2009 - Multi-Objective Optimization Using Evolutionary Algorithms

8/8/2019 2009 - Multi-Objective Optimization Using Evolutionary Algorithms

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Multi-objective Optimization using

Evolutionary Algorithms

Progress report by

Peter Dueholm Justesen

Department of Computer ScienceUniversity of Aarhus

Denmark

January 13, 2009

Supervisors: Christian N. S. Pedersen and Rasmus K. Ursem


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Abstract

This is a progress report describing my research during the last one and a half year,performed during part A of my Ph.D. study. The research field is multi-objectiveoptimization using evolutionary algorithms, and the reseach has taken place in acollaboration with Aarhus Univerity, Grundfos and the Alexandra Institute.

My research so far has been focused on two main areas, i) multi-objective evo-lutionary algorithms (MOEAs) with different variation operators, and ii) decreasingthe cardinality of the resulting population of MOEAs. The outcome of the formeris a comparative analysis of MOEA versions using different variation operators on asuite of test problems. The latter area has given rise to both a new branch of multi-objective optimization (MOO) called MODCO (Multi-Objective Distinct Candidates

Optimization) and a new MOEA which makes it possible for the user to directly setthe cardinality of the resulting set of solutions. To motivate and cover my previousand future work, the progress report is divided into three main parts:

1. Introduction to the research area

2. The contributions made by this author

3. Future work

Peter Dueholm JustesenAarhus University

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Table of contents

1 Introduction 11.1 Main concepts of multi-objective optimization . . . . . . . . . . . . . . 11.2 Multi-objective optimization using evolutionary algorithms . . . . . . . 3

1.2.1 The history of multi-objective evolutionary algorithms . . . . . 41.2.2 Goals of multi-objective evolutionary algorithms . . . . . . . . . 51.2.3 Basic operators of multi-objective evolutionary algorithms . . . 6

2 Contributions 8

2.1 Genetic algorithms versus Differential Evolution . . . . . . . . . . . . . 82.1.1 Content and ideas . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Evaluation and discussion . . . . . . . . . . . . . . . . . . . . . 132.1.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Multiobjective Distinct Candidate Optimization . . . . . . . . . . . . . 142.2.1 Introduction, arguments and goals . . . . . . . . . . . . . . . . . 142.2.2 Evaluation and discussion . . . . . . . . . . . . . . . . . . . . . 172.2.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Cluster-Forming Differential Evolution . . . . . . . . . . . . . . . . . . 172.3.1 Content and ideas . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Evaluation and discussion . . . . . . . . . . . . . . . . . . . . . 25

3 Future work 26

3.1 Testing MODCO algorithms on Grundfos problems . . . . . . . . . . . 263.1.1 Constrained problems . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Many-objective optimization . . . . . . . . . . . . . . . . . . . . 273.1.3 The effect of changing MODCO parameters . . . . . . . . . . . 283.1.4 A more efficient search? . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Inventing performance indices . . . . . . . . . . . . . . . . . . . . . . . 293.3 Inventing alternate MODCO algorithms . . . . . . . . . . . . . . . . . 303.4 Relevant conferences and journals . . . . . . . . . . . . . . . . . . . . . 31

List of figures 32

List of algorithms 32

References 32

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1 Introduction

This section will introduce the main concepts of multi-objective optimization, as wellas motivate the use of evolutionary algorithms for this.

Multi-objective problems are problems with two or more, usually conflicting, objec-tives. The main difference from single-objective optimization is that a multi-objectiveproblem does not have one single optimal solution, but instead has a set of optimalsolutions, where each represents a trade-off between objectives.

At Grundfos, a danish company manufacturing pumps, an example of a multi-objective problem is that of designing centrifugal pumps. The designs of centrifugalpumps are complex, and each design lead to a different efficiency rate and produc-tion price. Grundfos is naturally interested in making their pumps have maximum

efficiency, which may be measured as the throughput of water per second. However,Grundfos do not want the pumps to cost too much to manufacture and sell, so atthe same time they want to minimize the production cost. Typically, increasing theefficiency of a pump will also increase the production cost. This way optimal pumpdesigns will have trade-off between cost and efficiency, ranging from maximum effi-ciency at maximum cost to minimum efficiency at minimum cost. However, findingthe optimal designs is not easy, and the task calls for optimization procedures.

A way to avoid the complexities of multi-objective optimization is to convert themulti-objective problem into a single-objective problem by assigning weights to thedifferent objectives, calculating a single fitness value. The major problem with thisweighted sum approach, is that it is subjective, as it ultimately leaves it to a deci-sion maker (DM) to assign weights according to the subjective importance of thecorresponding objectives. The approach further assumes that the DM has a prioriknowledge of the importance of the different objectives, which is often hard or impos-sible to come by. The objective approach uses Pareto compliant ranking of solutions,as explained in the following section. This approach favours solutions which are betterin a true, multi-objective sense. Only the latter approach has been investigated in mywork, as this makes no assumptions and does not rely on higher level knowledge.

1.1 Main concepts of multi-objective optimization

The primary concept of multi-objective optimization, is the multi-objective problemhaving several functions to be optimized (maximized or minimized) by the solutionx, along with different constraints to satisfy, as seen in Equation 1.

x is a vector of decision variables: x = (x1, x2,...,xn)T, where each decision variable

xi R is bounded by the lower bound xLi and the upper bound xUi . These boundsconstitute the decision variable space or simply the decision space D, and the Mobjective functions fm(x) define a mapping from D to the objective space Z. Thesurjective mapping is between the n-dimensional solution vectors x D and the m-dimensional objective vectors fm(x) Z, such that each x D corresponds to onepoint y

Z, as illustrated in figure 1.

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The problem may be constrained, so Equation 1 also shows J inequality and Kequality constraints. Solutions satisfying these constraints are feasible, and belong to

the feasible part of the decision space; SD D which the constraint functions map tothe feasible part of objective space SZ Z.

Minimize/Maximize fm(x), m = 1, 2,...,M;

Subject to gj(x) 0, j = 1, 2,...,J;hk(x) = 0, k = 1, 2,...,K;

xLi xi xUi , i = 1, 2,...,N;

(1)

The transition from single-objective to multi-objective optimization problems intro-duce a challenge in comparison of solutions, since performance is then a vector of objective values instead of a single scalar. The concept ofPareto dominance addressesthis issue, enabling comparison of solutions. We say that a solution x dominatessolution y, written x y, if and only if the following two conditions hold: 1) Thesolution x is no worse than solution y in all objectives. 2) The solution x is betterthan y on at least one objective. Formally, assuming minimization on all objectives:

fm(x) fm(y)m i : fi(x) < fi(y) (2)This definition entails that the dominating solution is, in a true multi-objective

sense, the better choice. The dominance relation allows for Pareto based ranking,ranking solutions according to how dominating they are wrt. a domination basedcomparison to other solutions. Other performance indicators should assign the bestvalue to the most dominating solutions; such indicators are called Pareto compliant,in contrast to the weighted sum approach described above.

