Evolutionary Optimization Algorithms:

Biologically-Inspired and Population-Based Approaches to Computer Intelligence

 

Dan Simon, Professor

Cleveland State University

Department of Electrical and Computer Engineering

 

Evolutionary Optimization Algorithms: Biologically-Inspired and Population-Based Approaches to Computer Intelligence, John Wiley & Sons, 2013. This textbook is intended for the advanced undergraduate student, the beginning graduate student, or the practicing engineer who wants a practical but rigorous introduction to the use of evolutionary algorithms (EAs) for optimization. I’ve been working and publishing in the area of EAs since the early 1990s, originally in industry, and now in academia. I’ve also taught graduate-level courses on EAs. I like step-by-step explanations, so that is how I wrote the book. I tried to make it clear and simple for the beginning EA student, but I also included recent research results, so I think that the book has a nice balance of established principles and cutting-edge results. The book includes 103 worked examples and 262 end-of-chapter problems. A solution manual is available to course instructors from http://www.wiley.com/go/optimization . Matlab code for the examples in the book is available for download from this web site. The book is well-documented with over 700 references, ranging from historical references from hundreds of years ago, to papers published within the last few months.

 

Overview

 

There are already many fine books on EAs, which raises the question: Why yet another textbook on this topic? After teaching an EA course for many years and reading many books on the subject, I thought that I would be able to write a book that better met the needs of my students, and that also might better meet the needs of others. The reason I wrote this book is to offer a pedagogical approach, perspective, and material that is not available in any other single book. In particular, I hope that this book will offer the following.

·        A straightforward, bottom-up approach that assists the reader in obtaining a clear but theoretically rigorous understanding of EAs. Some books discuss a variety of EAs as cookbook algorithms without any theoretical support or detailed explanations. Other books read more like research monographs than textbooks, and are not very accessible to the average engineer. This book tries to strike a balance by presenting easy-to-implement algorithms along with some rigorous theory, and lots of discussion about tuning parameters, implementation issues, and trade-offs.

·        Simple examples that provide the reader with an intuitive understanding of EA math, software, equations, and theory. Some EA books present examples or problems that are not amenable to an intuitive understanding. However, it is possible to present simple examples and problems that require only paper and pencil to solve. These simple examples and problems allow the student to more directly see how the theory works out in practice, and more importantly, why it works.

·        MATLAB source code for all of the examples in the book are available on this web page. A number of other books supply source code, but it is often incomplete or outdated, which is frustrating for the reader. My email address is available on my home page at http://academic.csuohio.edu/simond, and I enthusiastically welcome feedback, comments, suggestions for improvements, and corrections. The book also contains algorithmic, high-level pseudocode listings that are more permanent than any specific software listings. Note that the examples and the MATLAB code are not intended to provide efficient, production-quality, or competitive optimization algorithms; they are instead intended to allow the reader to gain a basic understanding of underlying EA concepts. Any serious research or application should rely on the sample code only as a preliminary starting point.

·       Theory and recently-developed EAs that are not available in most other books. These topics include Markov models of EAs, dynamic system models of EAs, artificial bee colony algorithms, biogeography-based optimization, opposition-based learning, artificial fish swarm algorithms, shuffled frog leaping, bacterial foraging optimization, and many others. These topics are recent additions to the state of the art, and their coverage in this book is not matched in any other books. However, this book is not intended to survey the state-of-the-art in any particular area of EA research; that would be impossible, in view of the breadth and depth of current EA research. This book is instead intended to provide a high-level overview of many areas of EA research so that the reader can gain a broad understanding of EAs, and so that the reader can be well-positioned to pursue additional, in-depth  studies in the state-of-the-art.

·         There are other books on EAs that offer some of the above features, but no other books of which I am aware offer all of these features.

 

Summary Table of Contents

 

Part I: An Introduction to Evolutionary Optimization

1. Introduction

2. Optimization

 

Part II: Classic Evolutionary Algorithms

3. Genetic Algorithms

4. Mathematical Models of Genetic Algorithms

5. Evolutionary Programming

6. Evolution Strategies

7. Genetic Programming

8. Variations on Evolutionary Algorithms

 

Part III: More Recent Evolutionary Algorithms

9. Simulated Annealing

10. Ant Colony Optimization

11. Particle Swarm Optimization

12. Differential Evolution

13. Estimation of Distribution Algorithms

14. Biogeography-Based Optimization

15. Cultural Algorithms

16. Opposition-Based Learning

17. Other Evolutionary Algorithms

 

Part IV: Special Types of Optimization Problems

18. Combinatorial Optimization

19. Constrained Optimization

20. Multi-Objective Optimization

21. Expensive, Noisy, and Dynamic Fitness Functions

 

Appendices

A. Some Practical Advice

B. The No Free Lunch Theorem and Performance Testing

C. Benchmark Optimization Functions

 

Errata

 

A list of typos and errors in the book is available at http://academic.csuohio.edu/simond/EvolutionaryOptimization/errata.pdf.

