A Computationally Efficient Multi-Objective Optimization Procedure Using Successive Function Landscape Models

(1)

Optimization Procedure Using Successive

Function Landscape Models

by

Pawan

Kumar

Singh

Nain

KanGAL Report Number 2005015

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY KANPUR

(2)

Optimization Procedure Using Successive

Function Landscape Models

A Thesis Submitted

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

by

Pawan Kumar Singh Nain

to the

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY KANPUR

(3)

It is certified that the work contained in the thesis entitled“A Computationally Effi-cient Multi-Objective Optimization Procedure Using Successive Function Landscape

Models”, by Pawan Kumar Singh Nain, has been carried out under my supervision

and that this work has not been submitted elsewhere for a degree.

July, 2005 Kalyanmoy Deb Professor

Department of Mechanical Engineering Indian Institute of Technology

Kanpur -208016, India

(4)

Name of Student : Pawan Kumar Singh Nain Roll No. : 9910566

Degree for which submitted : Ph.D.

Department : Mechanical Engineering, IIT Kanpur, India

Thesis title : A Computationally Efficient Multi-Objective Optimization Procedure Using Successive Function Landscape Models

Name of Thesis Supervisor : Professor Kalyanmoy Deb ([email protected])

Month and Year of Thesis Submission: July, 2005

One of the main hurdles faced by an optimization algorithm in solving real-world optimization problems is the need of a reasonably large computational time in finding an optimal or a near-optimal solution. In order to reduce the overall computational time, researchers in the area of search and optimization look for efficient algorithms which demand only a few function evaluations to arrive at a near-optimal solution. Although successes in this direction have been achieved by using new and unortho-dox techniques (such as evolutionary algorithms, tabu search, simulated annealing etc.) involving problem-specific operators, such techniques still demand a consider-able amount of simulation time, particularly in solving computationally expensive problems. In such problems, the main difficulty arises in the large computational time required in evaluating a solution. This is because such problems either involve many decision variables or a computationally involved evaluation procedure, such as the use of finite element procedure or a network flow computation. Although faster computers are helpful in reducing the problem magnitude, still evolutionary

(5)

algorithms take many iterations with a significant number of solution evaluations to reach to optimum or near-optimum solution of the optimization problem. So, the total time required to reach the optimum solution becomes significant and cannot be ignored even with the faster computation machines available today. However, there are some remedies to this problem like using partial evaluation of a solution, using parallel computation or using approximation of the optimization problem. In the present work, the use of approximate models in evolutionary algorithm, with a purpose of making them suitable for multi-objective optimization, is basically explored.

A brief literature review for the use of approximation models in evolutionary al-gorithms is presented in Chapter 2. The approximate modeling technique is mainly used in problems requiring time consuming function evaluations, in noisy function landscapes and in multi-modal function landscapes. This work mainly concerns about use of approximate models in problems involving time consuming function evaluations. Basically, three types of approximation models are commonly used in evolutionary algorithms, namely problem approximation, functional approximation and algorithm approximations. The problem approximation refers to the decompo-sition of a problem into number of subproblems. In this technique, initially only most important subproblems are evaluated. Later, when search for optimum picks up meaningful directions, less important subproblems are also included in the solu-tion procedure. In the funcsolu-tional approximasolu-tion technique, the funcsolu-tion landscape is modeled using an approximation technique. A few most common approximate model building techniques are artificial neural networks, polynomial models and Kriging models. In algorithm approximation, two important sub-classes of fitness inheritance and fitness imitation are included. These type of approximation is only possible in the case of evolutionary optimization algorithms. The modality of in-corporating approximate models in the evolutionary algorithm is also an important issue. The approximate models can be used for guiding EA operators, such as crossover, mutation and for fitness evaluation. The use of approximate models in

(6)

island nodes involving migration models are also possible. In other studies, the original fitness function values at a few points are used in conjunction with approxi-mate model to guide the search for the optimal solution. This method is referred as model management in evolutionary algorithms. Three kinds of model management, namely no evolution control, fixed evolution control and adaptive evolution control are possible. The objective of the present work is to develop and test a optimiza-tion procedure which is a generic procedure capable of incorporating approximate models. It should be capable for multi-objective optimization, particularly suitable for computationally expensive optimization problems. It shall be equally applicable for unconstrained and constrained optimization. In the present work, a fundamen-tal assumption that the approximate model building time (such as constructing a neural network for approximate function evaluation) is negligible compared to the exact function evaluation time (each evaluation of a solution), is made. Hence, the comparative time study for proposed optimization procedure is not made in the present work.

An optimization methodology is proposed in Chapter 3. This methodology is based on integration of approximate models in evolutionary algorithm. In this work, we suggest a fundamental algorithmic change to the usual optimization procedure which can be used either serially or parallelly. Most search and optimization al-gorithms begin their search from one or more random guess solutions. Thus, the main task of a search algorithm in the initial few iterations is to provide a direction towards the optimal region in the search space. To achieve such a task, it may not be necessary to use an exact (or a very fine-grained) model of the optimization prob-lem early on. An approximate model of the probprob-lem may be adequate to provide a reasonably good search direction. However, as the iterations progress, finer models can be used successively to converge closer to the true optimum of the actual prob-lem. Although this idea of using an approximate model in the beginning of a search algorithm and refining the model with iterations is not new, we suggest a generic procedure which can be used in any arbitrary problem. Thus, an evolutionary

(7)

al-gorithm based on successive approximate model of function landscape is proposed here. For testing the proposed methodology, a multi-objective optimizer, namely the non-dominated sorting genetic algorithm-II (NSGA-II) (developed at Kanpur GA Laboratory) is selected. The approximate model of function landscape is built using artificial neural networks (ANN). So, in this work NSGA-II and ANN are integrated together. Two possible methods of training ANN are batch mode train-ing and incremental mode traintrain-ing. Thus, two different models, based on traintrain-ing method used for ANN, are suggested here for proposed NSGA-II-ANN optimization procedure.

In Chapter 4, the two models of NSGA-II-ANN optimization procedure are tested on seven standard test problems from the EA literature. The various aspects of opti-mization, which are tested on the NSGA-II-ANN procedure are, unconstrained and constrained optimization problem, problems having convex and non-convex Pareto-optimal front, problems having continuous and disconnected Pareto-Pareto-optimal front, problems having a 2D Pareto-optimal front and a 3D Pareto-optimal surface. Both models of NSGA-II-ANN have successfully modeled the function landscape of the optimization test problems. A good distribution and convergence to Pareto-front is observed in all the problems in both the models. However, the batch model of NSGA-II-ANN performed slightly better than the incremental model. The savings, ranging from 0% to 62% in function evaluations, are observed. However, the batch and incremental models are found sensitive to the permissible modeling error in ANN for fitness landscape approximation.

Proposed NSGA-II-ANN procedure is also tested on a CAD curve fitting prob-lem using B-splines in Chapter 5. This is a bi-objective, 39 real-valued variable, unconstrained optimization problem. This problem is solved successfully by both models of the proposed procedure. Both the models, i.e. the batch model and incre-mental model NSGA-II-ANN simulations have demonstrated savings ranging from 0% to 32% in the number of exact function evaluations. The good convergence and spread of final non-dominated solutions is obtained in simulations of the proposed

(8)

procedure. The batch model is found to be sensitive to ANN parameters like number of hidden neurons and the learning rate. Even without tuning such parameters, the model is found to deliver similar median performance with respect to convergence as that of NSGA-II working with exact function evaluations, while maintaining better distribution of non-dominated solutions.

