Frankenstein's PSO: A Composite Particle Swarm Optimization Algorithm

by Marco A. Montes de Oca, Thomas Stützle, Mauro Birattari, and Marco Dorigo
July 2008

Submitted to IEEE Transactions on Evolutionary Computation.

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. Experimental setup data
  3. Comparison of PSO algorithms
    1. Selected run-length distributions
      1. All variants (normal settings)
      2. Different Inertia weight schedules
    2. Solution quality development over time plots
      1. All variants (normal settings)
      2. Different Inertia weight schedules
    3. FIPS average distance to centroid
  4. Frankenstein's PSO: Parameterization effects
    1. Selected run-length distributions
    2. Solution quality development over time plots
  5. Performance Validation
    1. Selected algorithm configurations
    2. Selected run-length distributions
    3. Solution quality development over time plots

Paper Abstract

During the last decade, many variants of the original particle swarm optimization (PSO) algorithm have been proposed. In many cases, the difference between two variants can be seen as an algorithmic component being present in one variant but not in the other. In the first part of the paper, we present the results and insights obtained from a detailed empirical study of several PSO variants from a component difference point of view. In the second part of the paper, we propose a new PSO algorithm that combines a number of algorithmic components that showed distinct advantages in the experimental study concerning optimization speed and reliability. We call this composite algorithm Frankenstein's PSO in an analogy to the popular character of Mary Shelley’s novel. Frankenstein’s PSO performance evaluation shows that by integrating components in novel ways effective optimizers can be designed.

Keywords: Particle swarm optimization, continuous optimization, integration of algorithmic components, experimental analysis, run-time distributions, swarm intelligence.



Experimental setup data

The following table lists the benchmark functions used in our study.

Benchmark Problems
Function Name Definition Search Range ModalityDimensionality
Ackley Mathematical definition Search range Multimodal n = 30
Griewank Mathematical definition Search range Multimodal n = 30
Rastrigin Mathematical definition Search range Multimodal n = 30
Salomon Mathematical definition Search range Multimodal n = 30
Schwefel (sine root) Mathematical definition Search range Multimodal n = 30
Step Mathematical definition Search range Multimodal n = 30
Rosenbrock Mathematical definition Search range Unimodal n = 30
Sphere Mathematical definition Search range Unimodal n = 30

The following table lists the exact values we used for shifting and biasing all benchmark functions.