The binary dominance relation presented above is transitive, asymmetric and non-reflexive, in short meaning that if x y and y z, then x z, if x y then y x, and that x x. However, several relations between solutions exist. A list ofthe most commonly used relations between solutions, the corresponding notation andformal interpretation is presented in Table 1, listed according to strictness imposed.

relation notation interpretation

strictly dominates x y fm(x) < fm(y)mdominates x y fm(x) fm(y)m

i : fi(x) < fi(y)weakly dominates x y fm(x) fm(y)m

incomparable x y (x y) (y x)indifferent x y fm(x) = fm(y) m

Table 1: Solution relations - assuming minimization on all objectives

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From the definition of dominance, several other important definitions can be de-rived. When optimizing, we are interested in locating the non-dominated set of solu-

tions. Among a set of solutions P, the non-dominated set of solutions P are thosethat are not dominated by any other member of the set P.

Correspondingly, we define the globally Pareto-optimal set as the non-dominatedset of the entire feasible search space S D. This is also referred to simply as thePareto-optimal set, and our goal using multi-objective optimizers is to approximatethis set. In objective space, the mapping from the Pareto-optimal set is denoted thetrue Pareto-optimal front or simply the true Pareto front. See figure 1 for illustration.

Figure 1: Mapping from decision space to objective space - assuming maximization

1.2 Multi-objective optimization using evolutionary algorithms

One way to perform multi-objective optimization is by using an evolutionary algorithm(EA). Evolutionary algorithms are optimizers inspired by Darwenian evolution, andwith this the concept of survival of the fittest. In an EA, solutions to a given problemis considered individuals of a population, where the fitness of individuals are given by

how good they solve the problem at hand. In the population individuals may mateto create offspring, which makes parents and offspring compete for inclusion in thenext generation. As only the most fit will survive this fight, the full population isimproving iteratively in each passing generation.

More formally, the strength of EAs comes from their use of a set of solutions,not only improving on a single solution. This makes it possible to combine several(good) solutions, when creating a new one. An EA is actually a stochastic meta-heuristic, i.e. a general optimization method, basing itself on probabilistic operators.Thus, contrary to deterministic algorithms, EAs may produce different results fromdifferent runs.

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The greatest difference between single-objective EAs and multi-objective EAs(MOEAs), is that for single-objective optimization, it is simple to return the most

optimal solution in a population, as scalar based evaluation automatically implies atotal order on solutions. For MOEAs, the situation is very different. Due to the higherdimensionality of the objective space, all resulting individuals of the population maybe incomparable to each other, each representing an optimal trade-off between ob-jectives. That is, the result of running a MOEA is typically a set of non-dominatedsolutions. From this result set, it is up to the decision maker to find out whichsolution(s) to realise. The full process is illustrated in figure 2.

Minimize F2...Minimize Fn

Subject to constraints

Multiobjective optimization problem

Minimize F1

Chosen tradeoff solution

Step 2: Decision making

Step 1: Optimization

Multiple candidate solutionson true Pareto front

Multiobjective optimizerIDEAL

Higherlevelinformation

on all candidate solutions

Figure 2: Multi-objective optimization process

1.2.1 The history of multi-objective evolutionary algorithms

Here, we present a brief historic view of some of the algorithms that has been used inthe work of this author. Where single-objective EAs has been extensively researched

for many years now, the field of MOEAs is relatively new. Here, we consider onlyalgorithms, which in some way incorporates the elitism concept, which ensures thatthe number of non-dominating individuals in the population can only increase.

The first, very popular elitist genetic algorithm for multi-objective optimizationwas the Non-dominated Sorting Genetic Algorithm; NSGA-II, created by Deb et aland published in 2000 [5]. This was the first MOEA, which further incorporated adiversity preserving mechanism, to ensure population diversity. Another elitist geneticalgorithm, also incorporating a density measure, was the Strength Pareto EvolutionaryAlgorithm; SPEA2, created by Zitzler et al and published in 2001 [ 7]. Both of thesealgorithms were shown to solve 2D problems very efficiently [5; 6; 7].

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Later, an old idea made its way to the field of MOEAs. Differential Evolution isan alternative way of varying EA individuals based on vector difference, which was

published in 1997 by Price and Storn [8]. The approach was adopted in a numberof MOEAs, where we will consider only the most relevant to this research. Recently,Robic and Filipic combined NSGA-II and SPEA-2 selection with the Differential Evo-lution (DE) scheme for solution reproduction to create DEMO (DEMO: DifferentialEvolution for Multi-objective Optimization). The DE-based MOEAs were namedDEMONSII and DEMOSP2, and these have been shown to outperform both NSGA-IIand SPEA2 [9; 10].

As MOEAs have become increasingly popular over the last decade, it has beennoted that the cardinality of the resulting populations of MOEAs is often too high fordecision makers to make their final choice of candidate solutions. One way of dealing

with this problem is by applying clustering, such that solutions within the same areaof the objective space are reported as a single solution. Very recently, Knowles andHandl have suggested clustering by k-means (MOCK) [16], where solutions are beingassigned clusters after an optimization run. Alternatively, this author and fellowresearcher have suggested an approach for optimizing distinct candidates (MODCO)[2] during the run. The main difference in these approaches besides when applied isthat in MODCO, the user directly sets the number of returned solutions, whereas thenumber of clusters are automatically determined using MOCK.

In general, decreasing the result cardinality introduce some interesting options wrt.decreasing the population size along with the number of generations performed. Fur-

thermore, this drastically decreases the amount of post-processing of results needed,as will be discussed later in this report.

1.2.2 Goals of multi-objective evolutionary algorithms

As stated above, the goal of a multi-objective evolutionary algorithm, is to approxi-mate the Pareto-optimal set of solutions. However, this goal is often subdivided intothe following three goals for the resulting population of a MOEA:

1. Closeness towards the true Pareto-front

2. An even distribution among solutions

3. A high spread of solutions

First, we want all of our solutions to be as optimal as possible by making themget as close to the true Pareto front as possible, where closeness is measured asEuclidian distance in objective space. As most problems solved by MOEAs are NP-complete combinatorial problems, there is no way of guessing the decision vectormapping to some good point in objective space. MOEAs select the most dominating(or most non-dominated) individuals for survival to drive the population towards the

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true Pareto front, since these must be closest to it. Optimally, all returned solutionsfrom a MOEA lie on the true Pareto front.

The second goal of MOEAs is to cover as much of the Pareto front as possible, andthis goal is very specific to multi-objective optimization. Having an even distributionof solutions on the Pareto front ensures a diverse set of trade-offs between objectives.Having a set of solutions all equidistant to their nearest neighbour provides the DMwith an overview of the Pareto front, while also making the final selection possible,based on the range of trade-offs between objectives represented by the population, asillustrated in figure 2.

The third goal is very connected to the second. Having a high spread means tohave a high distance between the extreme solutions in objective space, and as before,this is to ensure coverage of the Pareto front. Ensuring population diversity is most

often done by applying a density or crowding measure which penalise individuals, thatare close to each other in objective space. Such measure also ensures a high spread,since it will force the full population to spread as far as possible.

From an application point of view, the first goal is by far the most important,since it directly determines how optimal the returned solutions are. The second goalis important, but not crucial, since most often only a few solutions from the finalpopulation will be chosen for further investigation. Finally, the third goal is practicallyirrelevant, since extreme solutions are rarely, if ever, implemented in reality.