In Chapter 6, the NSGA-II-ANN procedure is tested on a mechanical component design problem, namely cantilever beam design using FEM analysis. This is a bi-objective problem involving 240 binary variables and a single constraint. Since, the batch model is found to outperform incremental model in previous problems, this problem is attempted only by the batch model. This problem is solved successfully by batch model. Savings up to 60% in number of exact function evaluations is observed. A particular batch model is found to deliver better performance with respect to convergence and spread than that of original NSGA-II working with exact function evaluations.

Finally, the ANN models are replaced by a Kriging metamodel in Chapter 7. The NSGA-II-KRIGING procedure is tested on CAD curve fitting problem which is solved earlier in Chapter 5. The problem is solved with a partial success. Based on simulation results, some recommendations which could improve the performance of NSGA-II-KRIGING procedure, are made. It is concluded that fitting global Kriging metamodel is a computationally expensive process.

Conclusions about the performance of proposed NSGA-II-ANN procedure are made in Chapter 8. Some parameter values which are found to be performing bet-ter for the proposed procedure are also recommended. The proposed procedure is found to be performing well on test problems as well as on two real-world optimiza-tion problems. It is found that NSGA-II-ANN approach is computaoptimiza-tionally better and more powerful for fitness landscape approximation than NSGA-II-KRIGING approach. Hence, the use of proposed optimization procedure for solving computa-tionally expensive real-world problems is recommended. At the end, some directions for future work are discussed.

(9)

First thanks goes to my supervisor Prof. Kalyanmoy Deb, who introduced me to the field of genetic algorithms and soft computing. He always helped me in my research work. The quality of the present work has substantially improved by incorporation of many valuable suggestions made by Prof. Deb. The quality of Prof. Deb, which I liked most, is his easy accessibility and ever readiness for discussions. Without his constant help, I could never have completed this work.

I also thank Prof. P. M. Dixit for acting as my thesis caretaker at various stages during the course of my thesis. I am also thankful to Prof. A. Mukherjee, Prof. B. Deo, Prof. N. Chakroborty (now at IIT Kharagpur), Prof. P. Chakroborty, Prof. B. Sahay, Prof. N. N. Kishore for introducing me to new knowledge and subjects during my course work at IIT Kanpur. I am also thankful to Dr. C. M. Fonseca (University of Algarve, Portugal) for his suggestion to use attainment surfaces in the current work and to provide the computer code for it. I am also thankful to Dr. M. Farina, Dr. K. Rasheed, Dr. K. Giannakoglou and Dr. Yaochu Jin for helping me in literature survey for the current problem. I am thankful to Dr. J. Knowles for presenting my workshop paper at PPSN workshop. I am also thankful to my friends at CAD lab, namely Mukul, Saravana, Shiva, Puneet and Rahul for their help during my course work.

My friends at KanGAL, namely Shamik, Ashi, Dhiraj, Sachin, Sameer, Am-rit, Mary, Tushar, Vipin, Raji, Gulshan, Maheshwer, Divya, Prateek, Naveen, Manikanth, Santosh, Himanshu, Deepak, Dhish, Dilip and Ganesh made this stay memorable. The list of KanGAL lab-mates and friends is long and incomplete. The

(10)

kind of cooperation and research environment which exists in GA lab is truly re-markable and unparallel. Special thanks is due to Shamik and Dhish for reading and correcting the first draft of this work. I am thankful to my doctor, Dr. Ashok Agarwal for his valuable help.

I am also thankful to Mrs. Debjani Deb for all her affection and care which she takes for all of us at KanGAL. She has made stay at KanGAL most memorable. I will miss the KanGAL parties, outings and movie shows. I am thankful for the love and blessings which I received from Mrs. Ruby Sarkar. I am also thankful to my family, back in Delhi, for their patience and encouragement all along. I am thankful to my brothers Dr. Suresh Pal, Mr. Virendra Malik, Mr Sanjeev Singh and Mr. Lakshmi Raj Singh for their constant support. I am thankful to my sisters Dr. Lata, Mrs. Harsh Malik, Mrs. Lakshmi Singh and Mrs. Geeta Nain for all love and good wishes. Thanks is due to my wife Sonika for her constant support and patience. Lastly, thanks to the lady who taught me the very first lesson of my life and still is very keen that I shall finish, my mother Mrs. Vinod Nain. May be now on, we will share more time together.

Last but not the least, I would like to express my sincere gratitude to all of them who directly and indirectly helped me for successful completion of my thesis work.

(11)

(12)

Certificate ii

Synopsis iii

Acknowledgements viii

Dedication x

Contents xi

List of Figures xvi

List of Tables xx

Nomenclature xxi

1 Introduction 1

1.1 Prologue . . . 1

1.2 A Practical View on Evolutionary Algorithms . . . 4

1.2.1 The Flip Side . . . 5

1.3 Research Direction . . . 6 1.4 Work Overview . . . 7 1.5 Sum Up . . . 8 2 Literature Review 9 2.1 Introduction . . . 9 xi

(13)

2.2 Motivations . . . 9

2.3 Kinds of Approximations . . . 11

2.3.1 Problem Approximation . . . 11

2.3.2 Functional Approximation . . . 11

2.3.3 Evolutionary Approximation . . . 15

2.4 Approximate Model Realization Mechanisms . . . 16

2.4.1 Approximate Model for Guiding Crossover and Mutation . . . 16

2.4.2 Use of Approximate Model in Fitness Evaluation . . . 17

2.4.3 Approximate Model in Migration . . . 18

2.5 Approximate Model Management . . . 19

2.5.1 No Evolution Control . . . 19

2.5.2 Fixed Evolution Control . . . 20

2.5.3 Adaptive Evolution Control . . . 21

2.6 Objectives and Scope of Present Work . . . 21

2.7 Sum Up . . . 22

3 Proposed Methodology 23 3.1 Introduction . . . 23

3.2 Proposed Method . . . 23

3.2.1 A Sketch of Proposed Procedure . . . 27

3.3 Selection of Evolutionary Algorithm . . . 29

3.4 Selection of Approximate Modeling Technique . . . 30

3.5 Integration of NSGA-II and ANN . . . 31

3.6 Different Models of Proposed Procedure . . . 33

3.7 Sum Up . . . 33

4 Case Study-I: Test Problems 34 4.1 Introduction . . . 34

4.2 Test Problem BNH . . . 35

(14)

4.4 Test Problem ZDT2 . . . 47 4.5 Test Problem ZDT4 . . . 52 4.6 Test Problem ZDT6 . . . 57 4.7 Test Problem DTLZ2 . . . 61 4.8 Test Problem DTLZ8 . . . 67 4.9 Sum Up . . . 73

