Incremental Particle Swarm-Guided Local Search for Continuous Optimization

by Marco A. Montes de Oca, Ken Van den Enden, and Thomas Stützle

June 2008

Submitted to the International Workshop on Hybrid Metaheuristics (HM-2008).

This page contains all supplementary information that, for the sake of conciseness, was not included in the paper.

Table of Contents
  1. Paper abstract
  2. Complete results
    1. Ackley
      1. Fully connected topology
      2. Ring topology
    2. Rastrigin
      1. Fully connected topology
      2. Ring topology
    3. Rosenbrock
      1. Fully connected topology
      2. Ring topology
    4. Expanded Schaffer
      1. Fully connected topology
      2. Ring topology
    5. Schwefel
      1. Fully connected topology
      2. Ring topology
    6. Sphere
      1. Fully connected topology
      2. Ring topology
  3. Statistical significance tests
    1. Ackley
      1. Fully connected topology
      2. Ring topology
    2. Rastrigin
      1. Fully connected topology
      2. Ring topology
    3. Rosenbrock
      1. Fully connected topology
      2. Ring topology
    4. Expanded Schaffer
      1. Fully connected topology
      2. Ring topology
    5. Schwefel
      1. Fully connected topology
      2. Ring topology
    6. Sphere
      1. Fully connected topology
      2. Ring topology

Paper abstract

We present an algorithm for solving continuous optimization problems that is inspired by theoretical and empirical results in social learning and swarm intelligence research. The algorithm is based on a framework that we call incremental social learning. In practical terms, the algorithm is a hybrid between a local search procedure and a particle swarm optimization algorithm with growing population size. The local search procedure provides rapid convergence to good solutions while the particle swarm algorithm enables a comprehensive exploration of the search space. We provide experimental evidence that shows that the algorithm can find good solutions very rapidly without compromising its global search capabilities.

Keywords: Social Learning, Particle Swarm Optimization, Local Search.



Complete results

In the following plots, both the distribution of the solution obtained after a specific number of function evaluations (10^4, 10^5, 10^6) and the development of the median solution over time are shown.

Ackley (100 dimensions)

PSO component using a fully connected topolgy

Function evaluations
10^4 10^510^6
Ackley - 10^4 - FC Ackley - 10^5 - FC Ackley - 10^6 - FC
Solution development over time
Ackley - FC

 

PSO component using a ring topolgy

Function evaluations
10^4 10^510^6
Ackley - 10^4 - Ring Ackley - 10^5 - Ring Ackley - 10^6 - Ring
Solution development over time
Ackley - Ring

 

Rastrigin (100 dimensions)

PSO component using a fully connected topolgy

Function evaluations
10^4 10^510^6
Rastrigin - 10^4 - FC Rastrigin - 10^5 - FC Rastrigin - 10^6 - FC
Solution development over time
Rastrigin - FC

 

PSO component using a ring topolgy

Function evaluations
10^4 10^510^6
Rastrigin - 10^4 - Ring Rastrigin - 10^5 - Ring Rastrigin - 10^6 - Ring
Solution development over time
Rastrigin - Ring

 

Rosenbrock (100 dimensions)

PSO component using a fully connected topolgy

Function evaluations
10^4 10^510^6
Rosenbrock - 10^4 - FC Rosenbrock - 10^5 - FC Rosenbrock - 10^6 - FC
Solution development over time
Rosenbrock - FC

 

PSO component using a ring topolgy

Function evaluations
10^4 10^510^6
Rosenbrock - 10^4 - Ring Rosenbrock - 10^5 - Ring Rosenbrock - 10^6 - Ring
Solution development over time
Rosenbrock - Ring

 

Expanded Schaffer (100 dimensions)

PSO component using a fully connected topolgy

Function evaluations
10^4 10^510^6
Schaffer - 10^4 - FC Schaffer - 10^5 - FC Schaffer - 10^6 - FC
Solution development over time
Schaffer - FC

 

PSO component using a ring topolgy

Function evaluations
10^4 10^510^6
Schaffer - 10^4 - Ring Schaffer - 10^5 - Ring Schaffer - 10^6 - Ring
Solution development over time
Schaffer - Ring

 

Schwefel (100 dimensions)

PSO component using a fully connected topolgy

Function evaluations
10^4 10^510^6
Schwefel - 10^4 - FC Schwefel - 10^5 - FC Schwefel - 10^6 - FC
Solution development over time
Schwefel - FC

 

PSO component using a ring topolgy

Function evaluations
10^4 10^510^6
Schwefel - 10^4 - Ring Schwefel - 10^5 - Ring Schwefel - 10^6 - Ring
Solution development over time
Schwefel - Ring

 

Sphere (100 dimensions)

PSO component using a fully connected topolgy

Function evaluations
10^4 10^510^6
Sphere - 10^4 - FC Sphere - 10^5 - FC Sphere - 10^6 - FC
Solution development over time
Sphere - FC

 

PSO component using a ring topolgy

Function evaluations
10^4 10^510^6
Sphere - 10^4 - Ring Sphere - 10^5 - Ring Sphere - 10^6 - Ring
Solution development over time
Sphere - Ring

Statistical significance tests

The following tables report the p-values obtained after applying pairwise comparisons using the Wilcoxon rank sum test. In all cases, the Holm p-value adjustment method was used.