To pursue the three goals, traditional MOEAs employ two mechanisms directlymeant to promote Pareto-front convergence and a good solution distribution. The

first mechanism is elitism, which ensures that solutions closest to the true Pareto-front will never be eliminated from the population under evolution, i.e. the numberof non-dominated solutions in the population can only increase. The second is ameasure of crowding or density among solutions, a secondary fitness which is oftenincorporated into the Pareto-rank to form a single, final fitness.

1.2.3 Basic operators of multi-objective evolutionary algorithms

Here, we briefly introduce the MOEA operators, such that we have an overall ideaof their purpose before introducing more concrete mechanisms. The operators areapplied iteratively until some pre-determined termination criterion is met, usually

depending on the number of function evaluations performed, as this is where themain computational effort is spend. This is however depending on the population sizeN, the dimensionality of the problem M, and the number of generations performed;T. It is usual to create one new offspring per parent, such that M N T functionevaluations are performed in T generations. The basic operators used by MOEAs are:

Evaluation Selection

Variation

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Evaluation is typically based on the dominance relation, as described above. Itis intended to give an indication of the level of dominance for each individual by as-

signing a Pareto-rank, which bases itself on the number of other individuals in thepopulation that are dominated by the individual in question. This ranking must bePareto-compliant. Depending on which method used for Pareto ranking, it can bemore or less graded. Traditional MOEAs further incorporate a second fitness crite-rion when assigning the final fitness, in order to induce a total order on the qualityof individuals before selection. Evaluation is performed in objective space, and isdepending on the objective functions, which again are problem dependent.

Selection comes in two variants, and often base itself on the fitness assigned dur-ing evaluation. The first selection to be applied, is called mating selection or sexual

selection, where it is decided which individuals of the current generation get to mateand spawn offspring. This selection is often random or may include all individuals,in order to give all current generation individuals an equal chance to mate. Envi-ronmental selection is applied after variation is performed, by applying the famousrule of survival of the fittest, such that only the best of the combined set of parentsand offspring survive to the next generation. This typically truncates the expandedpopulation to its original size. Both kinds of selection are performed in objective space.

Variation is applied after mating selection to the individuals, who were selectedfor mating. These individuals then get to create offspring, which are a variation of

its parents. Here, two variation operators are typically applied; mutation and recom-bination. Mutation is intended to cause only small changes in the offspring, whereasthe recombination operator makes it possible to retain good parts of several parentsin the offspring. In general, we say that mutation enhances exploitation, whereasrecombination enhances exploration of the search space. Common for all variationoperators, is that they are applied in decision space.

Eva l u a t i o n

Ma t i n g

Se l e c t i o n

Var i a t i on

Env i r o nmen t a l

S e l e c t i o n

Figure 3: Multi-Objective Evolutionary Algorithm main loop

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2 Contributions

This section is devoted to the contributions made by this author and affiliates. To alarge part, my research so far has been focused on two main areas:

Genetic algorithms versus Differential Evolution Multi-objective Distinct Candidate Optimization (MODCO)We will go through these areas, first introducing the relevant articles submitted

for conferences in the MOEA society. We then go through the contributions with thefollowing taxonomy:

1. Content and ideas

2. Results

3. Evaluation and discussion

4. Future work

The content and idea section is intended to present the content and general ideasexpressed in the contribution, while the results section will present results, both in theform of data and in the form of implemented algorithms or operators. Evaluation anddiscussion is concerned with evaluating and discussing the contribution wrt. relatedresearch, and finally the section on future work intends to sketch future investigations.

2.1 Genetic algorithms versus Differential Evolution

The article Introducing the Strength Pareto Differential Evolution 2 (SPDE2) Algo-rithm - A Novel DE Based Approach for Multiobjective optimization [1] was submit-ted for the Parallel Problem Solving from Nature (PPSN) conference in may, 2008,but was rejected. The article was an investigation of the difference in performancebetween genetic algorithms and Differential Evolution, and introduces the idea of us-ing SPEA2 ranking and truncation with Differential Evolution as the variation andselection operator in a MOEA.

2.1.1 Content and ideas

EAs using variation operators that most often causes only small changes from parentsto offspring are called genetic algorithms, for historical reasons. Another branch ofEAs are differential evolution algorithms. The difference in the two kinds of algorithmslies in their way of applying mating selection and variation, along with an enhancedelitism concept in DE. The main idea of the contribution was to compare two popularmultiobjective genetic algorithms (MOGAs); NSGA-II and SPEA2 with their DE-enhanced counterparts. Here, we go through the main algorithmic differences, beforeintroducing the concrete algorithms implemented and tested; NSGA-II [5; 6], SPEA2[7] and the DEMO versions [9; 10].

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Genetic algorithms Using genetic algorithms, parents are stochastically selectedand copied to an offspring vector, and are then subject to recombination and mutation

operators in order to form offspring. After this, parents and offspring are joined in onepopulation, subject to truncation by environmental selection based on fitness. Thisis illustrated in figure 4, where it may be noted that the population size always variesfrom 2N to N during truncation.

V a r i a t i o n

M a t i n g s e l e c t i o n

E v a l u a t i o n

E n v i r o n m e n t a l s e l e c t i o n

Figure 4: Genetic algorithm

Differential Evolution Using Differential Evolution [8], we use vector differencesto create offspring, using several individuals to create a new candidate offspring tocompete against its parent. All individuals of the population are tried out as parents,generating one offspring each. The DE offspring creation algorithm is depicted inAlgorithm 1.

Algorithm 1 Differential Evolution

Require: Parent Pi, crossover factor CF, scaling factor F.Ensure: Offspring C.

1: Randomly select three individuals Pi1 , Pi2 , Pi3 from population P, where i, i1, i2and i3 are pairwise different.

2: Calculate offspring C as C = Pi1 + F (Pi2 Pi3).3: Modify offspring C by binary crossover with the parent Pi using crossover factor

CF.

Another mayor difference using Differential Evolution, is that DE enhances elitismby applying further rules after having created the candidate offspring. These rulesmakes the population size vary from N up to 2N before truncation, as can be seen infigure 5.

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Given parent x spawning offspring y, the elitist rules are:

If y x, x is replaced with y.

If y x, y is added to the population, as it is then possibly globally non-dominated. This is revealed during the competition with the parent part duringenvironmental selection.

If x y, y is discarded.

I n c o m p a r a b l e o f f s p r i n gD i f f e r en t i a l evo l u t i o n

E v a l u a t i o n

E n v i r o n m e n t a l s e l e c t i o n

R e p l a c e d b y d o m i n a n t o f f s p r i n g

Figure 5: Differential Evolution

NSGA-II The elitism mechanism in NSGA-II [5; 6] is based on non-dominatedsorting of the current population Pt. For each individual Pt, non-dominated sortingassigns a non-dominated rank equal to the non-dominated front label, which is used to

group the individuals into fronts. Here, the first front1

consists of the populations non-dominated solutions, the next front consists of solutions dominated only by the firstfront and so on. Different ways of performing non-dominated sorting are described in[6].