5 Case Study-II: Curve Fitting Problem 76 5.1 Introduction . . . 76

5.2 Curve Fitting Problem . . . 77

5.2.1 B-splines . . . 78

5.2.2 Problem Definition . . . 81

5.2.3 NSGA-II-ANN Parameters and Training Models . . . 84

5.3 Simulation Results . . . 85

5.3.1 Progressive Modeling Analysis of B-10-3 Model . . . 90

5.3.2 Effect of Permissible Normalized RMS Error . . . 93

5.3.3 Convergence and Spread Metric Results . . . 96

5.4 Attainment Surface Analysis ofB-10-3 Model . . . 101

5.5 Sum Up . . . 106

6 Case Study-III: Beam Design Problem 108 6.1 Introduction . . . 108

6.2 Shape Optimization . . . 109

6.2.1 Problem Definition . . . 110

6.2.2 NSGA-II-ANN Parameters . . . 114

6.3.1 Representative Solutions and Local Hill Climbing for B-20-3 Model . . . 119

6.3.2 Convergence and Spread Metric Results . . . 122

(15)

6.5 Sum Up . . . 132

7 Curve Fitting Problem Via Kriging Metamodel 135 7.1 Introduction . . . 135

7.2 Kriging: A Brief Introduction . . . 136

7.3.1 Some Implementation Issues . . . 141

7.4 Sum Up . . . 143

8 Conclusions and Future work 145 8.1 Conclusions . . . 146

8.1.1 Kriging Metamodel . . . 149

8.2 Future Work . . . 150

References 151 A Multi-Objective Optimization: NSGA-II 157 A.1 Introduction . . . 157

A.2 Definitions . . . 157

A.2.1 Concept of Pareto-optimal Solutions . . . 157

A.2.2 Domination Concept for Unconstrained Problem . . . 158

A.2.3 Domination Concept for Constrained Problem . . . 158

A.3 Multi-Objective Optimization Algorithm . . . 158

A.3.1 NSGA-II . . . 159

A.3.2 Crowded Tournament Selection Operator . . . 161

A.3.3 Elitism . . . 162

A.4 Sum Up . . . 163

B ANN: Error Back-propagation Algorithm 164 B.1 Introduction . . . 164

B.2 Error Back-propagation Training Algorithm . . . 164

(16)

(17)

3.1 Progressive approximate modeling. . . 25

3.2 A line diagram of the proposed technique. . . 28

3.3 A schematic diagram of ANN modeling . . . 30

3.4 A flow-chart of the proposed GA-ANN technique . . . 31

4.1 Batch model simulation results for BNH problem . . . 37

4.2 Incremental model simulation results for BNH problem . . . 37

4.3 Normalized spread metric plot for BNH problem . . . 40

4.4 Batch model simulation results for TNK problem . . . 42

4.5 Incremental model simulation results for TNK problem . . . 43

4.6 Normalized spread metric for top part of TNK Pareto-front . . . 44

4.7 Normalized spread metric for middle part of TNK Pareto-front . . . . 45

4.8 Normalized spread metric for bottom part of TNK Pareto-front . . . 45

4.9 Batch model simulation results for ZDT2 problem . . . 48

4.10 Incremental model simulation results for ZDT2 problem . . . 49

4.11 Normalized spread metric plot for ZDT2 problem . . . 51

4.18 NSGA-II exact model simulation results for DTLZ2 problem . . . 63

(18)

4.19 Batch model simulation results for DTLZ2 problem . . . 63

4.20 Incremental model simulation results for DTLZ2 problem . . . 64

4.21 Normalized sparsity measure plot for DTLZ2 problem . . . 66

4.22 NSGA-II exact model simulation results for DTLZ8 problem . . . 68

4.23 Batch model B-10-2 simulation results for DTLZ8 problem . . . 69

4.24 Batch model B-10-3 simulation results for DTLZ8 problem . . . 70

4.25 Incremental model I-10-2 simulation results for DTLZ8 problem . . . 71

4.26 Incremental model I-10-3 simulation results for DTLZ8 problem . . . 71

5.1 The B-spline curve fitting problem . . . 81

5.2 Batch model simulation results with two generation (2N) database for CAD problem . . . 85

5.3 Batch model simulation results with three generation (3N) database for CAD problem . . . 86

5.4 Incremental model simulation results with two generation (2N) database for CAD problem . . . 86

5.5 Incremental model simulation results with three generation (3N) database for CAD problem . . . 87

5.6 Best of both incremental and batch models simulation results for CAD problem . . . 88

5.7 Comparison of the overall best simulation result with exact solution at 1,100 generation for CAD problem . . . 89

5.8 Two extreme solutions and knee solution for CAD curve fitting problem 89 5.9 Modeling with 25% function evaluations for CAD problem . . . 90

5.10 Modeling with 50% function evaluations for CAD problem . . . 91

5.13 Different modeling error simulations for CAD problem . . . 93

(19)

5.15 Normalized convergence metric for different simulations on CAD prob-lem . . . 98 5.16 Normalized spread metric for different simulations on CAD problem . 100 5.17 0% Attainment surface plot for exact and approximate runs for CAD

problem . . . 103 5.18 50% Attainment surface plot for exact and approximate runs for CAD

problem . . . 104 5.19 100% Attainment surface plot for exact and approximate runs for

CAD problem . . . 104 5.20 Attainment surface plot for exact and approximate runs for CAD

problem . . . 105

6.1 A cantilever plate . . . 111 6.2 Different cases of smoothing . . . 112 6.3 Batch model results with two-generation (2N) database for FEM

problem . . . 115 6.4 Batch model results with three-generation (3N) database for FEM

problem . . . 115 6.5 Batch model results with four-generation (4N) database for FEM

problem . . . 116 6.6 Batch model results with five-generation (5N) database for FEM

problem . . . 116 6.7 Best of different batch model simulations for FEM problem . . . 117 6.8 Comparison of best batch simulation with exact NSGA-II runs for

FEM problem . . . 118 6.9 Representative Solutions for B-20-3 Model for FEM problem . . . 119 6.10 Nine trade-off shapes for the design of the cantilever plate of B-20-3

model for FEM problem . . . 120 6.11 Representative solutions of B-20-3 model after local search for FEM

(20)

6.12 Nine trade-off shapes for the design of the cantilever plate after local

search for FEM problem . . . 121

6.13 Convergence metric results for FEM problem . . . 125

6.14 Spread metric results for FEM problem . . . 127

6.15 0% Attainment surface plot for exact and approximate runs for FEM problem . . . 130

6.18 Attainment surface plot for exact and approximate runs for FEM problem . . . 131

7.1 Kriging model simulation results with two (2N) and three (3N) gen-eration database for CAD problem . . . 140

A.1 Flow-chart of NSGA-II . . . 160

A.2 Crowding distance calculation . . . 161

A.3 Formation of children from parent population keeping elitism . . . 162

(21)

4.1 Spread metric results obtained on BNH problem . . . 39

4.2 Spread metric results obtained on TNK problem . . . 44

4.3 Spread metric results obtained on ZDT2 problem . . . 50

4.6 Sparsity measure results obtained on DTLZ2 problem . . . 65

4.7 Summary of savings and best performing model for test problems . . 74

5.1 Best results obtained for CAD problem . . . 87

5.2 Convergence metric for different modeling error in CAD problem . . . 94

5.3 Convergence metric results for CAD curve fitting problem . . . 97

5.4 Spread metric results for CAD curve fitting problem . . . 99

6.1 Best results obtained for FEM problem . . . 117

6.2 Exact function evaluations for the cantilever plate . . . 122

6.3 Convergence metric results for FEM problem . . . 123

6.4 Spread metric results for FEM problem . . . 127

(22)