The alternative hypothesis is "less than". Significance level = 0.05

Ackley (100 dimensions)

PSO component using a fully connected topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1005.2e-08- - - - - - -
3 - PSO-10001.000001.00000- - - - - -
4 - IPSO0.001091.000002.1e-15- - - - -
5 - PSOLS-105.9e-141.00000< 2e-161.00000- - - -
6 - PSOLS-1006.0e-141.00000< 2e-161.000001.00000- - -
7 - PSOLS-10001.2e-131.00000< 2e-161.000001.000001.00000- -
8 - RLS7.4e-151.00000< 2e-161.000001.000001.000001.00000-
9 - IPSOLS1.0e-150.84332< 2e-160.049510.003920.003540.000170.00443
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1002.0e-15- - - - - - -
3 - PSO-10004.8e-151.00000- - - - - -
4 - IPSO1.4e-140.446461.00000- - - - -
5 - PSOLS-10< 2e-162.0e-051.8e-053.5e-05- - - -
6 - PSOLS-1001.6e-159.7e-059.8e-050.001631.00000- - -
7 - PSOLS-10002.0e-159.7e-050.000100.002001.000001.00000- -
8 - RLS1.2e-151.9e-056.0e-050.000261.000001.000001.00000-
9 - IPSOLS< 2e-161.6e-15< 2e-163.6e-164.6e-112.7e-132.5e-131.0e-13
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1007.2e-16- - - - - - -
3 - PSO-10003.0e-162.6e-09- - - - - -
4 - IPSO3.8e-160.031 1.000 - - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-163.8e-16< 2e-161.000 - - -
7 - PSOLS-1000< 2e-16< 2e-161.0e-15< 2e-161.000 1.000 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161.000 1.000 1.000 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.000 1.000 < 2e-161.2e-14

 

PSO component using a ring topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001.00000- - - - - - -
3 - PSO-10001.000001.00000- - - - - -
4 - IPSO0.246456.1e-16< 2e-16- - - - -
5 - PSOLS-100.005142.9e-15< 2e-161.00000- - - -
6 - PSOLS-1000.008152.9e-15< 2e-161.000001.00000- - -
7 - PSOLS-10000.019363.9e-15< 2e-161.000001.000001.00000- -
8 - RLS0.000593.7e-15< 2e-161.000001.000001.000001.00000-
9 - IPSOLS2.9e-086.8e-16< 2e-160.001810.003550.003220.000160.00399
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10001.0000 1.0000 - - - - - -
4 - IPSO< 2e-161.0000 < 2e-16- - - - -
5 - PSOLS-103.2e-131.0000 < 2e-160.0073 - - - -
6 - PSOLS-1001.0e-111.0000 < 2e-160.1631 1.0000 - - -
7 - PSOLS-10001.0e-111.0000 < 2e-160.1631 1.0000 1.0000 - -
8 - RLS4.5e-121.0000 2.6e-160.1631 1.0000 1.0000 1.0000 -
9 - IPSOLS< 2e-161.5e-14< 2e-164.4e-154.0e-112.8e-132.6e-138.0e-14
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-1000< 2e-161.00 - - - - - -
4 - IPSO< 2e-161.00 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-161.3e-157.2e-158.8e-151.00 - - -
7 - PSOLS-1000< 2e-163.0e-15< 2e-168.2e-161.00 1.00 - -
8 - RLS< 2e-161.2e-138.8e-151.1e-131.00 1.00 1.00 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.00 0.78 < 2e-167.2e-15

Rastrigin (100 dimensions)

PSO component using a fully connected topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 1 <2e-16- - - - -
5 - PSOLS-10<2e-16<2e-16<2e-16<2e-16- - - -
6 - PSOLS-100<2e-16<2e-16<2e-16<2e-161 - - -
7 - PSOLS-1000<2e-16<2e-16<2e-16<2e-161 1 - -
8 - RLS<2e-16<2e-16<2e-16<2e-161 1 1 -
9 - IPSOLS<2e-16<2e-16<2e-16<2e-161 1 1 1
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10003.7e-121.00000- - - - - -
4 - IPSO< 2e-161.000005.1e-14- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-162.2e-10- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-162.2e-101.00000- -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-160.000631.000001.00000-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-165.3e-165.3e-168.0e-16
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-1000< 2e-16< 2e-16- - - - - -
4 - IPSO< 2e-16< 2e-161.3e-06- - - - -
5 - PSOLS-10< 2e-16< 2e-161.2e-050.95 - - - -
6 - PSOLS-100< 2e-16< 2e-169.8e-161.1e-089.8e-16- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-161.0e-11< 2e-166.9e-10- -
8 - RLS< 2e-16< 2e-16< 2e-164.3e-12< 2e-166.4e-060.79 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16