 

Solution manual

 

A solution manual is available from the publisher at http://www.wiley.com/go/optimization . It includes solutions to all the problems at the end of the chapters, including Matlab source code for all of the computer exercises.

 

Feedback

 

My email address is listed on my home page at http://academic.csuohio.edu/simond. I enthusiastically welcome feedback, comments, suggestions for improvements, and corrections.

 

Matlab Code

 

Here are some caveats about the Matlab code.

·        I am aware that the code is not as efficient as it could be. This is due to two reasons. First, I am not a software engineer, and so my coding is less than perfect. Second, I intentionally sacrifice efficiency for the sake of readability. My goal in writing this software is education, not efficiency. (If I were looking for efficiency I would have written the code in C.)

·        I will not be able to explain the code to you in an email. In order for this code to be useful to you, you really need to be proficient in Matlab programming. So please do not email me with requests to help you modify the code for your research problem. The best way to learn how to use and modify the code is to compare it with the psuedo-code in the book, step through the code one line at a time, inspect the variables one at a time, and study the results so that you understand exactly what each line is doing, and why. If you are able to get to that point with this software, then you will be able to easily modify it for your own particular optimization problem and algorithmic variations. Also, I hope that the many comments in the code will be helpful.

·        I will be glad to hear from you if you find a bug in the code. Just like a paper always has one more typo, a computer program always has one more bug. So I will appreciate it if you tell me about any bugs that you find.

 

The examples in the book can be reproduced by running the code that is available here. Here is what you need to do.

1.     The first thing to do is to download the benchmarks and other generic routines under the “Common Matlab Routines” heading below. These routines include common code that is called by many different optimization algorithms. This common code is re-used by many algorithms and so it is available in separate routines for the sake of efficiency. On my computer, I put all of this common code in separate directories on my computer and add those directories to my Matlab path.

2.     After you download the benchmarks and the common code, you can download and run the code for whatever chapter you want, which is available under the “Chapter-by-Chapter Matlab Code” heading below.

 

All of the Matlab routines are described below and are available in hyperlinked ZIP files.

 

Common Matlab Routines

 

Benchmark Functions

Given an independent vector variable, these routines return the cost function value.

Ackley.m – Ackley benchmark function

AckleyDisc.m – Discretized Ackley benchmark function

Fletcher.m – Fletcher benchmark function

FourPeaks.m – Four peaks benchmark function

Griewank.m – Griewank benchmark function

Pairs.m – Pairs benchmark function

Penalty1.m – Penalty #1 benchmark function

Penalty2.m – Penalty #2 benchmark function

Quartic.m – Quartic benchmark function

Rastrigin.m – Rastrigin benchmark function

Rosenbrock.m – Rosenbrock benchmark function

Schwefel12.m – Schwefel 1.2 benchmark function

Schwefel221.m – Schwefel 2.21 benchmark function

Schwefel222.m – Schwefel 2.22 benchmark function

Schwefel226.m – Schwefel 2.26 benchmark function

Sphere.m – Sphere benchmark function

Step.m – Step benchmark function

 

Constrained Benchmark Functions

Benchmarks g01.m through g24.m are from the CEC 2006 competition

Benchmarks c01.m through c18.m are from the CEC 2010 competition

Readme.txt provides the references for the benchmarks

 

Multi-objective Benchmark Functions

Benchmarks u01.m through u10.m are from the CEC 2009 competition

 

Other Common Functions

ClearDups.m – Replaces duplicate individuals in the population with randomly-generated individuals

ComputeCostAndConstrViol.m – Computes the cost and the constraint violation level of each indivdiual

ComputeRandomShift.m – Computes a random shift for a benchmark function (see Appendix C.7.1)

Conclude.m – Displays data about the population and plots results

createRotMatrix.m – Creates a random rotation matrix for a benchmark function (see Appendix C.7.2)

Init.m – Initializes the population and common EA tuning parameters

PopSort.m – Sorts the population from best to worst

ResetPlotOptions.m – Resets Matlab plot options to default values

SetPlotOptions.m – Sets Matlab plot options to values that give nice-looking plots

 

Traveling Salesman Benchmarks

These files, along with many others, are available at the TSPLIB web site.