B Batch mode of training artificial neural network

I Incremental mode of training artificial neural network

K Kriging

N Population size

n Database size parameter (in terms of number of generations)

N0 _{Database size (in terms of number of individuals)}

Q Model updating frequency parameter (in terms of number of generations)

R Covariance matrix

Abbreviations

ANN Artificial neural networks

CAD Computer aided design

CFD Computational fluid dynamics

CPU Central processing unit

CST Constant strain triangle

EA Evolutionary algorithm

EMO Evolutionary multi-objective optimization

(23)

FEM Finite element method

FFT Fast Fourier transform

GA Genetic algorithm

HGA Hierarchical genetic algorithm

iiGA Injection island genetic algorithm

MLE Maximum likelihood estimate

MOEA Multi-objective evolutionary algorithm

MOGA Multi-objective genetic algorithm

MSSE Mean of sum of squared errors

NSGA-II Non-dominated sorting genetic algorithm - II

RBFN Radial basis function networks

RGA Real-coded genetic algorithm

(24)

Introduction

1.1 Prologue

Right from the ancient times, humans are discovering new tools to assist them in their day to day activities. The first revolutionary discovery is the invention of the wheel. Since then, in their endeavor to make quality of human life better new discoveries are continually made. The inventions gave birth to a specialized field in science, namely engineering design. Engineering design is basically an art to make something which assists humans in their daily life and improves the quality of human life. However, it is in the basic nature of humans to search for perfection and continually improving what they have designed. While improving quality of life, the dependence of humans on resources like natural and human is continuously increasing. As human is becoming more dependent on resources, the issue of mak-ing there efficient use became important, and then idea of optimization came in. However, it in no way limits the use of optimization to only resource management. It is now playing an important role in various kind of applications. But here we will limit ourself to engineering applications. The optimization plays a significant role in manufacturing and engineering design. At present, a large community of mathematicians and engineers is devoted to research in the field of optimization.

Though there exist a number of algorithms which are popular in optimization,

(25)

the optimization algorithms based on the search techniques can broadly be classified in two categories:

• Point-by-point search based optimization algorithms,

• Population based optimization algorithms.

The classical optimization algorithms generally work on point to point search meth-ods i.e. they start with some initial guess of the solution of the optimization problem and then move to next solution in the search space based on some mathematical principles. How these algorithms move to next point in the search space is governed by the set of specific rules which are characteristic of that particular algorithm em-ployed to solve the problem. This process of moving in search space from one point to another point completes one iteration of the optimization algorithm. This new point in the search space is hopefully a better point in terms of the objective(s) of optimization. The optimization algorithm is permitted to make many such itera-tions till it reaches better regions in the search space where it can not improve any further in terms of the objective(s) of optimization or the improvement becomes too small to be of any practical significance or a prespecified number of iterations are over. These algorithms in general have a mathematical proof which guarantees them to find a local optimum of the problem. But these mathematical proof comes at the cost of some significant assumptions which restrict their use to only a limited class of problems. Convexity of the search space and objective is one such major assumption. Another drawback of most classical optimization algorithms is choice of initial solution guess which is to be supplied by the user. The final outcome of optimization is sometimes dependent on the initial guess, particularly so in the case of problems with complex search space. In such problems, some classical optimiza-tion algorithms try to approximately model the original optimizaoptimiza-tion problem by some simple quadratic functions and then direct their search to find the optimum of the approximated function. Hence such classical algorithms may also find a so-lution which may not be true optimum of the original problem. The essence of this

(26)

discussion is that the idea of using approximate model is also important in classical optimization algorithms.

The other class of optimization algorithms i.e. population based optimization algorithms work with a number of solutions simultaneously. One of the most popu-lar heuristic based approach is evolutionary algorithms (EA). The basic manner in which evolutionary algorithms differ from classical algorithms is the way in which they make search. The evolutionary algorithms start with number of random solu-tions in the search space. This random initial set of solusolu-tions is first evaluated for the objective(s) of the optimization problem. Then using some stochastic operations which are based on concepts derived from nature are performed on the initial set of solutions. This results in a new set of solutions for the optimization problem. The new set is also evaluated for objective(s) of the optimization. This completes one iteration of the evolutionary algorithm. Then, once again, the stochastic operations are performed on this new set of solutions. Many such iterations are performed in order to reach optimal or near-optimal solutions. The basic difference from classical algorithms is that in the case of evolutionary algorithms, the collective information about the search space based on a set of solutions direct the search for the optimum. The fact that the collective information from a set of solutions is used to guide the search, makes the EA search difficult to model mathematically. There are some theoretical proofs based on Markov chain analysis which gives convergence proofs under some restrictive assumptions (Eiben et al., 1990). There are many positive aspects about the use of evolutionary algorithms (no gradient information or con-vexity requirement) but the basic fact that here we are simultaneously working with a number of solutions, and many such iterations are required to reach to the final set of useful solutions remains a point of concern. In the current work, this issue is primarily addressed.

(27)

1.2 A Practical View on Evolutionary Algorithms

The evolutionary algorithms are widely used to solve practical optimization prob-lems. It is not only the manner of conducting search which makes them outstanding, there are several other positive aspects. Here we list a few of them.

• No requirement for gradient information.

• Equal applicability to convex or non-convex search space.

• Can handle non-linearities.

• Can handle real, binary, discrete or mixed variables.

• Applicability to constrained optimization.

• Single and multi-objective optimization.

• Can work with noise.

• Simultaneous presence of optimal solutions.

• Multi-modal optimization.

In some of the real-world problem, the gradient of the objective is either difficult to compute or does not exist. In such cases, the gradient based optimization algo-rithms fail to perform. The evolutionary algoalgo-rithms do not use gradient information in directing search for optimal solutions. Hence they are useful to a wider class of optimization problems. A wide range of traditional optimization algorithms perform better for convex problems, while in case of evolutionary algorithms no such require-ment is present. In traditional optimization algorithms, there is separate class of algorithms which can handle non-linear optimization problems, but in evolution-ary algorithms no such restriction is present. Non-linearities in objective as well as constraints can easily be handled by evolutionary algorithms. The constraints are easily handled by a penalty-less constraint handling approach in evolutionary algorithms (Deb, 2000). While in classical algorithms an additional penalty factor

(28)

have to be introduced to handle such situations. Both single and multi-objective optimization is possible with evolutionary algorithms. Evolutionary algorithms can easily handle the presence of noise in functions to be optimized without affecting their performance much. Such algorithms can find multiple solutions in a single run, thereby providing flexibility to the decision maker. It is a useful aspect of these algorithms. Furthermore, these algorithms can be made to discover multiple local or global optima of the optimization problem in a single run. It is very use-ful property for multi-modal optimization problems and it can not be achieved by traditional optimization algorithms. Hence, the evolutionary algorithms are very useful for real-world optimization problems.