The density measure in NSGA-II is called crowding-sort, and gives an indicationof the degree of any solutions distance to its nearest neighbours wrt. the M differentobjectives. The crowding measure assigned to individual i is the average side length ofthe cuboid given by its nearest neighbours in M dimensions, normalized with respectto the maximal and minimal function value of the respective objectives.

1Where all individuals are assigned rank 1.

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A full NSGA-II run is to perform non-dominated sorting on the combined pop-ulation and offspring population. From here, as many fronts as possible are accom-

modated in the next population. The last front to be inserted is sorted using thecrowding-sort procedure, and only the best individuals wrt. crowding distance is cho-sen for inclusion. This truncation mechanism favors rank first, and good crowdingdistance next, just as the binary crowded tournament selection operator which NSGA-II applies to the parent part of the population to fill the offspring vector. Finally allindividuals in the offspring vector are subject to crossover and mutation operators.

SPEA2 SPEA2 [7] differs from NSGA-II by maintaining an archive At of size N ofthe best solutions found so far, and from this archive, a population Pt of size M iscontinuously generated to compete against it in generation t. As in NSGA-II, SPEA2

uses the dominance concept to promote elitism, but here a Pareto-strength value isassigned to each individual i in both population and archive according to how manysolutions in both archive and population i dominates:

S(i) = |{j|j Pt At i j}|, (3)where || denote cardinality of a set, is multiset union and i j means that idominates j. After all individuals i At have been assigned a Pareto-strength, araw fitness is assigned to individual i At equal to the sum of the strength of itsdominators:

R(i) =

jPtAtji

S(j). (4)

The density estimation in SPEA2 is based on the k-th nearest neighbour method,using Euclidean distance in objective space. In short, density for an individual i iscalculated as an inverse to its distance to its k-th nearest neighbour ki , where k = 1corresponds to the closest neighbour:

D(i) =1

ki + 2. (5)

Finally, each individual Pt At is assigned a fitness value equal to the sum of itsraw fitness and its density value; F(i) = R(i) + D(i) to provide a single value to judge

by.After fitness assignment, non-dominated solutions Pt At with F(i) < 1 are

first archived, i.e. placed in At+1. In case the archive is then too small, N |At+1|dominated individuals Pt At are archived, where individuals are chosen accordingto their assigned fitness value. Otherwise, if the archive is too large, it is truncatedby recursively removing the individual with the worst density from archive, i.e. theindividual who is closest to its k-th nearest (non-deleted) neighbour is repeatedlychosen for deletion until the archive size is reached. After this, mating selection isapplied using binary tournament on the archive, in order to fill the offspring vector,which is then subject to recombination and crossover operators.

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DEMO versions The Differential Evolution versions of the above genetic algo-rithms are called DEMONSII and DEMOSP2 [9; 10]. In the article [1], DEMOSP2 is

denoted SPDE2 (Strength Pareto Differential Evolution 2), as we thought this was anovel approach. However, the alternate DEMO version had already been investigatedunder the name DEMOSP2, and so we will use this.

The DE versions of NSGA-II and SPEA2 are using the corresponding Pareto-ranking, secondary fitness assignment and truncation mechanism as their origins.The difference lies in applying the elitist rules, and in using several individuals foroffspring creation, as seen in Algorithm 1 and in figure 5. Using DE, the elitist rulesensure the spread of good decision variables, while using several individuals increasethe chance of retaining good parts of the decision variables and decrease the risk ofpremature stagnation by introducing more diversity than genetic variation operators.

2.1.2 Results

Here, we summarize the results of our contribution. Overall, the results from thiscontribution have been:

1. Implementation of NSGA-II and SPEA2 with genetic operators.

2. Implementation of DEMONSII and DEMOSP2.

3. Implementation of 5 ZDT (Zitzler, Deb, Thiele) test problems.

4. Implementation of 5 different performance indices (PIs)5. Data from runs using all algorithms on all test problems with all PIs.

Implemented algorithms All algorithms described in the contents section havebeen implemented in a C++ framework supplied by Rasmus Ursem working at Grund-fos R & T. The framework features classes for numerical individuals along with anabstract class for multi-objective EAs, enabling implementation of different concreteversions. All algorithms have been implemented enabling several options regardingvariation operators, populations size, number of generations performed, etc.

As a part of implementing the genetic MOEAs, several genetic variation operators

were implemented. The simulated binary crossover (SBX) operator was implemented,along with arithmetic crossover operators, and a Gaussian annealing mutation oper-ator. For the DEMO versions, DE was implemented as in Algorithm 1 with theenhanced elitism rules, along with a few other DE variants, which showed no im-provement over the rand/1/bin scheme presented.

Test problems To test the different algorithms, 5 2-dimensional ZDT problemswere implemented. These problems have different true Pareto-front characteristica,which attempts to make MOEAs prematurely converge in various ways. These prob-lems are described further in [6].

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Performance indices As the true Pareto-fronts of the ZDT problems are known, itwas possible to implement performance indices directly minded on giving an indication

of the MOEA performance wrt. the 3 goals of MOEAs described earlier. 2 PIs oncloseness, 2 PIs on distribution and 1 on spread were implemented.

Data To gather data, each algorithm was run 10 times on each test problem. Fromthe resulting populations and the known true Pareto-fronts, the PIs were calculated,averaged, analyzed and discussed. It was clear, that the DEMO versions of NSGA-II and SPEA2 performed better than their original versions on all indices. Further,DEMOSP2 was observed to perform slightly better than DEMONSII, possibly due toa more fine-grained Pareto-ranking and a recursive density assignment procedure.

I further made a few investigations into the differences in distribution achieved by

the genetic and the DEMO algorithms, and found a discrepancy between DE and thecrowding/density measure of NSGA-II and SPEA2, as DE itself does not promote agood distribution. Thus, DE - based algorithms seem to be more prone to generateoffspring, which are removed due to crowding and thus does not contribute to thesteady qualitative improvement in the population.

2.1.3 Evaluation and discussion

As mentioned above, the article was submitted to the PPSN conference in may 2008,but was rejected. Some of the points of criticism were:

1. Low news value2. Test problems were outdated

The article was developed on background on the first DEMO-article by Robicand Filipic [9], but as the PPSN committee pointed out, a newer article had beenreleased meanwhile. In their second DEMO article [10], Robic and Filipic made aninvestigation very similar to the one performed by this author. Especially they alsoincorporated SPEA2 selection and truncation in DEMO. However, Robic and Filipictested on higher - dimensional problems, and with more general performance indices,which made their article superior to my work and further decreased the news valueof my article. Unfortunately, the article by Robic and Filipic was not available when

I started working on my article, which is the reason of the overlap.The problem with my chosen test suite for the article, was that MOEAs need

to be tested on problems with more than two objectives. It has been shown thatMOEAs working well on two - dimensional problems do not necessarily work well onhigher-dimensional problems. Especially there seems to be a blowup in complexitywhen moving from two dimensions to three.

On the more positive side, my article and the article by Robic and Filipic [10]agreed on the main point, that DE based MOEAs seem to outperform genetic algo-rithms, especially with respect to closeness towards the true Pareto-front. Thus, eventhough a part of my research was not up to date, it makes an interesting, valid point.

