GEATbx: List of Figures

GEATbx Tutorial

Fig. 2-1: Output in Matlab command window at start of optimization run (used options)

Fig. 2-2: Status information displayed in command window during optimization (some lines removed)

Fig. 2-3: Graphical output during optimization

Fig. 2-4: Result information displayed in command window at the end of the optimization (some parts have been removed)

Fig. 2-5: Demonstration demogeatbx: option selection in menu (top: objective function, bottom: evolutionary algorithm to apply)

Fig. 2-6: Definition of objective function objexample1

Fig. 2-7: Definition of options, size of the problem (number of variables) and execution of optimization

Fig. 2-8: Definition of a larger number of variables and with extended boundaries

Fig. 2-9: Graphical output during optimization of first own objective function

Fig. 3-1: Definition of an objective function

Fig. 3-2: Definition of special return values of an objective function

Fig. 5-1: Layer model of the GEATbx

Fig. 5-2: Calling tree of the Genetic and Evolutionary Algorithm Toolbox (GEATbx)

Fig. 7-1. Procedure for solving optimization problems using evolutionary algorithms

Fig. 7-2. Structure of the system to be optimized as objective function

GEATbx Options
Parameter Options

Fig. 1-1: Output of all used options at the beginning of an optimization (Header of status information)

Fig. 1-2: Status information displayed in command window during optimization (some lines removed)

Fig. 1-3: Result information displayed in command window at the end of the optimization

GEATbx Introduction
Evolutionary Algorithms: Overview, Methods and Operators

Fig. 1-1: Problem solution using evolutionary algorithms

Fig. 2-1: Structure of a single population evolutionary algorithm

Fig. 2-2: Structure of an extended multipopulation evolutionary algorithm

Fig. 3-1: Fitness assignment for linear and non-linear ranking

Fig. 3-2: Properties of linear ranking

Fig. 3-3: Roulette-wheel selection

Fig. 3-4: Stochastic universal sampling

Fig. 3-5: Linear neighborhood: full and half ring

Fig. 3-6: Two-dimensional neighborhood; left: full and half cross, right: full and half star

Fig. 3-7: Properties of truncation selection

Fig. 3-8: Properties of tournament selection

Fig. 3-9: Dependence of selection parameter on selection intensity

Fig. 3-10: Dependence of loss of diversity on selection intensity

Fig. 3-11: Dependence of selection variance on selection intensity

Fig. 4-1: Possible positions of the offspring after discrete recombination

Fig. 4-2: Area for variable value of offspring compared to parents in intermediate recombination

Fig. 4-3: Possible area of the offspring after intermediate recombination

Fig. 4-4: Possible positions of the offspring after line recombination

Fig. 4-5: Possible positions of the offspring after extended line recombination according to the positions of the parents and the definition area of the variables

Fig. 4-6: Single-point crossover

Fig. 4-7: Multi-point crossover

Fig. 5-1: Effect of mutation of real variables in two dimensions

Fig. 6-1: Scheme for elitist insertion

Fig. 7-1: Classification of population models by range of selection (selection pool)

Fig. 7-2: Global population model (master-slave-structure)

Fig. 7-3: Local model (diffusion evolutionary algorithm)

Fig. 7-4: Unrestricted migration topology (Complete net topology)

Fig. 7-5: Scheme for migration of individuals between subpopulation

Fig. 7-6: Ring migration topology; left: distance 1, right: distance 1 and 2

Fig. 7-7: Neighbourhood migration topology (2-D grid)

Fig. 8-1. Division of resources for subpopulations: linear and non-linear ranking and different values of division pressure

Fig. 8-2. Application of different strategies, order of subpopulations; left: beginning of optimization run, middle: middle phase, right: final phase

Fig. 8-3. Competing subpopulations; left: size of subpopulations, middle: relative size of subpopulations, right: objective values of all individuals at the beginning of the optimization run

GEATbx Examples
Examples of
Objective Functions

Fig. 2-1: Visualization of De Jong's function 1 using different domains of the variables; however, both graphics look similar, just the scaling changed; left: surf plot of the function in a very large area from -500 to 500 for each of both variables, right: the function at a smaller area from -10 to 10

Fig. 2-2: Visualization of Axis parallel hyper-ellipsoid function; surf/mesh plot of the function in an area from -5 to 5

Fig. 2-3: Visualization of Rotated hyper-ellipsoid function; surf/mesh plot of the first two variables in an area from -50 to 50

Fig. 2-4: Visualization of Moved axis parallel hyper-ellipsoid function; surf/mesh plot of the of the first and fourth variable, the objective values were calculated from the 4-dimensional function with second and third variable set to 0

Fig. 2-5: Visualization of Rosenbrock's function; left: full definition range of the function, right: focus around the area of the global optimum at [1, 1]

Fig. 2-6: Visualization of Rastrigin's function; left: surf plot in an area from -5 to 5, right: focus around the area of the global optimum at [0, 0] in an area from -1 to 1

Fig. 2-7: Visualization of Schwefel's function; surf plot in an area from -500 to 500

Fig. 2-8: Visualization of Griewangk's function; top left: full definition area from -500 to 500, right: inner area of the function from -50 to 50, bottom left: area from -8 to 8 around the optimum at [0, 0]

Fig. 2-9: Visualization of Sum of different power function; surf plot in an area from -1 to 1

Fig. 2-10: Visualization of Ackley's Path function; left: surf plot in an area from -30 to 30, right: focus around the area of the global optimum at [0, 0] in an area from -2 to 2

Fig. 2-11: Visualization of Langermann's function; left: surf plot in an area from 0 to 10 for the first and second variable, right: same as left, but for the second and third variable

Fig. 2-12: Visualization of Michalewicz's function; top left: surf plot in an area from 0 to 3 for the first and second variable, right: area around the optimum, bottom left: same as top left for the third and fourth variable, variable 1 and 2 are set 0

Fig. 2-13: Visualization of Branins's rcos function; surf plot of the definition range

Fig. 2-14: Visualization of Easom's function; left: surf plot of a large area around the optimum the definition range, right: the direct area around the optimum

Fig. 2-15: Visualization of Goldstein-Price's function; surf plot of the definition range

Fig. 2-16: Visualization of Six-hump camel back function; left: surf plot of the area surrounding the minima, right: smaller area around the minima