1.2.1 The Flip Side

Despite of the facts highlighted above, which is the outcome of many years of test-ing evolutionary algorithms on various test problems and performtest-ing applications to many real life optimization problems, there still remains clinching problem associ-ated with them. Often, researchers do some modifications on the basic evolutionary algorithms which make them suitable for application to practical problems. The ob-jective of these modifications is to make search process of evolutionary algorithms faster i.e. to speed-up the convergence to the optimal or near-optimal solutions. These modifications are problem specific i.e. they vary form problem to problem. This is in contrast to the fact which is stated above. The normal evolutionary algo-rithms will still be able to solve the problem for which the modifications are made but are slower to locate the optimum solutions. Hence these modifications just en-hance the speed with which the optimum solutions to the optimization problem is achieved. Though very recently, few generic evolutionary algorithms for optimiza-tion have come which have shown the potential to solve major class of problems (Deb and Tiwari, 2005). As we all know that the computation power and speed of processors is increasing rapidly, the evolutionary optimization take a few minutes to solve most of the problems. Still for some real-world problems, the evaluation

(29)

proce-dure of a single solution takes a few hours to several days. In such cases evolutionary algorithms which may have the potential to solve to optimality face difficulties in terms of computational time. In real-world optimization problems the search space is quite complex. The presence of constraints further complicates the problem. In some cases the feasible search space where the optimum solution is supposed to lie can be as small as 1% (or less) of the total search space. In such cases, the robust search capability of evolutionary algorithms is highly desirable. On the other hand, if the solution evaluation is time consuming, it becomes a bottle neck in using the evolutionary algorithms. Evolutionary algorithms take many iterations with a sig-nificant number of solutions to reach the optimum or near-optimal solution of the optimization problem. So the total time required to reach the optimum solution becomes significant and remains as a problem even with the faster computation machines. Though there are some remedies to this problem like:

• Use of partial evaluation of a solution.

• Use of parallel computation.

• Use of approximation of the optimization problem.

Each of the possible remedies pointed above has some limitations and cannot have general applicability. However, the detailed discussion will be done later. The final point is that lack of the convergence proof for evolutionary algorithms makes them less acceptable to mathematically-oriented users. Though it is important to point that still evolutionary algorithms have successfully solved a large class of optimization problems which were either not tractable with traditional algorithms or were difficult to solve.

1.3 Research Direction

From the discussion above it can be concluded that the evolutionary algorithms have many positive points for their practical use in solving optimization problems. There

(30)

are three main directions in which there is a need for further exploration:

• Theoretical convergence proof.

• Development of generic evolutionary algorithms.

• Reduction of computational load for computationally expensive problems. The need of working out a theory which could explain the working of evolution-ary algorithms is self evident. Little research is reported in this direction. Few research groups are actively trying to develop the new evolutionary optimization algorithms (Deb and Tiwari, 2005; Knowles, 2005). Mostly, the performance of the new algorithms is based on standard test problems. They either solve the test problems in lesser number of function evaluations or demonstrate solving the dif-ficult test problems which were previously unsolvable. Such algorithms are then tested on real-world optimization problems. However, while applying these generic and powerful optimizers on real-world optimization problems, users encounter com-putational time limitation in evolutionary algorithms. In real-world optimization problems, even a single function evaluation can take CPU time ranging from few hours to days. Hence, such cases form an important area for research. The main focus for research in such cases is to somehow reduce the computational load for computationally expensive problems. The scheme for reducing the computational load should be generic in nature, that it shall be applicable to any popular evolu-tionary algorithm based optimization procedure. The present work focuses on the last point, that it addresses the applicability of evolutionary algorithms for com-putationally expensive problems. However, a detailed discussion on the proposed procedure is left till Chapter 3 of this work.

1.4 Work Overview

In the next chapter, a brief presentation of available literature for handling compu-tationally expensive problem in evolutionary algorithms is made. In chapter 3, the

(31)

generic procedure for optimization with evolutionary algorithms using approximate models is proposed. The proposed procedure is validated on standard test problems in chapter 4. The proposed procedure is employed to solve the curve fitting problem using B-Splines in chapter 5. Chapter 6 presents solution for a finite element analy-sis based shape optimization problem. In all the problems, artificial neural networks are used to generate the approximate model. In chapter 7, the curve fitting prob-lem is revisited and an attempt is made for replacing artificial neural networks by Kriging metamodel for approximate model generation. Lastly some conclusions and future work is discussed in chapter 8.

1.5 Sum Up

In this chapter, first a brief introduction for need of optimization in human life in gen-eral, and for engineering design in particular, is discussed. For solving optimization problems in engineering design, engineers use optimization algorithms. Such opti-mization algorithms can be classified in two classes, namely point-by-point search methods and population based optimization algorithms. These two types of opti-mization algorithms are compared for their relative merits and demerits highlighting the basic differences in their working procedure. The evolutionary algorithms are new optimization algorithms which are currently gaining popularity. So, positive aspects which make evolutionary algorithms popular are discussed. Next, the limi-tations of the evolutionary algorithms are discussed. Based on this discussion, the possible research directions in evolutionary optimization algorithms are identified. One such research direction, i.e. to reduce computational load in evolutionary algo-rithms for computationally expensive problems is selected for the current work. At the end, a brief overview of the present work is discussed.

(32)

Literature Review

2.1 Introduction

As pointed out in the previous chapter, evolutionary algorithms outperform conven-tional optimization algorithms in a certain class of problems which are discontinuous, non-differentiable, multi-modal, noisy and not well defined. Evolutionary algorithms are also quite suitable for multi-objective problems. Here, we will explore the avail-able literature which deals with the function/fitness approximation in evolutionary algorithms. The use of approximate models has certainly caught the attention of various research groups, as in science and engineering applications a large number of fitness evaluations are required for solving them. A good survey of fitness ap-proximation in evolutionary computation is provided by Jin and Sendhoff, (2002) and Jin, (2005). The structure of the rest of the sections in literature review part follows closely the classifications suggested by Jin, (2005).

2.2 Motivations

There are mainly three basic reasons for the use of function/fitness landscape ap-proximations. The reasons are:

1. Time consuming function evaluation.

(33)

2. Noisy function landscape.

3. Multi-modal function landscape.

If the fitness evaluation is time consuming, the idea of using an approximate model for the function evaluation can help substantially. The structural design optimiza-tion is one such example. In aerodynamic design optimizaoptimiza-tion, computaoptimiza-tional fluid dynamics (CFD) simulations are used to evaluate the performance of a given struc-ture. A complete three-dimensional CFD simulation is very costly and may take several hours to complete one such function evaluation. Therefore, approximate models are an appropriate solution for their use in structural optimization. Sim-ilarly in finite element simulations, a single function evaluation is very costly and hence is an ideal case for the use of approximate models. If the function landscape is noisy, then two possible methods exist in literature. However, it is worth men-tioning that evolutionary algorithms can work in the noisy environment as well. The first method needs repeated sampling of the fitness and then use its average. This method needs additional function evaluations. The other method makes use of neighborhood information. This method calculates the fitness of any individual by averaging the value of this individual and other individuals in the neighborhood. To reduce the computational cost, the individuals which participate in averaging are chosen from current and previous generations. The better idea is to construct an ap-proximate model from the history of obtained data and then to evaluate neighbor’s fitness from this model. In case of multi-modal functions, an approximate global model can be constructed which may represent the function landscape. The basic assumption is made that the approximate model smoothen out the local optima of the original multi-modal function without changing the global optima and its loca-tion. Two possible approaches exist in literature for such cases. The first makes use of Gaussian kernel for coarse-to-fine smoothing of original multi-modal function. In the other approach, global polynomial models are used for smoothing multi-modal functions.