 

PSO component using a ring topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 2.0e-08< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 1
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1002.3e-11- - - - - - -
3 - PSO-10001.000001.00000- - - - - -
4 - IPSO1.000001.00000< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-162.8e-10- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-162.8e-101.00000- -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-160.000841.000001.00000-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-165.7e-165.7e-168.7e-16
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-1000< 2e-161.00 - - - - - -
4 - IPSO< 2e-163.0e-061.6e-12- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-166.2e-16- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-16< 2e-164.9e-10- -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-164.8e-060.79 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16

Rosenbrock (100 dimensions)

PSO component using a fully connected topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1009.9e-15- - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO< 2e-16< 2e-16< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-169.8e-113.9e-12< 2e-16-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-169.8e-113.9e-12< 2e-161
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10003.9e-151 - - - - - -
4 - IPSO< 2e-161 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-169.3e-129.3e-129.3e-121.3e-12
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10003.9e-151 - - - - - -
4 - IPSO< 2e-161 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-169.3e-129.3e-129.3e-121.3e-12

 

PSO component using a ring topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 < 2e-16< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161.2e-104.7e-12< 2e-16-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.2e-104.7e-12< 2e-161
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1003.8e-05- - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 1 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.2e-111.2e-111.2e-111.6e-12
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1002.3e-16- - - - - - -
3 - PSO-10002.6e-061.00 - - - - - -
4 - IPSO6.0e-131.00 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161.00 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161.00 1.00 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161.00 0.46 0.46 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-160.85 < 2e-16< 2e-16< 2e-16

Expanded Schaffer (100 dimensions)

PSO component using a fully connected topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 4.7e-09< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS1 5.3e-08< 2e-161 1 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.3e-12< 2e-16< 2e-16< 2e-16
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO2.3e-121 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS1 1 1.1e-131 1 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-1000< 2e-161.6e-12- - - - - -
4 - IPSO< 2e-16< 2e-167.0e-05- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161.00000- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161.000001.00000- -
8 - RLS0.000131.000001.000001.000001.000001.000001.00000-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-162.5e-14< 2e-16< 2e-16< 2e-16

 

PSO component using a ring topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001.00000- - - - - - -
3 - PSO-10001.000001.00000- - - - - -
4 - IPSO1.000001.00000< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161.00000- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161.000001.00000- -
8 - RLS1.000003.9e-07< 2e-160.000131.000001.000001.00000-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.3e-12< 2e-16< 2e-16< 2e-16
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001.00 - - - - - - -
3 - PSO-10001.00 1.00 - - - - - -
4 - IPSO1.00 1.00 <2e-16- - - - -
5 - PSOLS-10<2e-16<2e-16<2e-16<2e-16- - - -
6 - PSOLS-100<2e-16<2e-16<2e-16<2e-161.00 - - -
7 - PSOLS-1000<2e-16<2e-16<2e-16<2e-161.00 1.00 - -
8 - RLS1.00 1.00 <2e-160.74 1.00 1.00 1.00 -
9 - IPSOLS<2e-16<2e-16<2e-16<2e-16<2e-16<2e-16<2e-16<2e-16
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1004.3e-12- - - - - - -
3 - PSO-10007.6e-051.00 - - - - - -
4 - IPSO4.1e-140.14 4.3e-14- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161.00 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161.00 1.00 - -
8 - RLS1.00 1.00 1.00 1.00 1.00 1.00 1.00 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-163.2e-14< 2e-16< 2e-16< 2e-16

Schwefel (100 dimensions)

PSO component using a fully connected topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 1 <2e-16- - - - -
5 - PSOLS-10<2e-16<2e-16<2e-16<2e-16- - - -
6 - PSOLS-100<2e-16<2e-16<2e-16<2e-161 - - -
7 - PSOLS-1000<2e-16<2e-16<2e-16<2e-161 1 - -
8 - RLS<2e-16<2e-16<2e-16<2e-161 1 1 -
9 - IPSOLS<2e-16<2e-16<2e-16<2e-161 1 1 1
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10009.4e-091.0000 - - - - - -
4 - IPSO9.8e-151.0000 0.0076 - - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-160.0096 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-160.0155 1.0000 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-160.0093 1.0000 1.0000 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.6e-156.7e-142.3e-141.7e-14
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-1000< 2e-16< 2e-16- - - - - -
4 - IPSO< 2e-163.5e-160.458 - - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-163.0e-16- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-16< 2e-162.0e-07- -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-161.3e-090.019 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16