*.tsp and *.opt.tour – Note that * is the name of the problem:

       ulysses16 (a 16-city problem)

       ulysses22 (a 22-city problem)

       pr76 (a 76-city problem)

       berlin52 (a 52-city problem)

*.tsp file is a text file that defines the problem

*.opt.tour is a text file that specifies the globally optimal solution

 

Common Functions for TSPs

CalcDistance.m – Calculate the distance of a TSP tour

ConcludeTSP.m – Display data about a TSP population and plot results

CreateDistanceArray.m – Calculate the array of distances between each pair of cities in a TSP

GetCoordinates.m – Retrieve latitude and longitude from a .TSP file

GetLongLat.m – Convert .TSP-format data to latitude and longitude

MutateTSP.m – Mutate a closed TSP tour using one of several possible mutation methods

PlotBestTour.m – Plot the best TSP tour from a *.opt.tour file

PlotTour.m – Plot a TSP tour

PopSortTSP.m – Sort TSP individuals from best to worst

ReplaceDupsTSP.m – Replace duplicate individuals in a TSP population

 

Chapter-by-Chapter Matlab Code

 

Chapter 1: Introduction

There is no software for this chapter

 

Chapter 2: Optimization

AdaptiveHillClimbing.m – Adaptive hill climbing

NextHillClimbing.m – Next ascent hill climbing

RandomHillClimbing.m – Random mutation hill climbing

SteepestHillClimbing.m – Steepest ascent hill climbing

MonteHill.m – Monte carlo simulation software to obtain the results in Example 2.7

 

Chapter 3: Genetic Algorithms

GA.m – Genetic algorithm for discrete or continuous optimization (Example 3.3)

PlotContour.m – Plots individuals on top of the Ackley contour plot (called from GA.m)

AckleyContour.m – Create a contour plot of the two-dimensional Ackley function (called from PlotContour.m)

GAContVsDisc.m – Compare a continuous GA with a discrete GA (Example 3.4)

 

Chapter 4: Mathematical Models of Genetic Algorithms

GAMarkovTheory.m – Uses a Markov model to calculate probabilities of GA population distributions (Example 4.9 and 4.10)

GAMarkovSim.m – Simulates a simple GA and plots the proportion of various population distributions (Example 4.9 and 4.10)

EnumPops.m – Recursively generate a list of all possible EA populations (called by GAMarkovSim.m and GAMarkovTheory.m)

GADyn1.m – Uses a dynamic system model to calculate the proportion of each individual in a selection-only  GA (Example 4.11)

GADyn2.m – Uses a dynamic system model and a simulation to calculate the proportion of each individual in a GA with only selection and mutation (no crossover) (Example 4.12)

GADynEx3.m – Uses a dynamic system model to calculate the percentage of GA population distributions (Example 4.14)

 

Chapter 5: Evolutionary Programming

EP.m – Evolutionary programming for continuous optimization

EPMonte.m – Comparision of EP with and without adaptation of mutation variance (Example 5.1)

FSMPrediction.m – EP to optimize a finite state machine to output a desired bit pattern (Example 5.2)

PrimePrediction.m – EP to optimize a finite state machine to predict prime numbers (Example 5.3)

PrimePredictionMonte.m – Monte Carlo simulation of PrimePrediction.m (Example 5.3)

Prisoner.m - EP to optimize a finite state machine for the prisoner’s dilemma problem (Example 5.4)

SanteFe32.m - EP to optimize a finite state machine for the 32 x 32 Sante Fe trail (Section 5.5)

SanteFe32Monte.m – Monte Carlo simulation of SanteFe32.m (Section 5.5)

 

Chapter 6: Evolution Strategies

ES.m – Evolution strategy for continuous optimization (Example 6.1 and 6.3)

MonteES1plus1.m – Compare an ES with standard deviation adaptation and an ES without it (Example 6.1)

MonteESmulambda.m – Compare a (mu+lambda)-ES with a (mu,lambda)-ES (Example 6.2)

MonteESmulambdaAdapt.m – Compare an ES with mutation rate adaptation and an ES without it (Example 6.3 and 6.4)