(34)

2.3 Kinds of Approximations

There are three types of approximation levels possible in evolutionary algorithms.

1. problem approximation

2. functional approximation

3. evolutionary approximation- fitness inheritance and fitness imitation

In the following subsections, a brief description about these different types of ap-proximation is presented.

2.3.1 Problem Approximation

In problem approximation, the original problem is restated in other approximated problem which is easier to solve. Such type of approximation is very common in the computational fluid dynamics problem where by making certain assumptions, the solution procedure can be simplified to great extent (Anderson, 1995). In problem approximation, the original problem is replaced by the one which is approximately the same to the original statement of the problem but is easier to solve. In some cases, the true objective function can only be established by experimentation. How-ever, with some levels of approximation, computerized simulation can also predict the results which can then be used for fitness assignment. There is also possibility of using different levels of approximation like three-dimensional model analysis or two-dimensional model analysis or quasi-three-dimensional solvers, which are com-putationally more efficient.

2.3.2 Functional Approximation

In functional approximation, instead of using experimental data, an alternate and explicit expression is constructed for the objective function or the fitness function. An explicit mathematical model is constructed, the input to the model are decision

(35)

variables and output is the corresponding fitness value. Recently, functional land-scape approximation is reported in limited number of exact function evaluations using ParEGO algorithm (Knowles, 2005; Knowles and Hughes, 2005). Local func-tion approximafunc-tion for costly funcfunc-tions in EA is reported by Regis and Shoemaker, (2004). In evolutionary optimization, the following methods are employed to built approximate models:

1. Polynomial model.

2. Kriging model.

3. Artificial neural networks, radial basis function networks.

Polynomial Models

The polynomial models are the most commonly used approximate models. In poly-nomial models, for a given data set (xi, yi), i = 1, . . . , n, the regression coefficients

βi can be determined in a quadratic model of the form

ˆ

y(x) =β0 +β1x+β2x2,

where x is the input variable, ˆy(x) is the predicted value at that input value. This model is fit by minimizing the mean of sum of squared errors (MSSE) of the predicted output value at xi. M SSE = 1 n n X i=1 (yi−yˆ(xi))2.

With the optimal values of the coefficients chosen, a polynomial equation relating the input and output is built. Such type of approximate models can be built for any degree of polynomial equation by simply changing the number of regression coefficients included in the model. This polynomial can also be readily extended to multivariate expressions. The common name for this approach is response surface methodology (Myers and Montgomery, 1995). In advanced applications of polyno-mial models, only statistically significant terms of the polynopolyno-mial are used to build

(36)

the model.

Kriging Model

In the Kriging metamodels, a fundamental assumption is thaty(x) = ˆy(x) +εwhere

ε assumes an independent, identically distributed normal distribution ε_∼N(0, σ2). The main idea behind the Kriging metamodels is that the error in the predicted values are not independent, instead they follow a systematic function of x. The Kriging metamodel, ˆy(x) in multi-dimensions consists of two parts:

1. a polynomial f(x), and

2. a functional departure Z(x) from that polynomial.

Hence, the Kriging metamodel is expressed as ˆy(x) = f(x) + Z(x). The details of Kriging metamodel are discussed in Chapter 7. El-Beltagy et al., (1999) have suggested the use of Kriging metamodels to reduce the computational burden on the evolutionary algorithms. This paper points toward the use of local metamodels instead of global metamodels, as building a global metamodel will be a computa-tionally expensive process. The local metamodels is built by using a limited number of neighbours of the solution to be evaluated from the database. While in global metamodels, all solutions in the database are used to construct an approximate model. A reconstruction algorithm is proposed by Ratle, (1998) which uses a Krig-ing metamodel for fitness landscape approximation. The algorithm shows an overall reduction in the number of fitness calls. The algorithm is tested successfully for a two-dimensional problem and for a difficult twenty-dimensional multi-modal prob-lem. Emmerich et al., (2002) have proposed a metamodel assisted evolution strat-egy for reducing the computational cost for computationally expensive optimization problems. Local Kriging metamodel is built which uses fixed number of nearest neighbors. The method is tested on Kean’s function and airfoil shape optimization problem. The results are quite encouraging for single objective optimization. In other work by Emmerich and Naujoks, (2004), the Kriging metamodels are used

(37)

in multi-objective optimization. The local metamodels are used to decide the po-tential of a new population member, i.e. to decide whether it should be evaluated precisely or rejected. Three different rejection principles are tested over three differ-ent optimizers. Recdiffer-ently, Willmes et al., (2003) have compared Kriging and neural networks for fitness approximations in evolutionary algorithms. They have tested these procedure on three test functions, namely the Ackley function, the Rosenbrok function and Keane function. The online learning mode showed significantly better results than offline learning mode.

Artificial Neural Networks and Radial Basis Function Networks

Most modeling methods assume some underlying functional relationship between input and output variables while fitting the approximate model. Artificial neural networks (ANN) avoid this assumption and fit the data in a free form manner. For ANN, only the set of input and corresponding output values are required. The ANN emulate functioning of the human brain. The input values are passed through layers of interconnected neurons which alter the data using sigmoidal transforms. The multi-layer neural network is trained on a data set by using some standard training algorithms which develop a model within some permissible error limit. The use of multiple neurons in different layers can model highly nonlinear functions. A detailed discussion about ANN is given in Appendix B and in Chapter 3 of this work. Some researchers have also used radial basis function networks (RBFN) for developing approximate model (Gutmann, 2001). The theory of RBFN can be tracked back to interpolation problems. In generalized RBFN, the number of hidden nodes is smaller than the number of samples. Here, the hidden unit output is obtained by calculating the closeness of the input and the centers of basis functions. The centers of basis functions needs to be determined for this purpose. Farina, (2002) has also used radial basis neural network for objective function approximation. The algorithm described elsewhere (Farina, 2002) has been successfully tested on test problem ZDT3 which has a typical discrete Pareto-optimal front. The same research group has proposed

(38)

number of variations in basic optimization algorithms with a objective of reducing the number of function evaluation calls (Farina, 2001; Farina and Sykulski, 2001; Barba et al., 2001; Rashid et al., 2001; Farina et al., 2002). Giotis and Giannakoglou have also proposed the use of ANN and RBFN (Giannakoglou, 2002) to reduce computational cost of GA by a surrogate model based inexact pre-evaluations. The other important work in this area is reported by Poloni et al., (2000). They have developed a methodology which uses a multi-objective genetic algorithm (MOGA) for exploring the design space to find a local Pareto-optimal set of solutions. Next, they train a neural network with the database of solutions obtained thus far to get the global approximation of the objective function. Finally, by defining the proper weights to combine the objectives, a single objective optimizer using the conjugate gradient method is run on the globally approximated objective function obtained earlier. They tested this methodology for the design of sailing yacht fin keel problem, coupling their optimization code to a three-dimensional Navier-Stokes equation solver.