Displacement Vectors and Biases
Dimension Ackley Griewank Rastrigin Salomon Schwefel Step Rosenbrock Sphere
X_1 -1.68230e+1 -2.762684e+2 1.90050e+0 -1.68230e+1 0.00000e+0 0.00000e+0 8.10232e+1 -3.93119e+1
X_2 1.49769e+1 -1.191100e+1 -1.56440e+0 0.00769e+0 0.00000e+0 0.00000e+0 -4.83950e+1 5.88999e+1
X_3 6.16900e+0 -5.787884e+2 -9.78800e-1 6.16900e+0 0.00000e+0 0.00000e+0 1.92316e+1 -4.63224e+1
X_4 9.55660e+0 -2.876486e+2 -2.25360e+0 9.55660e+0 0.00000e+0 0.00000e+0 -2.52310e+0 -7.46515e+1
X_5 1.95417e+1 -8.438580e+1 2.49900e+0 1.95417e+1 0.00000e+0 0.00000e+0 7.04338e+1 -1.67997e+1
X_6 -1.71900e+1 -2.286753e+2 -3.28530e+0 -1.71900e+1 0.00000e+0 0.00000e+0 4.71774e+1 -8.05441e+1
X_7 -1.88248e+1 -4.581516e+2 9.75900e-1 -1.88248e+1 0.00000e+0 0.00000e+0 -7.83580e+0 -1.05935e+1
X_8 8.51100e-1 -2.022145e+2 -3.66610e+0 8.51100e-1 0.00000e+0 0.00000e+0 -8.66693e+1 2.49694e+1
X_9 -1.51162e+1 -1.058642e+2 9.85000e-2 -1.51162e+1 0.00000e+0 0.00000e+0 5.78532e+1 8.98384e+1
X_10 1.07934e+1 -9.648980e+1 -3.24650e+0 1.07934e+1 0.00000e+0 0.00000e+0 -9.95330e+0 9.11190e+0
X_11 7.40910e+0 -3.957468e+2 3.80600e+0 7.00000e+0 0.00000e+0 0.00000e+0 2.07778e+1 -1.07443e+1
X_12 8.61710e+0 -5.729498e+2 -2.68340e+0 8.61710e+0 0.00000e+0 0.00000e+0 5.25486e+1 -2.78558e+1
X_13 -1.65641e+1 -2.703641e+2 -1.37010e+0 -1.65641e+1 0.00000e+0 0.00000e+0 7.59263e+1 -1.25806e+1
X_14 -6.68000e+0 -5.668543e+2 4.18210e+0 -6.68000e+0 0.00000e+0 0.00000e+0 4.28773e+1 7.59300e+0
X_15 1.45433e+1 -1.524204e+2 2.48560e+0 1.45433e+1 0.00000e+0 0.00000e+0 -5.82720e+1 7.48127e+1
X_16 7.04540e+0 -5.883819e+2 -4.22370e+0 7.04540e+0 0.00000e+0 0.00000e+0 -1.69728e+1 6.84959e+1
X_17 -1.86215e+1 -2.828892e+2 3.36530e+0 -1.86215e+1 0.00000e+0 0.00000e+0 7.83845e+1 -5.34293e+1
X_18 1.45561e+1 -4.888865e+2 2.15320e+0 1.45561e+1 0.00000e+0 0.00000e+0 7.50427e+1 7.88544e+1
X_19 -1.15942e+1 -3.469817e+2 -3.09290e+0 -1.05942e+1 0.00000e+0 0.00000e+0 -1.61513e+1 -6.85957e+1
X_20 -1.91531e+1 -4.530447e+2 4.31050e+0 -1.91531e+1 0.00000e+0 0.00000e+0 7.08569e+1 6.37432e+1
X_21 -4.73720e+0 -5.065857e+2 -2.98610e+0 -4.73720e+0 0.00000e+0 0.00000e+0 -7.95795e+1 3.13470e+1
X_22 9.25900e-1 -4.759987e+2 3.49360e+0 9.25900e-1 0.00000e+0 0.00000e+0 -2.64837e+1 -3.75016e+1
X_23 1.32412e+1 -3.620492e+2 -2.72890e+0 1.32412e+1 0.00000e+0 0.00000e+0 5.63699e+1 3.38929e+1
X_24 -5.29470e+0 -2.332367e+2 -4.12660e+0 -5.29470e+1 0.00000e+0 0.00000e+0 -8.82249e+1 -8.88045e+1
X_25 1.84160e+0 -4.919864e+2 -2.59000e+0 1.84160e+0 0.00000e+0 0.00000e+0 -6.49996e+1 -7.87719e+1
X_26 4.56180e+0 -5.440898e+2 1.31240e+0 4.56180e+0 0.00000e+0 0.00000e+0 -5.35022e+1 -6.64944e+1
X_27 -1.88905e+1 -7.344560e+1 -1.79900e+0 -1.88905e+1 0.00000e+0 0.00000e+0 -5.42300e+1 4.41972e+1
X_28 9.80080e+0 -5.269011e+2 -1.18900e+0 9.80080e+0 0.00000e+0 0.00000e+0 1.86826e+1 1.83836e+1
X_29 -1.54265e+1 -5.022561e+2 -1.05300e-1 -1.54265e+1 0.00000e+0 0.00000e+0 -4.10061e+1 2.65212e+1
X_30 1.27220e+0 -5.372353e+2 -3.10740e+0 1.27220e+0 0.00000e+0 0.00000e+0 -5.42134e+1 8.44723e+1
Bias -140.0 -180.0 -330.0 -100.0 100.0 -200.0 390.0 -450.0

Comparison of PSO algorithms

Selected run-length distributions

All variants (normal settings)

Since run-length distributions can be generated for any solution quality level, we present here only some selected plots.

The plots are grouped by benchmark problem.

Note: The inertia weight in the decreasing and increasing inertia weight PSO variants reaches its minimum (maximum) value at the end of the run (1,000,000 function evaluations).