MonteESmulambdaAdaptAll.m – Save Matlab figure files from MonteESmulambdaAdapt.m for all benchmarks

 

Chapter 7: Genetic Programming

test1.lisp – A simple Lisp program to see how Lisp works

test1Instructions.txt – Instructions for running test1.lisp

test2.lisp – Another simple Lisp program to see how Lisp works

test2Instructions.txt – Instructions nfor running test2.lisp

GPCartControl.lisp – Genetic programming routine for the minimum-time control problem (Section 7.3)

*.lisp – Various auxiliary Lisp routines that are called by GPCartControl.lisp

PhasePlane.lisp – Creates a file of controls as a function of position and velocity for a given switching strategy

EvalCartControl.lisp – Evaluate the cost of a given switching strategy

PhasePlane.m – Generate the theoretically optimal switching curve and sample trajectory (Figures 7.10 and 7.11)

PlotPhasePlane.m – Plot the phase plane based on input files that were created with PhasePlane.lisp

AddNodes.m – An implementation of recursive syntax tree generation (Figures 7.6 and 7.7)

Readme.txt – Instruction file

 

Chapter 8: Evolutionary Algorithm Variations

EPMonteDirectedInit.m – Directed initialization in an evolutionary program (Example 8.1)

SuddenJump.m – An example of a sudden jump in an EA cost function (Figure 8.2)

GrayLandscape.m – Show the difference between a binary-code and gray-code landscape (Example 8.2)

MonteEAVarGA.m – Explore the effect of binary-coding vs. gray-coding in a GA (Examples 8.3 and 8.5). The Matlab command “MonteEAVarGA(@AckleyDisc)” reproduces the results of Example 8.3, and “MonteEAVarGA(@WorstCaseProblem)” reproduces the results of Example 8.5.

EAVarGA.m – Genetic algorithm for Examples 8.3 and 8.5

WorstCaseProblem.m – Cost function file for the worst-case problem of Example 8.5

MonteGAElite.m – Explore the effect of elitism on a GA (Example 8.6)

MonteStudGA.m, GAStud.m – Explore the effect of stud selection on a GA (Example 8.11)

 

Chapter 9: Simulated Annealing

SACooling.m – Generate the cooling schedule plots of Figures 9.3, 9.4, 9.6, and 9.7

SA.m – Simulated annealing for continuous optimization

SAMonteBeta.m – Monte Carlo simulation of SA.m (Example 9.1)

CauchyGaussian.m – Generate the Cauchy and Gaussian PDFs of Figure 9.8

AckleyScaledPlot.m – Generate the scaled Ackley plot of Figure 9.9

SADimension.m – Modified version of SA.m to use different cooling schedules for different dimensions

AckleyScaled.m – Initialization and cost functions the scaled Ackley benchmark function (Example 9.2)

SAMonteBetaDim.m – Monte Carlo simulation of SADimension.m (Example 9.2)

 

Chapter 10: Ant Colony Optimization

ACOInitial.m – Generate the ant simulation plot of Figure 10.5

AS.m – Ant system code for TSP optimization (Example 10.1)

ASCont.m – Ant system code for continuous optimization

ASContMonte.m – Monte Carlo ant system simulation to explore the effect of the number of pheromone bins (Example 10.2)

ASContNumBestMonte.m – Monte Carlo ant system simulation to explore the effect of the number of pheromone contributors (Example 10.3)

ASContMonte1.m – Monte Carlo ant system simulation to explore the effect of the local pheromone decay constant (Example 10.4)

ASContMonte2.m – Monte Carlo ant system simulation to explore the effect of the exploration constant (Example 10.5)

 

Chapter 11: Particle Swarm Optimization

DeltaPlot.m – Generate the discriminant plot of Figure 11.3

ConstrictionLambda.m – Generate the eigenvalue plots of Figures 11.4 and 11.5

PSO.m – Particle swarm optimization for continuous functions (Example 11.1)

PSOMonte.m – Monte Carlo simulation of PSO (Example 11.1)

PSOFully.m – Fully informed particle swarm optimization (Example 11.2)

PSOFullyMonte.m – Monte Carlo simulation of fuzzy informed PSO (Example 11.2)

NPSO.m – Negative reinforcment PSO (Example 11.3)

NPSOMonte.m – Monte Carlo simulation of negative reinforcement PSO (Example 11.3)

 

Chapter 12: Differential Evolution

DE.m – Differential evolution

DEMonteLbin.m – Compare the “/L” and the “/bin” versions of DE (Example 12.1)