2.3.3 Evolutionary Approximation

Evolutionary approximation is special kind of approximation which is applicable to evolutionary algorithms. The two sub-classes are fitness inheritance and fitness imitation. In fitness inheritance, the fitness evaluations can be spared by estimating the fitness value of the offspring individual from the fitness value of their parents. The attempt to reduce number of function evaluations using fitness inheritance (Chen et al., 2002; Sastry et al., 2001) is also reported. Sastry et al., (2001) have used inheritance combined with population sizing models and have reported a saving of 20% in function evaluations. In case of fixed population size, authors have reported a saving of 70% by employing a simple inheritance technique. Chen et al., (2002) have extended the fitness inheritance concept to multi-objective optimization. The authors have reported a 40% saving in terms of function evaluations for the case of fitness inheritance without fitness sharing, while in the case of fitness inheritance

(39)

with fitness sharing, a saving of 25% is claimed. Goldberg’s research group has also worked on efficiency enhancement of optimization algorithms (Pelikan and Sastry, 2004; Sastry and Goldberg, 2002a; Sastry and Goldberg, 2002b).

In fitness imitation, the individuals are clustered into several groups. One rep-resentative from each cluster is chosen and is evaluated using the fitness function. The fitness value for rest of the individuals in the same cluster will be estimated from the representative individual based on the distance measure.

2.4 Approximate Model Realization Mechanisms

Approximation models are embedded in almost every aspect of EAs including mi-gration, initialization, recombination, mutation and fitness evaluations. Basically, by means of using approximate model in evolutionary algorithms, one incorporates knowledge into the optimization process. The use of approximate models based on their incorporation mechanism can be divided in three broad categories. These categories are:

1. Use of approximate model in guiding crossover and mutation.

2. Use of approximate model in fitness evaluation.

3. Use of approximate model in migration models.

In the next subsections, these different methods of incorporating approximate mod-els in evolutionary algorithms is explored in detail.

2.4.1 Approximate Model for Guiding Crossover and

Mu-tation

The use of approximate fitness models for population initialization and for guiding crossover and mutation can significantly improve the performance of evolutionary algorithms (Rasheed, 2000; Rasheed and Hirsh, 2000; Rasheed et al., 2002; Rasheed

(40)

et al., 2005). In normal evolutionary algorithms, the population initializations and other genetic operations like crossover and mutation is done stochastically. The approximate fitness models can guide the genetic operations in evolutionary algo-rithms. Hence, the use of some prior information from approximate fitness models can considerably improve the performance of optimization algorithms. A significant speedup can be obtained by the use of this strategy.

2.4.2 Use of Approximate Model in Fitness Evaluation

It is the most direct place where the use of approximate model can be helpful in re-ducing the number of function evaluations in evolutionary algorithms, though most of such applications are restricted to single objective optimization problems. Most applications differ in the type of technique used for approximate model construc-tion. The most commonly used approximation techniques are polynomial models, Kriging models, artificial neural networks and radial basis function networks, as discussed earlier. Approximate models can also be subjected to simultaneous esti-mation of error bounds in the model predictions. It is found to be useful in the case of multi-modal fitness function optimization. The use of approximate models in fit-ness evaluations is common in the literature of evolutionary algorithms. Nair et al., (1998) have combined approximation concept with genetic algorithms for structural optimization applications. Investigators have tried to reduce the number of exact function evaluations while ensuring to converge to the optima of the original prob-lem. They have employed a dynamic optimization technique, wherein the fitness function changes over successive generations. They have controlled the generational delay before the approximate model is updated along with an adaptive selection operator. This technique is applied to solve a 10-bar truss design problem. They have used a strategy by which the fitness function is changed during the run but granularity of the optimization model is not changed.

(41)

2.4.3 Approximate Model in Migration

The island model based architecture is commonly used in parallel evolutionary algo-rithms. In such cases, different subpopulations evolve separately for optimization of a function. In this case, it is possible that each subpopulation evolves making use of a different level of approximation. Hence, different approximate models are used by different subpopulations. Intermittently, these sub-populations exchange individu-als, which result in speeding the convergence of the optimization algorithms. In this case, an individual from a low resolution approximate model based subpopulation can go to a higher level resolution based subpopulation. Also, the reverse migration is possible. The elegance of these techniques lie in the fact that fast but approximate models are successfully mixed with slow and accurate models. One such applica-tion is the optimal design of elastic flywheels using the injecapplica-tion island GA (iiGA) suggested by Eby et al., (1998). It uses a finite element code to assist the iiGA to evaluate the solutions to find the specific energy density of flywheels. The iiGA seek solutions simultaneously at different level of refinements of the problem represen-tation, thereby using different approximations of the problem. Solutions are first sought at low levels of the refinement for a quick exploration of the coarse design space. Next, individuals are injected into population with a higher level of resolution to fine tune the flywheel designs. The implementation involves a topological ring structure with different subpopulations (islands) working at different levels of re-finement of the mesh. Similar work using the hierarchical genetic algorithm (HGA) for a computational fluid dynamics problem is reported by Sefrioui and Periaux, (2000). They used a multi-layered hierarchical topology to solve a classical explo-ration/exploitation dilemma while using multiple models for optimization problems. When using three layer hierarchical topology, the lower-most layer performed the search space exploration employing large mutation rates, while the top-most layer used a more precise model employing low mutation rate. Investigators have success-fully mixed simple yet faster models with more complex models with slow solvers for a nozzle reconstruction problem. They have reported to achieve the same quality

(42)

results as that obtained by a simple GA, but spending only about one-third of the computational time.

2.5 Approximate Model Management

The most common approximation incorporation mechanism is to use it in fitness evaluation. But there are certain concerns which need to be addressed. In order to reduce the number of function evaluations, the approximate model needs to be developed from a small database size. The higher dimensional optimization prob-lems are difficult to model. In such cases, it is difficult to generate a global model of the function landscape which has the same global optima as that of the origi-nal optimization problem. The scarcity of training data is also a point of concern. Therefore, it is very important to use the original fitness function in conjunction to approximate model for the optimization problem. This is termed as model manage-ment or evolution control. The evolution control refers to the use of original fitness function for some individuals at every generation or for all the individuals at a cer-tain generation. If any individual is evaluated using original fitness function, it is referred as controlled individual. Similarly, if all the individuals are evaluated using original fitness function, that particular generation is called controlled generation. There are three possible approaches for controlled evolution. These are:

1. No evolution control.

2. Fixed evolution control.

3. Adaptive evolution control.

2.5.1 No Evolution Control

If the approximate model is assumed to be of high-fidelity, the original fitness func-tion is not at all used in evolufunc-tionary computafunc-tion. Such cases are classified as having no evolution control.

(43)

2.5.2 Fixed Evolution Control

It is important to use both original fitness function and approximate model simul-taneously in optimization. There are two possible type of fixed evolution controls (Jin et al., 2000):

• Individual based.

• Generation based.