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2.1.4 Future work

No future work is intended to follow up on this contribution. The implemented algo-rithms all have been tested to work well on a range of problems, especially the DEbased ones. However, there seems to be little to gain in making further research inthis direction, as much is already thoroughly covered in [10]. Still, the algorithms havebeen a basis for experimenting with new algorithms, partly based on the techniquesof NSGA-II, SPEA2 and the DEMO versions. Especially the pareto ranking methodsand the elite preserving truncation mechanism have been used in newly developedalgorithms, along with Differential Evolution.

2.2 Multiobjective Distinct Candidate Optimization

We decided early that we wanted to research in which extent it was possible to decreasethe cardinality of the resulting population of traditional MOEAs. After some discus-sion, we named the research area Multiobjective Distinct Candidate Optimization(MODCO), and submitted two papers on the area to the Evolutionary MultiobjectiveOptimization (EMO) 2009 conference.

The first paper Multiobjective Candidates Optimization (MODCO) - A NewBranch of Multiobjective Optimization Research [2] was mainly developed by co-researcher Rasmus Kjr Ursem as an argumentation and quantification of the MODCOarea, based on his experience in real world optimization at Grundfos. This paper wasrejected, mostly due to its non-technical content.

The second paper Multiobjective Candidates Optimization (MODCO): A ClusterForming Differential Evolution Algorithm [3] was mainly developed by this authoras a first attempt to create a MOEA which complied with the MODCO goals. Thispaper was accepted, mostly due to its novel approach to decrease result set cardinality.

Here, we want to briefly present the main arguments and goals described in theMODCO article before evaluating and discussing, as these are the basis of the secondarticle on Cluster-Forming Differential Evolution (CFDE). That is, the concrete al-gorithm suggested in the CFDE paper is evaluated wrt. the goals of this paper. TheCFDE article is presented in Section 2.3. Note that the article on MODCO holds noalgorithms or results, and thus does not fully comply with the taxonomy from the

introduction to section 2.

2.2.1 Introduction, arguments and goals

The concept of MODCO is the optimization of a user-defined low number of dis-tinct candidates, with a user-defined degree of distinctiveness. This is in contrast tothe traditional MOEA goal of covering as much as possible of the true Pareto-front.However, having a full population of alternatives wrt. the trade-offs between objec-tives, just introduce more choices to be made by the DM. Basically, a standard EApopulation is simply of too high cardinality to be directly usable.

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MODCO basics MODCO tries to circumvent this problem by incorporating gen-eral, higher level information to the search, replacing step 2 in figure 2. This info

includes relevant practical information such as:

1. How much time and money can be spend on post-processing the solution set?

2. How many solutions is it feasible to inspect and compare?

3. How distinct must solutions be in order to be distinguishable in tests?

This information is relevant, and these questions must be answered, for most real-life applications. The goal of MODCO is to use this information as a guide towardsa small, final set of distinct candidates, replacing step 2 in figure 2, such that this

final selection is not only the responsibility of the DM, but is incorporated in theoptimization process.

Arguments for the soundness of MODCO To argue for the soundness ofMODCO, four main categories were identified and argued about in detail in [2]. Here,we briefly review these categories of arguments:

1. Post-processing of many Pareto-optimal solutions.

2. Physical realization of a solution.

3. Decision making among large sets of solutions.

4. Algorithmic and theoretical perspectives.

Post-processing many Pareto-optimal solutions is essential in most real worldapplications, where it is necessary to further investigate the most promising solutionsfrom an optimization process, in order to to figure out which one(s) to physicallyimplement. This process is expensive, time consuming, and it may only address asmall part of the full final design. Therefore it would be preferable to only have asmall set of solutions to perform post-processing on.

Post-processing is expensive, since this typically involves prototyping and testing,

where both may be very costly. Especially prototyping can be expensive, due to ma-terials. Post-processing also includes more detailed simulations, which is normallyvery time consuming. For example, a conducting a full computational fluid dynam-ics (CFD) calculation may take days, making it infeasible to simulate hundreds ofsolutions, which is not an unusual cardinality of MOEA results. Finally, when theoptimization process is only concerned with a part of a design, it is not feasible to havehundreds of alternatives, since the impact on the full design is expensive to calculate.

The physical realization of solution sets returned from a traditional MOEA alsosuffers some disadvantages due to the high cardinality. This is due to the problem ofhaving a 100 % accurate simulation, which is often far too costly. Normally, a few

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percent of tolerance is acceptable in simulation. However, if the simulator inaccuracyis greater than the difference between neighbor solutions from the MOEA resulting

population, a lot of these will not be distinguishable from each other. Again, a smaller,more distinct set of candidate solutions would be preferable.

Choosing among large sets of solutions implies problems for the DM, wrt.human factors. The DM may not have the technical background, or the knowledgeon optimization to be able to select among 100-500 different Pareto-optimal solutionsto a problem with high dimensionality. The amount of solutions makes it infeasibleto inspect and evaluate the different trade-offs, and further it is often not possible tostate explicit preference rules in order to guide this final selection. The only generalrule is, that solutions in knee regions in the objective space are preferred. Knee regionsare areas in the objective space, where a small improvement in one objective leads to

a high deterioration in another, or put another way, a convex part of the Pareto front.Hence, a small set of solutions in knee regions would be preferable from a DM pointof view.

Algorithmic perspectives on MODCO reveal a few points where the searchfor only small set of solutions should have advantages over traditional MOEA resultsets. First of all, allowing a decrease in the local population diversity may result ina more focused search, in some ways similar to local search, which results in a betterconvergence toward the true Pareto front. Secondly, the approach makes it possibleto use a smaller population size, since we are now no longer interested in covering thefull Pareto front, but only in locating a few distinct candidates. This naturally implies

fewer computational steps, especially function evaluations, but without compromisingthe quality of the found distinct candidates. Finally, if we are able to locate kneeregions, these are indeed the interesting areas, so no other solutions need be reported.The search for knees may further improve convergence towards the true Pareto front,as will be argued later.

MODCO goals The goals of MODCO are related to, but different from the goalsof traditional MOEAs, and are inspired by the desire of finding a small set of optimal,distinct candidates, preferably in knee sections. The features of the ideal MODCOalgorithm are described in more detail in [2], but in general, the goals of a MODCO

algorithm are:

1. Closeness towards the true Pareto front. Ideally, solutions are placed on thetrue Pareto front, but otherwise they should be as close as possible.

2. Global distinctiveness in the returned solutions, i.e. that solutions are dis-tinguishable wrt. performance or design. Ideally, this feature is parametrized,in order for the user to set how distinct returned solutions should be.

3. Local multiobjective optimality in returned solutions, i.e. it is preferred toreturn solutions in knee regions.

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The first goal corresponds to the first goal of traditional MOEAs, as we are stillinterested in optimizing our results. The two last goals are different, and based on the

idea of a decreased cardinality in the result set. That is, if we want to decrease theresult set cardinality, we need other ways of controlling the global diversity. Further,as argued earlier, results in knee regions are more interesting for a DM, and so theseshould be reported.


The paper on MODCO was rejected for the EMO 2009 conference, mostly due to itsnon-technical nature. Much of the argumentation is based on experience, and thusdeals with many non-technical issues related to decision making and the quantification

of DM preferences. The connection to the more concrete paper on the CFDE algorithmis strong and is pointed out several places, but this was not enough to be accepted.