 

PSO component using a ring topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 1 <2e-16- - - - -
5 - PSOLS-10<2e-16<2e-16<2e-16<2e-16- - - -
6 - PSOLS-100<2e-16<2e-16<2e-16<2e-161 - - -
7 - PSOLS-1000<2e-16<2e-16<2e-16<2e-161 1 - -
8 - RLS<2e-16<2e-16<2e-16<2e-161 1 1 -
9 - IPSOLS<2e-16<2e-16<2e-16<2e-161 1 1 1
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1000.015 - - - - - - -
3 - PSO-10001.000 1.000 - - - - - -
4 - IPSO1.000 1.000 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-160.014 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-160.021 1.000 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-160.013 1.000 1.000 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.6e-157.3e-142.5e-141.8e-14
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001.9e-15- - - - - - -
3 - PSO-10006.1e-150.98630- - - - - -
4 - IPSO3.4e-160.000407.7e-07- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-164.5e-16- - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-16< 2e-163.3e-07- -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-161.9e-090.01949-
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16< 2e-16

Sphere (100 dimensions)

PSO component using a fully connected topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001.2e-14- - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO< 2e-16< 2e-16< 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161 1 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161 1 1 1
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-10001.6e-151.0000 - - - - - -
4 - IPSO< 2e-160.0019 < 2e-16- - - - -
5 - PSOLS-10< 2e-16< 2e-16< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-16< 2e-16< 2e-16< 2e-161.0000 - - -
7 - PSOLS-1000< 2e-16< 2e-16< 2e-16< 2e-161.0000 1.0000 - -
8 - RLS< 2e-16< 2e-16< 2e-16< 2e-161.0000 1.0000 1.0000 -
9 - IPSOLS< 2e-16< 2e-16< 2e-16< 2e-161.0000 1.0000 1.0000 1.0000
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-100< 2e-16- - - - - - -
3 - PSO-1000< 2e-161 - - - - - -
4 - IPSO< 2e-161 2.6e-16- - - - -
5 - PSOLS-10< 2e-161.0e-05< 2e-16< 2e-16- - - -
6 - PSOLS-100< 2e-161.0e-05< 2e-16< 2e-161 - - -
7 - PSOLS-1000< 2e-161.0e-05< 2e-16< 2e-161 1 - -
8 - RLS< 2e-161.0e-05< 2e-16< 2e-161 1 1 -
9 - IPSOLS< 2e-161.0e-05< 2e-16< 2e-161 1 1 1

 

PSO component using a ring topolgy

10^4 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1001 - - - - - - -
3 - PSO-10001 1 - - - - - -
4 - IPSO1 <2e-16<2e-16- - - - -
5 - PSOLS-10<2e-16<2e-16<2e-16<2e-16- - - -
6 - PSOLS-100<2e-16<2e-16<2e-16<2e-161 - - -
7 - PSOLS-1000<2e-16<2e-16<2e-16<2e-161 1 - -
8 - RLS<2e-16<2e-16<2e-16<2e-161 1 1 -
9 - IPSOLS<2e-16<2e-16<2e-16<2e-161 1 1 1
10^5 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1000.28 - - - - - - -
3 - PSO-10001.00 1.00 - - - - - -
4 - IPSO1.00 1.00 <2e-16- - - - -
5 - PSOLS-10<2e-16<2e-16<2e-16<2e-16- - - -
6 - PSOLS-100<2e-16<2e-16<2e-16<2e-161.00 - - -
7 - PSOLS-1000<2e-16<2e-16<2e-16<2e-161.00 1.00 - -
8 - RLS<2e-16<2e-16<2e-16<2e-161.00 1.00 1.00 -
9 - IPSOLS<2e-16<2e-16<2e-16<2e-161.00 1.00 1.00 1.00
10^6 function evaluations
1 - PSO-102 - PSO-1003 - PSO-10004 - IPSO5 - PSOLS-106 - PSOLS-1007 - PSOLS-10008 - RLS
2 - PSO-1002.2e-14- - - - - - -
3 - PSO-10000.24 1.00 - - - - - -
4 - IPSO2.5e-091.00 < 2e-16- - - - -
5 - PSOLS-102.2e-141.00 < 2e-16< 2e-16- - - -
6 - PSOLS-1002.2e-141.00 < 2e-16< 2e-161.00 - - -
7 - PSOLS-10002.2e-141.00 < 2e-16< 2e-161.00 1.00 - -
8 - RLS2.2e-141.00 < 2e-16< 2e-161.00 1.00 1.00 -
9 - IPSOLS2.2e-141.00 < 2e-16< 2e-161.00 1.00 1.00 1.00