665.539	−30521.434	−30665.538599
g05	Min	5126.498	5249.825	5534.610	6117.660	6112.198192
g06	Min	−6961.814	−6961.814	−6954.434	−6933.264	−6961.802327
g07	Min	24.306	56.055	24.330	103.5507	26.764343
g08	Max	0.095825	0.095825	0.095825	0.095825	0.095825
g09	Min	680.63	680.631	680.631	701.344	681.069672
g10	Min	7049.25	7894.812	9264.886	16395.51	8355.855430
g11	Min	0.75	0.749	1	0.749346	0.749034
g12	Max	1.000	0.994	1.000	1.000000	1.000000
g13	Min	0.053950	1.793361	0.990	20.784428	10.560852

TABLE 5 Error Index of the Mean Solutions Obtained by IPSA and the Other Three Methods

Problem	PSO	DE	EM	IPSA
g1	0.3673	1.0000	0.8552	0.0000
g2	1.0000	0.2789	0.9772	0.0000
g3	1.0000	0.0000	0.0014	0.0000
g4	0.0000	0.0000	1.0000	0.0000
g5	0.0000	0.1647	0.2601	1.0000
g6	0.0000	2.0000	1.0000	0.0002
g7	0.2813	0.0000	1.0000	0.0288
g8	0.0000	0.0000	0.0000	0.0000
g9	0.0000	0.0000	1.0000	0.0177
g10	0.0248	0.0000	1.0000	0.2377
g11	0.0000	1.0000	0.0005	0.0000
g12	1.0000	0.0000	0.0000	0.0000
g13	0.0000	0.1999	1.0000	0.5998

FIGURE 1 Comparing the average of the error index values of IPSA and those of three other methods.

Muñoz Zavala, A. E., A. H. Aguirre, and E. R. Villa Diharce. 2005. Constrained optimization via particle evolutionary swarm optimization algorithm (PESO). Paper presented at Proceedings of the 2005 Conference on Genetic and Evolutionary Computation, Washington, DC, USA, June.

 Google Scholar

Rocha, A. M. A. C., and E. M. G. P. Fernandes. 2008. Feasibility and dominance rules in the electromagnetism-like algorithm for constrained global optimization. In Computational science and its applications–ICCSA 2008, LNCS. Berlin, Heidelberg: Springer.

 Google Scholar