Ackley function (0.01%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Square RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Ring RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Griewank function (0.001%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Griewank at 0.01% RLD Griewank at 0.01% RLD Griewank at 0.01%
Square RLD Griewank at 0.01% RLD Griewank at 0.01% RLD Griewank at 0.01%
Ring RLD Griewank at 0.01% RLD Griewank at 0.01% RLD Griewank at 0.01%
Rastrigin function (20.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Square RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Ring RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Salomon function (1.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Square RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Ring RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Schwefel function (4000.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Square RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Ring RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Step function (1.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Square RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Ring RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Rosenbrock function (20.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Square RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Ring RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Sphere function (0.000001%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
Square RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
Ring RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%

Different Inertia weight schedules

The following run-length distributions show the effects of different inertia weight schedules in the decreasing inertia weight PSO algorithm. The schedules studied were 100, 1,000, 10,000, 100,000, and 1,000,000 function evaluations.

The plots are grouped by benchmark problem.

Ackley function (0.01%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Square RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Ring RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Griewank function (0.001%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
Square RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
Ring RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
Rastrigin function (20.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Square RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Ring RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Salomon function (1.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Square RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Ring RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Schwefel function (4000.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Square RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Ring RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Step function (1.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Square RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Ring RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Rosenbrock function (20.0%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Square RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Ring RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Sphere function (0.000001%)
Topology\No. of Particles 20 40 60
Fully Connected RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
Square RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
Ring RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%

Solution quality development over time plots

All variants (normal settings)

Solution quality development over time plots are based on the median number of function evaluations needed to find a solution of a certain solution quality.

The plots are grouped by benchmark problem.

Note: The inertia weight in the decreasing and increasing inertia weight PSO variants reaches its minimum (maximum) value at the end of the run (1,000,000 function evaluations).

Ackley function
Topology\No. of Particles 20 40 60
Fully Connected SQT Ackley SQT Ackley SQT Ackley
Square SQT Ackley SQT Ackley SQT Ackley
Ring SQT Ackley SQT Ackley SQT Ackley
Griewank function
Topology\No. of Particles 20 40 60
Fully Connected SQT Griewank SQT Griewank SQT Griewank
Square SQT Griewank SQT Griewank SQT Griewank
Ring SQT Griewank SQT Griewank SQT Griewank
Rastrigin function
Topology\No. of Particles 20 40 60
Fully Connected SQT Rastrigin SQT Rastrigin SQT Rastrigin
Square SQT Rastrigin SQT Rastrigin SQT Rastrigin
Ring SQT Rastrigin SQT Rastrigin SQT Rastrigin
Salomon function
Topology\No. of Particles 20 40 60
Fully Connected SQT Salomon SQT Salomon SQT Salomon
Square SQT Salomon SQT Salomon SQT Salomon
Ring SQT Salomon SQT Salomon SQT Salomon
Schwefel function
Topology\No. of Particles 20 40 60
Fully Connected SQT Schwefel SQT Schwefel SQT Schwefel
Square SQT Schwefel SQT Schwefel SQT Schwefel
Ring SQT Schwefel SQT Schwefel SQT Schwefel
Step function
Topology\No. of Particles 20 40 60
Fully Connected SQT Step SQT Step SQT Step
Square SQT Step SQT Step SQT Step
Ring SQT Step SQT Step SQT Step
Rosenbrock function
Topology\No. of Particles 20 40 60
Fully Connected SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Square SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Ring SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Sphere function
Topology\No. of Particles 20 40 60
Fully Connected SQT Sphere SQT Sphere SQT Sphere
Square SQT Sphere SQT Sphere SQT Sphere
Ring SQT Sphere SQT Sphere SQT Sphere

Different Inertia weight schedules

The following plots show the effects of different inertia weight schedules on the median solution quality improvement achieved by the decreasing inertia weight PSO algorithm. The schedules studied were 100, 1,000, 10,000, 100,000, and 1,000,000 function evaluations.

The plots are grouped by benchmark problem.