DEMonteBase.m – Compare DE using different base vectors (Example 12.2)

DEMonteDiff.m – Compare DE using one or two difference vectors (Example 12.2)

DEMonteF.m – Compare DE using dithered, jittered, or constant F (Example 12.3)

 

Chapter 13: Estimation of Distribution Algorithms

UMDABinary.m – Simulation of the binary univariate marginal distribution algorithm

MonteUMDABinary.m – Monte Carlo simulation of UMDABinary.m (Example 13.1)

cGABinary.m – Simulation of the binary compact genetic algorithm

MonteCGABinaryAlpha.m – Monte Carlo simulation of cGABinary.m with various values of alpha (Example 13.2)

MonteCGABinaryPopSize.m – Monte Carlo simulation of cGABinary.m with various values for population size (Example 13.3)

Kullback.m – Calculation and optimization of mutual information (Examples 13.5 and 13.6)

MIMICBinary.m – Simulation of binary MIMIC and COMIT algorithms

MonteCOMITBinary.m – Monte Carlo simulation of MIMICBinary.m (Example 13.7)

MonteCOMIT_MIMICBinary.m – Monte Carlo simulation of MIMICBinary.m (Example 13.7)

EDAContEx1.m – Generate the PDF plot of Figure 13.18

PBILCont1.m – Generate the PDF plots of Figure 13.20

PBIL.m – Simulation of PBIL algorithm

PBILEta.m – Monte Carlo simulation of PBIL.m with various learning rates (Example 13.10)

PBILUpdateCount.m – Monte Carlo simulation of PBIL.m with various values of Nbest and Nworst (Example 13.10)

PBILSigma.m – Monte Carlo simulation of PBIL.m with various values of k0 and kf (Example 13.10)

 

Chapter 14: Biogeography-Based Optimization

BioSim.m – Calculate species count probabilities (Example 14.1)

SinusoidMigration.m – Generate the migration curves of Figure 14.5

BBO.m – Simulation of the biogeography-based optimization algorithm

MonteBBOSinusoidVsLinear.m – Monte Carlo simulation of BBO.m with linear and sinusoidal migration (Example 14.3)

MonteBBOBlendedVsStandard.m – Monte Carlo simulation of BBO.m with and without blended migration (Example 14.4)

InitialImmigration.m – Generate the immigration curve of Figure 14.10

 

Chapter 15: Cultural Algorithms

CAEPMutate1.m – Generate the PDFs of Figure 15.2

CAEP.m – Simulation of a cultural algorithm with evolutionary programming (Example 15.2)

CAEPMonte.m – Monte Carlo simulation of CAEP.m with and without a belief space (Example 15.2)

SampleACMGrid.m – Generate a random sample grid for the adaptive cultural model (Figure 15.5)

CATSP.m – Simulation of an adaptive cultural model to solve the traveling salesman problem (Example 15.3)

PlotCATSPNumBest.m – Generate the plot of Figure 15.10 (Example 15.3)

 

Chapter 16: Opposition-Based Learning

OBBO.m – Oppositional biogeography-based optimization for optimizing a continuous function

MonteOBBOJumpRate.m – Monte Carlo simulation of OBBO.m with various jump rates (Examples 16.2 and 16.3)

OBLTSP.m – Oppositional biogeography-based optimization for optimizing the traveling salesman problem

MonteOBLTSP.m – Monte Carlo simulation of OBLTSP.m with various jump rates and jumping ratios (Example 16.5)

 

Chapter 17: Other Evolutionary Algorithms

GSO.m – Group search optimizer algorithm for optimizing a continuous function (Section 17.3)

MonteGSO.m – Monte Carlo simulation of GSO.m on various benchmarks

Results from this software are not in the book. This software was contributed to this web page by Steve Szatmary.