On afterthought, the MODCO article could have been compressed, and maybeinserted in the beginning of the CFDE paper, simply giving the goals and a shortargumentation. This could have increased the correlation between the theory andthe practical appliance of MODCO. However, we wanted to separate the theory frompractical experiments in order to be able to argue for the soundness and use of both.

2.2.3 Future work

The future work for the MODCO article includes a rewrite, incorporating some moretechnical material from the technical report [4] also related to MODCO. This is dueto the criticism of the articles non-technical nature, and in order to make correlationsto CFDE more clear. As the MODCO paper gives much of the argumentation forwhy this class of algorithms should even be considered, it is further important for theCFDE article. Thus we need the article to be publicly available, and as such it mustbe well founded.

Further, we are currently discussing the MODCO branch of MOO with a few wellknown writers within the MOEA community, in order to establish the usability andrelated issues of the MODCO branch of MOO. A few answers have so far been positive

towards the idea and have suggested further reading.

2.3 Cluster-Forming Differential Evolution

The article Multiobjective Candidates Optimization (MODCO): A Cluster FormingDifferential Evolution Algorithm [3] was mainly developed by this author as an em-pirical investigation on the first concrete algorithm which conforms to the MODCOgoals. The article was accepted at the EMO (Evolutionary Multiobjective Optimiza-tion) 2009 conference. Due to this fact and the correlation to the MODCO paper [2],future work of this paper is presented in the section devoted to this; Section 3.

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2.3.1 Content and ideas

The article [3] demonstrates the first instance of a MODCO algorithm; the Cluster-Forming Differential Evolution (CFDE) algorithm. It is intended as a sequel to thearticle introducing MODCO as described above, implementing and evaluating a con-crete instance of the suggested algorithm class. The algorithm follows the goals ofMODCO, resulting in the following features:

1. User defined result set cardinality, parametrized as KNC.

2. User defined performance distinctiveness, parametrized as KPD.

3. The ability of converging towards knee regions.

These features corresponds to the MODCO goals, also used for evaluating theCFDE algorithm performance. For reference, the CFDE algorithm is depicted inAlgorithm 2. As usual we have a population P of size N, but the primary data struc-ture is now a vector of subpopulations Pi P of size N/KNC each, assuming WLOGthat N mod KNC = 0, as seen in figure 6. Further, a vector holding subpopulationcentroids, and a temporary offspring vector are used. minDist(Ci) is the function re-turning the minimum distance from centroid Ci to the nearest other centroid, whereasthe calculation of is depending on both KPD and the current problem.

Algorithm 2 Cluster-Forming Differential Evolution

Require: Population size N, KNC, KPDEnsure: KNC different non-dominated individuals.

1: Initialize KNC subpopulations with N/KNC random individuals in each2: while Halting criterion has not been met do3: Perform global DE-based mating - store incomparable offspring4: Calculate subpopulation centroids Ci5: Migrate incomparable offspring to nearest subpopulation wrt. centroid6: for All Pi P do7: if minDist(Ci) < then8: Assign nearest other centroid distance to each individual xi,j Pi9: else

10: Assign knee utility function value to each individual xi,j Pi11: end if

12: end for

13: Assign final fitness wrt. global pareto rank, then secondary fitness14: Truncate subpopulations wrt. final fitness15: end while

16: Return KNC solutions, by returning the non-dominated solution closest to thesubpopulation centroid from each subpopulation.

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N /

Of fsp r ing pa r t

Pa ren t pa r t

KN C

KN C

N /KN C

Figure 6: Population after migration

Main algorithm To initialize, the CFDE algorithm creates N random individuals,and insert these in KNC subpopulation with N/KNC individuals in each, which isdepicted as the parent part in figure 6. For problems with two objectives, an ini-

tial sorting is performed before insertion into subpopulations, in order to enhanceclustering.

Then the CFDE algorithm proceeds by performing global mating with replace-ment as in usual DE, and as seen in Algorithm 1 and in figure 5. However, it storesthe incomparable offspring in a temporary offspring vector, until it can be determined,which subpopulation they should belong to. From the parent part of the subpopula-tions, a centroid for each is then calculated. Following this, the incomparable offspringare migrated to the subpopulations with the nearest centroid.

At this point, the CFDE algorithm determines which of the two secondary fit-ness measures to use for each subpopulation. The secondary fitness measure is each

individuals distance to the nearest other centroid, if the subpopulations centroid istoo close to its nearest neighboring centroid2. This makes subpopulations reject eachother by favoring the individuals furthest away from other subpopulation centroids.Further, this most likely penalizes solutions created far from the subpopulation cen-troid, as these are most likely to be close to other subpopulation centroids, effectivelyenhancing clustering. In case the centroid of the subpopulation is sufficiently far awayfrom its neighbours, the secondary fitness measure is Branke et al.s utility function,favoring individuals in knee regions [12]. Thus, a subpopulation will search for knees,if not too close to another subpopulation centroid.

2Wrt. which depends on KPD and the problem, see Section 2.3.2 for an example calculation.

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Still, CFDE maintain focus on convergence towards the true pareto-front, so itthen assigns to each individual a global pareto rank using the NSGA-II non-dominated

sorting. This is used for assigning each individual a final fitness, such that the finalfitness incorporates both rank and secondary fitness measure in a total order.

Finally each subpopulation is truncated to the original size of N/KNC using thetruncation mechanism of NSGA-II. Here, the main differences are, that truncationis done locally in subpopulations, and that the subpopulations are truncated usingone of the two secondary fitness measures, which is incorporated in the final fitness.Hence, some subpopulations may be truncated using distance, and others using theknee utility function. This way subpopulations may be attracted to different kneeregions, while forming clusters during the evolutionary process. As we return onlyone solution from each subpopulation, we get the wanted number of distinct solutions

returned.

Novel contributions In the article [3], a more thorough walktrough of all themechanisms of the CFDE algorithm is found, but here we will only go through thenovel contributions and their compliance to the MODCO goals. This is due to manymechanisms used in the CFDE algorithm having already been explained in this report.The novel contributions in the CFDE algorithm are:

1. Flexible subpopulation based Differential Evolution.

2. Centroid calculation for each subpopulation allowing for migration.

3. Subpopulation centroid distance based secondary fitness.

4. Alternating secondary fitness assignment to subpopulations.

The idea of subpopulations is not new in itself, and has also been used in single-objective optimization, e.g. in multi-modal optimization, where decision spaces con-tains several global optima to be discovered, not unlike multi-objective optimization.However, to this authors knowledge, this is the first algorithm with a parametrizedamount of subpopulations subject to Differential Evolution. The subpopulation ap-proach is naturally crucial for the full algorithm, in that it allows for reporting backthe wanted number of distinct candidates.

The centroid calculation for each subpopulation is simple and based on the place-ment of the parent part of each subpopulation in objective space. It gives the averageplacement of the elite (parent) part of each subpopulation, and is used for both migra-tion and the centroid distance calculation used in the secondary fitness assignment.Thus, calculating centroids is essential for the clustering ability of CFDE. For eachsubpopulation Pi, we calculate the centroid Ci = [Ci,1, Ci,2...Ci,M] as the average pointof the elite in objective space:

Ci,m =

N/KNCj=1 fm(xi,j)

N/KNC, m = 1..M. (6)

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The centroid distance calculation enhances clustering by penalizing the individualsof any subpopulation with the distance to the nearest other subpopulation centroid.