Ackley function
Topology\No. of Particles 20 40 60
Fully Connected SQT Ackley SQT Ackley SQT Ackley
Square SQT Ackley SQT Ackley SQT Ackley
Ring SQT Ackley SQT Ackley SQT Ackley
Griewank function
Topology\No. of Particles 20 40 60
Fully Connected SQT Griewank SQT Griewank SQT Griewank
Square SQT Griewank SQT Griewank SQT Griewank
Ring SQT Griewank SQT Griewank SQT Griewank
Rastrigin function
Topology\No. of Particles 20 40 60
Fully Connected SQT Rastrigin SQT Rastrigin SQT Rastrigin
Square SQT Rastrigin SQT Rastrigin SQT Rastrigin
Ring SQT Rastrigin SQT Rastrigin SQT Rastrigin
Salomon function
Topology\No. of Particles 20 40 60
Fully Connected SQT Salomon SQT Salomon SQT Salomon
Square SQT Salomon SQT Salomon SQT Salomon
Ring SQT Salomon SQT Salomon SQT Salomon
Schwefel function
Topology\No. of Particles 20 40 60
Fully Connected SQT Schwefel SQT Schwefel SQT Schwefel
Square SQT Schwefel SQT Schwefel SQT Schwefel
Ring SQT Schwefel SQT Schwefel SQT Schwefel
Step function
Topology\No. of Particles 20 40 60
Fully Connected SQT Step SQT Step SQT Step
Square SQT Step SQT Step SQT Step
Ring SQT Step SQT Step SQT Step
Rosenbrock function
Topology\No. of Particles 20 40 60
Fully Connected SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Square SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Ring SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Sphere function
Topology\No. of Particles 20 40 60
Fully Connected SQT Sphere SQT Sphere SQT Sphere
Square SQT Sphere SQT Sphere SQT Sphere
Ring SQT Sphere SQT Sphere SQT Sphere

FIPS average distance to centroid

The following plots show the average distance of a swarm's previous best positions to their centroid as a function of the number of particles and the topology used.

Problem/No. of Particles 20 40 60
Ackley Convergence of FIPS on Ackley Convergence of FIPS on Ackley Convergence of FIPS on Ackley
Griewank Convergence of FIPS on Griewank Convergence of FIPS on Griewank Convergence of FIPS on Griewank
Rastrigin Convergence of FIPS on Rastrigin Convergence of FIPS on Rastrigin Convergence of FIPS on Rastrigin
Salomon Convergence of FIPS on Salomon Convergence of FIPS on Salomon Convergence of FIPS on Salomon
Schwefel Convergence of FIPS on Schwefel Convergence of FIPS on Schwefel Convergence of FIPS on Schwefel
Step Convergence of FIPS on Step Convergence of FIPS on Step Convergence of FIPS on Step
Rosenbrock Convergence of FIPS on Rosenbrock Convergence of FIPS on Rosenbrock Convergence of FIPS on Rosenbrock
Sphere Convergence of FIPS on Sphere Convergence of FIPS on Sphere Convergence of FIPS on Sphere

Frankenstein's PSO: Parameterization effects

Selected run-length distributions

The plots presented below correspond to some run-length distributions obtained by Frankenstein's PSO with different parameter settings.

The results are grouped by topology adaptation schedule and number of particles.

Ackley function (0.01%)
Topology schedule\No. of Particles n = 20 n = 40 n = 60
1 x n RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
2 x n RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
3 x n RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
4 x n RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Griewank function (0.001%)
Topology schedule\No. of Particles 20 40 60
1 x n RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
2 x n RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
3 x n RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
4 x n RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
Rastrigin function (20.0%)
Topology schedule\No. of Particles 20 40 60
1 x n RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
2 x n RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
3 x n RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
4 x n RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Salomon function (1.0%)
Topology schedule\No. of Particles n = 20 n = 40 n = 60
1 x n RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
2 x n RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
3 x n RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
4 x n RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Schwefel function (4000.0%)
Topology schedule\No. of Particles n = 20 n = 40 n = 60
1 x n RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
2 x n RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
3 x n RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
4 x n RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Step function (1.0%)
Topology schedule\No. of Particles n = 20 n = 40 n = 60
1 x n RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
2 x n RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
3 x n RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
4 x n RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Rosenbrock function (20.0%)
Topology schedule\No. of Particles 20 40 60
1 x n RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
2 x n RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
3 x n RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
4 x n RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Sphere function (0.000001%)
Topology schedule\No. of Particles 20 40 60
1 x n RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
2 x n RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
3 x n RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%
4 x n RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%

Solution quality development over time plots

The following plots show the effects of different parameter settings on the median solution quality improvement achieved by Frankenstein's PSO algorithm. The schedules studied were 100, 1,000, 10,000, 100,000, and 1,000,000 function evaluations.

The plots are grouped by benchmark problem.