 

Chapter 18: Combinatorial Optimization

TSP.m – Simulation of combinatorial evolutionary optimization to solve traveling salesman problems

TSPMonte.m – Monte Carlo simulation of TSP.m with various crossover, mutation, and initialization methods (Example 18.1)

 

Chapter 19: Constrained Optimization

InteriorExample.m – Generate Figure 19.1

BBO.m – Constrained biogeography-based optimization (same routine as in Chapter 14)

MonteBBOConstrained.m – Monte Carlo simulation of BBO.m (Section 19.6)

 

Chapter 20: Multi-Objective Optimization

Pareto1.m – Generate the Pareto set and Pareto front for a multi-objective problem (Example 20.2)

Pareto2.m – Use the aggregation method to generate the Pareto set and Pareto front for a multi-objective problem (Example 20.5)

Pareto3.m – Use a brute force search, along with the aggregation method, to generate the Pareto set and Pareto front (Example 20.6)

MultiBBO.m – Multi-objective biogeography-based optimization

MonteMOEA.m – Monte Carlo simulation of MultiBBO.m with various multi-objective strategies and for various benchmarks (Section 20.5.5 and Table 20.1)

 

Chapter 21: Expensive, Noisy, and Dynamic Fitness Functions

DACE.m – Use the design of computer experiments (DACE) algorithm to approximate the two-dimensional Branin or Goldstein-Price function (Examples 21.1 , 21.2, and 21.3)

Overfitting.m – Generate Figure 21.14

BBODynamic.m – Biogeography-based optimization for optimizing a time-varying function

BBODynamicMonte1.m – Monte Carlo simulation of BBODynamic.m (Example 21.4)

BBODynamicMonte2.m – Monte Carlo simulation of BBODynamic.m with various types of dynamic function changes and various dynamic adaptation strategies (Examples 21.5 and 21.6)

DynamicAckley.m – Dynamic Ackley benchmark function (Examples 21.4, 21.5, 21.6)

DynamicSphere.m – Dynamic Sphere benchmark function (not used in any examples)

GaussianNoise.m – Generate Figure 21.23

Resample.m – Generate Figure 21.24

 

Appendix A: Some Practical Advice

There is no software for this appendix

 

Appendix B: The No Free Lunch Theorem and Performance Testing

Irregular.m – Generate a random function (Figure B.1) or a deceptive function (Figure B.2)

IrregularTest.m – Generate a random function (Example 2.1) or a deceptive function (Example 2.2) and see how long it takes, on average, for hill descending, random search, and hill ascending algorithms to find the minimum

BoxPlotExample.m – Generate the box plot of Figure B.4 (requires the Statistics Toolbox)

TTest.m – T test example (Example 2.5)

FTest.m – F test example (Example 2.6)

 

Appendix C: Benchmark Optimization Functions

SpherePlot.m – Plot the two-dimensional sphere function (Figure C.1)

AckleyPlot.m – Plot the two-dimensional Ackley function (Figure C.2)

AckleyTestPlot.m – Plot the two-dimensional Ackley Test function (Figure C.3)

RosenbrockPlot.m – Plot the two-dimensional Rosenbrock function (Figure C.4)

FletcherPlot.m – Plot the two-dimensional Fletcher function (Figure C.5)

GriewankPlot.m – Plot the two-dimensional Griewank function (Figure C.6)

Penalty1Plot.m – Plot the two-dimensional Penalty 1 function (Figure C.7)

Penalty2Plot.m – Plot the two-dimensional Penalty 2 function (Figure C.8)

QuarticPlot.m – Plot the two-dimensional Quartic function (Figure C.9)

TenthPlot.m – Plot the two-dimensional Tenth Power function (Figure C.10)

RastriginPlot.m – Plot the two-dimensional Rastrigin function (Figure C.11)

Schwefel12Plot.m – Plot the two-dimensional Schwefel Double Sum function (Figure C.12)

Schwefel221Plot.m – Plot the two-dimensional Schwefel Max function (Figure C.13)

Schwefel222Plot.m – Plot the two-dimensional Schwefel Absolute function (Figure C.14)

Schwefel226Plot.m – Plot the two-dimensional Schwefel Sine function (Figure C.15)

StepPlot.m – Plot the two-dimensional Step function (Figure C.16)

AbsPlot.m – Plot the two-dimensional Absolute function (Figure C.17)

ShekelPlot.m – Plot the two-dimensional Shekel Foxhole function (Figure C.18)

MichalewiczPlot.m – Plot the two-dimensional Michalewicz function (Figure C.19)

SineEnvPlot.m – Plot the two-dimensional Sine Envelope function (Figure C.20)

EggholderPlot.m – Plot the two-dimensional Eggholder function (Figure C.21)

WeierstrassPlot.m – Plot the two-dimensional Weierstrass function (Figure C.22)

SphereShiftedPlot.m – Plot the shifted Sphere function (Figure C.26)

Schwefel221RotatedPlot.m – Plot the rotated Schwefel Max function (Figure C.28)

 


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Cleveland State University


Last Update November 1, 2013