As discussed earlier, in individual based controlled evolution, a fixed percentage of the population at every generation is evaluated using original fitness function. The rest of the population is evaluated using approximate model. In individual based controlled evolution, two types of strategies are possible. In one approach, the best individual based on approximate model evaluation at every generation is reevalu-ated using the original fitness function. In the other approach, the individual to be evaluated using original fitness function is selected randomly. The best individual based controlled evolution strategy generally performs better. Here, the approx-imate model gets updated at every generation. The idea of taking less function evaluations in order to reach the optima using controlled individuals with the help of clustering and neural network is explored by Jin and Sendhoff, (2004). The in-dividual near the center of the cluster is evaluated using expensive fitness function evaluation to create neural network ensemble which is used for fitness values of re-maining individuals. The structure and parameters of the neural network ensemble are also optimized using a standard evolution strategy. Branke and Schmidt, (2005) have used two estimation methods, namely, regression and interpolation, to achieve faster convergence to the optima. In their technique, at every generation, a fixed percentage of the population is evaluated with exact objective function. The fitness of the rest of the population is estimated. The individuals which are evaluated ac-curately are determined based on their estimated fitness and uncertainty. Savings in accurate function evaluations up to 50% are reported.

(44)

In generational based controlled evolution, all individuals after fixed number of generations are evaluated using original fitness function. So the approximate model gets updated after a fixed number of generations. It is assumed that generational based controlled evolution is carried out when the evolutionary algorithm converges to the approximate model. This is an important assumption and may not hold during course of optimization. Jin et al., (2002) has demonstrated the controlled evolution in evolution strategy. A framework which guarantees the correct conver-gence while reducing the computational cost is established.

2.5.3 Adaptive Evolution Control

The frequency of evolution control shall depend on fidelity of the approximate model. In generation based approach, a method to adjust the frequency of evolution control based on trust region framework is possible. The approximate model quality should be considered before going for a model update. An adaptive generation based frame-work for model management is successfully applied for two-dimensional aerodynamic optimization. Recently, Jang et al., (2002) has proposed adaptive approximation in single-objective optimization and adaptive approximation in multi-objective opti-mization.

2.6 Objectives and Scope of Present Work

The objective of the present study is to develop and test an optimization procedure having following properties:

• It is a generic procedure.

• It incorporates approximate models.

• It is capable for solving single as well as multi-objective optimization problems.

(45)

• It is suitable for unconstrained and constrained optimization.

The proposed procedure is a generic procedure which can work with a population-based evolutionary optimization procedure. It can incorporate approximate models which may change during the course of optimization. The proposed procedure can solve single and multi-objective optimization problems. The optimization procedure can reduce the number of function evaluations required for reaching the optimal or near-optimal solutions, compared to the original exact procedure. It can solve stan-dard test problems as well as the real-world optimization problems. It is equally applicable for constrained and unconstrained optimization problems. It can solve real and/or binary variable problems. The optimization procedure is able to han-dle continuous or discrete Pareto-optimal fronts. The proposed procedure can at least maintain good convergence and spread among the final non-dominated solu-tions. There are less number of additional user-defined parameters introduced in the proposed optimization procedure. The parameters which are characteristic of the proposed procedure should be studied and recommendations about their gen-eral values is to be made. In the present work, a fundamental assumption that the approximate model building time (such as constructing a neural network model) is negligible compared to the exact function evaluation time, is made.

2.7 Sum Up

In the present chapter, a literature review on the use of approximate models in evo-lutionary algorithms is presented. At first, the motivation for the use of approximate models is discussed. Next, three possible kinds of approximations in optimization problems are briefly discussed. It is followed by a discussion on the incorporation mechanism for approximate model in evolutionary algorithm. The use of approxi-mate model for fitness function evaluation is most commonly used. Hence, the issue of approximate model management in such applications is discussed. At the end, the objective and scope of the present work is discussed.

(46)

Proposed Methodology

3.1 Introduction

It is now amply clear that the researchers are trying to figure out different possi-ble methods to reduce computational loads on evolutionary optimization methods. Here, we shall focus on a generic method which can work with approximate models. At first, the basic idea of using successive approximate models will be presented. It is followed by a brief description of the method and some discussion over the integration scheme. The validation of the proposed scheme requires selection of a multi-objective EA and an approximate modeling technique. Hence, a popular multi-objective genetic algorithm, namely NSGA-II will be discussed. The approxi-mate models of function landscape is done by artificial neural networks. At the end, issues which arise out of the integration of the optimization method and the ANN will be discussed.

3.2 Proposed Method

We propose to combine EA with the approximation technique which permits EA to take a reduced number of function evaluations for computationally expensive problems. In order to reduce number of function evaluations during optimization, one can use several methods. These methods can be categorized as:

(47)

1. Use of a partial solution evaluation.

2. Use of a parallel computer.

3. Use of an approximate model of the optimization problem.

The first method, i.e. the partial evaluation of the solution, can be done on a spe-cial class of optimization problems which can be functionally decomposable into a number of subproblems. During initial optimization, only the most important subproblems may be considered for guiding the search. Since a less number of sub-problems are used, the computation cost is less for such solutions. Obviously such a solution is an approximate solution to the optimization problem. However, as the generations of EA progress, more and more of less important subproblems can be included. This results in getting more and more accurate solutions as generations of EA progresses. Towards the end of search, all subproblems are included and hence the final solutions are the most accurate solutions. Hence the computational burden on the optimizer is least in the early generations and it gradually increases and becomes maximum towards the end of the search. In such cases, the savings in function evaluations comes from the initial generations only because we are using less number of subproblems to guide the search. No savings in function evaluations will come from later generations when almost all subproblems are included in solv-ing the optimization problem. Hence basically an approximate solution guides the search and optimization procedure during initial generations and as generations of EA progresses, the solution procedure slowly moves for more and more accurate solutions. It must be noticed that such a scheme does not work for problems which are not functionally decomposable.

Because of the availability of parallel computers, it may be plausible to take advantage of parallel computing of different tasks involved in a function evaluation. For example, to evaluate a solution involving FFT or finite element computations, the solution can be sent to multiple processors for a faster computation. Since EAs use a population of solutions in each generation, most parallel EA applications

(48)

per-form a distributed computing of allocating a complete solution to each available processor, thereby reducing the overall computational time to complete one gener-ation. In such applications, although each solution can be evaluated with the help of multiple processors, usually this is not followed. Although faster solution proce-dures are developed using parallel migration and island models, most such studies have found a lower bound on the computational complexity achievable in terms of resorting to an optimum number of processors. Beyond the optimum number of pro-cessors, the computational advantage is overshadowed by the communication time involved among the processors.

Exact function X Model 1 Model N 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 3.1: Progressive approximate modeling.

The focus of this study is to use a successive approximation of the optimization problem. Starting with a coarsely approximated model of the problem, EAs use successively fine-grained models as generations progress. Figure 3.1 depicts this procedure. The figure shows a hypothetical one-dimensional objective function for minimization in a solid line. Since this problem has a number of local minimum solutions (which is one of the difficulties often exist in a real-world optimization problem), it would be a difficult problem for any optimization technique. It is concluded elsewhere (Goldberg et al., 1992) that to find the global optimum in such a problem using a GA, a population of size O(γ2_{), where} _γ _{is the inverse of the}