Ackley function
Topology\No. of Particles 20 40 60
1 x n SQT Ackley SQT Ackley SQT Ackley
2 x n SQT Ackley SQT Ackley SQT Ackley
3 x n SQT Ackley SQT Ackley SQT Ackley
4 x n SQT Ackley SQT Ackley SQT Ackley
Griewank function
Topology\No. of Particles 20 40 60
1 x n SQT Griewank SQT Griewank SQT Griewank
2 x n SQT Griewank SQT Griewank SQT Griewank
3 x n SQT Griewank SQT Griewank SQT Griewank
4 x n SQT Griewank SQT Griewank SQT Griewank
Rastrigin function
Topology\No. of Particles 20 40 60
1 x n SQT Rastrigin SQT Rastrigin SQT Rastrigin
2 x n SQT Rastrigin SQT Rastrigin SQT Rastrigin
3 x n SQT Rastrigin SQT Rastrigin SQT Rastrigin
4 x n SQT Rastrigin SQT Rastrigin SQT Rastrigin
Salomon function
Topology\No. of Particles 20 40 60
1 x n SQT Salomon SQT Salomon SQT Salomon
2 x n SQT Salomon SQT Salomon SQT Salomon
3 x n SQT Salomon SQT Salomon SQT Salomon
4 x n SQT Salomon SQT Salomon SQT Salomon
Schwefel function
Topology\No. of Particles 20 40 60
1 x n SQT Schwefel SQT Schwefel SQT Schwefel
2 x n SQT Schwefel SQT Schwefel SQT Schwefel
3 x n SQT Schwefel SQT Schwefel SQT Schwefel
4 x n SQT Schwefel SQT Schwefel SQT Schwefel
Step function
Topology\No. of Particles 20 40 60
1 x n SQT Step SQT Step SQT Step
2 x n SQT Step SQT Step SQT Step
3 x n SQT Step SQT Step SQT Step
4 x n SQT Step SQT Step SQT Step
Rosenbrock function
Topology\No. of Particles 20 40 60
1 x n SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
2 x n SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
3 x n SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
4 x n SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Sphere function
Topology\No. of Particles 20 40 60
1 x n SQT Sphere SQT Sphere SQT Sphere
2 x n SQT Sphere SQT Sphere SQT Sphere
3 x n SQT Sphere SQT Sphere SQT Sphere
4 x n SQT Sphere SQT Sphere SQT Sphere

Performance validation

The following plots show the relative performance difference between the best configurations of each of the compared algorithms after 1000, 10 000, 100 000 and 1 000 000 function evaluations.
Frankenstein's PSO configurations are shown in black.

Selected algorithm configurations

Selected configurations

Selected run-length distributions

Ackley function (0.01%)
10^3 10^4 10^5 10^6
RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01% RLD Ackley at 0.01%
Griewank function (0.001%)
10^3 10^4 10^5 10^6
RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001% RLD Griewank at 0.001%
Rastrigin function (20.0%)
10^3 10^4 10^5 10^6
RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0% RLD Rastrigin at 20.0%
Salomon function (1.0%)
10^3 10^4 10^5 10^6
RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0% RLD Salomon at 1.0%
Schwefel function (4000.0%)
10^3 10^4 10^5 10^6
RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0% RLD Schwefel at 4000.0%
Step function (1.0%)
10^3 10^4 10^5 10^6
RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0% RLD Step at 1.0%
Rosenbrock function (20.0%)
10^3 10^4 10^5 10^6
RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0% RLD Rosenbrock at 20.0%
Sphere function (0.000001%)
10^3 10^4 10^5 10^6
RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001% RLD Sphere at 0.000001%

Solution quality development over time plots

Ackley function
10^3 10^4 10^5 10^6
SQT Ackley SQT Ackley SQT Ackley SQT Ackley
Griewank function
10^3 10^4 10^5 10^6
SQT Griewank SQT Griewank SQT Griewank SQT Griewank
Rastrigin function
10^3 10^4 10^5 10^6
SQT Rastrigin SQT Rastrigin SQT Rastrigin SQT Rastrigin
Salomon function
10^3 10^4 10^5 10^6
SQT Salomon SQT Salomon SQT Salomon SQT Salomon
Schwefel function
10^3 10^4 10^5 10^6
SQT Schwefel SQT Schwefel SQT Schwefel SQT Schwefel
Step function
10^3 10^4 10^5 10^6
SQT Step SQT Step SQT Step SQT Step
Rosenbrock function
10^3 10^4 10^5 10^6
SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock SQT Rosenbrock
Sphere function
10^3 10^4 10^5 10^6
SQT Sphere SQT Sphere SQT Sphere SQT Sphere