As this is to be maximized, solutions created far from the subpopulation centroid aremost prone for penalty, as they are most likely to be close to the centroid of anothersubpopulation. This further makes subpopulations reject each other, as solutions gen-erated the furthest away from other centroids are then favored. Let dist(x, y) denotethe distance in objective space between point x and point y, each of dimension M.Further, let min(S) denote the minimal element of the set S. For the subpopulationPi, we then assign to each individual x Pi a secondary fitness SF as:

SF(x) = min({dist(f(x), Cj), j = 1..KNC, j = i}) (7)The subpopulation approach of CFDE also makes it possible to assign alternating

secondary fitnesses, according to case. To this authors knowledge, this is also anew idea, facilitating the use of an arbitrary amount of secondary fitness functions.However, this calls for a priority among secondary fitnesses. In the case of CFDE, weare only interested in distinct candidates, and thus the two secondary fitness measuresconflicts only if knee regions are located too close wrt. , i.e. we may not report backsolutions in knee regions located too close too each other, as they are not considereddistinct. Otherwise, knee search will not deteriorate clustering, as knee regions aretypically small, and likewise, the centroid distance assignment will not prevent kneesfrom being found when distinctiveness is achieved for any subpopulation.

The alternate secondary fitness measure, the utility function proposed in [12], is

intended to discover knee regions by calculating an average fitness value for a largenumber of randomly sampled weight vectors. If this average fitness is good, theindividual is more likely to reside in a knee region. Knee regions are characterized bythe fact that a small improvement in one objective, will result in large deterioration inanother objective. The utility function takes only one argument, precision, denotingthe number of sample weight vectors to apply. Let denote the weight vector ofdimension M, with

m m = 1. Then we calculate the secondary fitness SF with

precision precision of each individual x Pi as:

SF(x) =

precisionp=1 p f(x)

precision(8)

Lastly, let us see how CFDE comply with the goals of MODCO, and how this issupported by the novel contributions. Goal 1 of convergence is ensured by elitismwithin subpopulations, i.e. the number of globally non-dominated solutions within asubpopulation can only increase. Further this is enhanced by the elitism rules usingDifferential Evolution. Goal 2 of distinctiveness is achieved using the subpopulationapproach, which together with the centroid calculation, migration and centroid dis-tance penalty forms clusters from the initial random subpopulations, each reportingback a distinct solution. Further, the subpopulation approach is easily parametrizedto enable user setting. Goal 3 of detecting knees is achieved using Branckes kneeutility function when distinctiveness is reached.

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2.3.2 Results

To provide results, the CFDE algorithm was demonstrated on 3 kinds of problemsfrom MOEA literature; the 2D ZDT problems [6; 5], the knee problems of Branke etal. [12], and all non-constrained 3D DTLZ problems [11], all with problem settingsas suggested in the respective papers. The experiments were performed with respectto the MODCO goals:

Convergence performance (MODCO goal 1) Global distinctiveness (MODCO goal 2) User-defined performance distinctiveness (MODCO goal 2) Local multiobjective optimality (MODCO goal 3)

First, we check CFDE convergence against the DEMO versions, which have demon-strated good performance on many problems [9; 10]. Next, we want to demonstrateconvergence to KNC clusters, how we may change solution diversity by setting KPD,and finally that CFDE are able to locate knees.

We demonstrate the use of KPD only on knee problems, as this is only relevantfor such problems. For ZDT and DTLZ problems, we therefore always set = ,effectively disabling knee search. For the 3D DTLZ problems, we have used a higherKNC to ensure that we find both extreme and intermediate trade-offs. All othersettings of the algorithms investigated can be found in [3], Section 3, Table 1 and 2.

Convergence performance To deal with the different cardinality of more standardMOEAs and the CFDE algorithm, we use the universal notion of dominance. Here, wecompare the CFDE algorithm against DEMONSII and DEMOSP2. One algorithmicargument for MODCO is that the low number of returned solutions allows for a morefocused search because MODCO does not aim at an even distribution. Consequently,a MODCO algorithm should be able to return solutions closer to the true pareto front.

So we wish to investigate in which extent the returned solutions from the CFDEalgorithm dominate the most similar solutions from the returned population of the

competing MOEAs, where similarity is measured as distance in objective space. Thisway we see if the CFDE approach is competitive to simply picking KNC solutionsfrom the resulting populations of the DEMO versions.

For all results presented in [3], these are generated using the NSGA-II version ofglobal ranking and truncation in the CFDE algorithm. We have used 20 runs for boththe DEMO versions and for CFDE on each problem. For each generated populationof CFDE, we have compared each resulting individual to its most similar counterpartfrom each of the DEMO populations. This indicator gives a percentage of the amountof dominating, dominated and incomparable individuals CFDE was able to produceand is independent of KNC and KPD.

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F2

F1

20 runs - 5 clusters

Figure 7: ZDT1 plot - 20 runs - 5 clusters

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

F2

F1



Figure 8: ZDT3 plot - 20 runs

The results on each of the test problems can be found in Table 3 and 4 in [ 3] alongwith a more thorough walkthrough, but here we will only note that the CFDE resultsare almost never dominated by results from the DEMO versions, in fact only on asingle test problem with a high number of local Pareto fronts near the global one,the DEMO versions seems to outperform CFDE. On the rest of the 15 test problems,CFDE demonstrates equal or superior convergence, with up to 87 % dominatingsolutions produced compared to the most similar DEMO counterparts. Overall, theCFDE algorithm appears to be superior to the DEMO versions wrt. convergence.

Global distinctiveness Global distinctiveness is achieved by the CFDE algorithmusing the centroid distance to repel subpopulations. Figure 7 and 8 displays thereturned results of 20 runs of the CFDE algorithm on ZDT1 and ZDT3, using KNC = 5and KNC = 10. As mentioned above, we here set = . Similar robust convergenceis seen for the other test problems, i.e., CFDE found roughly the same set of distinctcandidates in repeated runs.

As can be seen in figure 7, all of the 20 runs of CFDE returned similar distinctsolutions. In figure 8, we see that using KNC = 5 ensures a result returned fromeach of the 5 patches of the true pareto-front and again we see only a small variation.However, using KNC = 10 makes the returned results be much more spread in the20 runs, since there are now more clusters to be formed than there are discontinuouspatches. As can be observed from the density, solutions will here most often seek the

most outer part of the patches making the returned solutions as distinct as possible.

User-defined performance distinctiveness The MODCO parameter KPD [0, 1] allows the DM to set how distinct the returned solutions should be. A low valuecorresponds to a low distinctiveness and a high value to a high distinctiveness, butas mentioned, KPD is problem dependent. So to demonstrate the effects of changingKPD, we have chosen DEB3DK as an illustrative example, as it allows us to visualizethe effect of altering the balance between knee search and subpopulation repelling.

We first demonstrate the calculation of used in the CFDE algorithm. First, wewill assume settings KPD = 1 and KNC = 5. For DEB3DK, we may use reference

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points z = (0, 0, 0) and zI = (8, 8, 8), which span the interesting part of the objectivespace. Then we can calculate:

= KPD/KNC ||z zI|| = 1/5

192 2.77 (9)In this setting, subpopulations will repel each other if they get within a distance

of 2.77 of each others centroids. Setting KPD = 1 should ensure maximum globaldistinctiveness, such that we get clusters uniformly spread across the objective spacespanned by the reference points. Contrary, setting KPD close to zero enables clustersto get closer to each other while searching for knees.

01

2

3456

78

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

F3

True frontKPD=1.0 - 20 runs

KPD=0.5 - 20 runsKPD=0.2 - 20 runs

F1

F2

F3

Figure 9: DEB3DK plot - investigating user defined performance distinctiveness

Figure 9 illustrates the results of setting KPD to 0.2, 0.5 and 1.0. For KPD = 1,the 5 clusters are equidistant around the single knee, where one cluster is placed. Thefour clusters not in the knee are repelled from each other as they reach a distance of2.77 between centroids, as was demonstrated in the example calculation above. The

four clusters not in the knee are located in the partial knees on the lines forming across. For KPD = 0.5, we always hit the knee with one cluster. Further, it can beseen that some solutions has found other knee regions, crawling towards the one inthe middle, but not being allowed to get too close. Setting KPD = 0.2 results in allclusters getting very close to the single knee region. Overall, it is clear that increasingKPD indeed makes clusters repel each other more.

Local multiobjective optimality Figure 10 and 11 illustrates the knee problemsDO2DK and DEB2DK, with the resulting CFDE individuals of 20 runs. In theDO2DK problem, we set K = 4 and s = 1.0, such that we have exactly the same

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0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

F2

F1

true front


Figure 10: DO2DK plot - 20 runs - 5 clus-ters - 4 knees

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

F2

F1

true front

20 runs

Figure 11: DEB2DK plot - 20 runs - 4 clus-ters - 4 knees

settings as has been used for creating the results illustrated in figure 4 in [12]. Forthe 20 runs depicted in figure 10, it may be noticed, that the density of solutions nearknee regions is very high. When using KNC = 5, CFDE finds the 4 knee regions veryprecisely, while one cluster typically hits an outer solution, or is caught in-betweenknee regions. For DO2DK, we have used KPD = 0.75 corresponding to = 1.5. Thisway we keep clusters separated, while still allowing for knees to be found.

For the DEB2DK problem, we have used K = 4 to replicate the results illustratedin figure 5 in [12]. In figure 11, we again see that for the 20 runs the density ofsolutions near knees are very high. Here, we have to set KNC to be equal to thenumber of knees, and it is clear that all knees are discovered in all runs. Here, we

have used KPD = 0.2 corresponding to = 0.5. This is low, so the centroid distanceassignment is rarely used. Hence, subpopulations converge to knees, and as long as > 0, the clusters formed will not overlap.

From the figures 10 and 11, and further figure 9, it is clear, that CFDE is indeedable to locate knee regions. Further, it has been demonstrated how to balance thesearch using KPD resulting in different values.


The paper on the CFDE algorithm [3] was accepted at EMO 2009, and this authoris to present it in Nantes, France, in April 2009. The referees were impressed with

the novel approach, the presentation and the experiments, even though there wassome complaints about submitting both an abstract MOEA class and a concreteimplementation in two different papers.

However, many usable comments were given in the reviews, especially consideringthat decreasing the result set cardinality is a quite novel approach in MOEA research,and some have given rise to future work. Given that we invented the binary per-formance indicators for MOEAs with different result set cardinality, the experimentswere still said to be illustrative and usable. This, along with a somewhat simple andeasily understandable parametrization are said to provide a potentially very usefulfunctionality, very competitive to pruning the result sets of standard MOEAs.

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3 Future work

This section is devoted to my future work, which mainly will be concerned with theappliance of MODCO algorithms in relation to Grundfos problems, with the CFDEalgorithm being the first. The major challenge will be applying the newly developedalgorithms to the real world problems supplied by Grundfos, which are typically muchmore complex than test problems, along with being constrained.

Apart from working on Grundfos problems, new performance indices, possiblyfacilitating comparison between MOEAs of different result set cardinality are to beinvented, along with new versions of MODCO algorithms. Lastly, we will investigatewhich conferences and journals could be relevant to submit to, and how this couldcoincide with my research.

In general, the hope is a synergy between the new knowledge acquired from testingon real world problems, and the general appliance of MOEAs. Grundfos have bothproblems and decision makers, which are both required in order to investigate the fullmultiobjective optimization process, which is of general interest to computer scientistsin the MOO area. On the other hand, the resulting designs of these investigationsshould be of interest to Grundfos, as they are presumably both distinct and optimal,wrt. user settings and the precision used in simulations.

3.1 Testing MODCO algorithms on Grundfos problems

Testing algorithms on real world problems are usually more interesting, in that theresults have a concrete, physical interpretation. Grundfos has a general interest inMOO, since they have parametrized design spaces of their products, including simu-lators to give the performance of designs. That is, the objective functions are actuallysimulations of varying degree of precision. This is one reason for applying MODCOalgorithms, as discussed in Section 2.2.

Considering MOEAs, the decision space of test problems holds no meaning, andneither does the objectives. Test problems are fine for their purpose; revealing ifand to which extent algorithms may solve them, in spite built-in traps and pitfalls- e.g. many local Pareto fronts in objective space. Test problems further facilitate

performance comparison of algorithms, which is of general interest to the community.Real world problems are contrary prone to more advanced investigations, in that

we may derive meaning and sense from e.g. the design space of centrifugal pumps.Likewise, the objectives are measurable in a physical sense, and are as such much moreinteresting. Working with specialists should enable new knowledge to be extractedfrom designs proposed by e.g. the CFDE algorithm, which are of natural interestto Grundfos. Likewise, we are more likely to uncover new knowledge on algorithmicbehavior, when both design space and results are comprehendable.

From testing on Grundfos problems, we hope to investigate several interestingfacets related to MODCO research:

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An approach for constrained problems.

Whether MODCO is well suited for many-objective optimization. How changing MODCO parameters will affect the result.

Whether the subpopulation approach could result in less computations.

3.1.1 Constrained problems

Most real world problems are constrained, i.e. solutions belonging to a part of thedecision space are not feasible, since they violate constraints. The constraint space

is defined by the constraint functions as seen in Equation 1, which maps the feasiblepart of decision space SD to the feasible part of objective space, SZ independentlyof the objective functions. This part is of varying shape and size, depending on theproblem at hand. However, for real world problems, the constrained space is often avery large part of the objective space, and as such feasible solutions are much harderto generate than unfeasible ones.

So the first challenge connected to the appliance of MODCO algorithms to Grund-fos problems, is to find an approach for constrained problems. One idea so far is touse the approach of Generalized Differential Evolution (GDE3) [14], where additionalrules are incorporated in the comparison of solutions. A feasible solution will alwaysbe preferred over an unfeasible one, but if two solutions are both infeasible, compar-ison then takes place in constraint space, such that the most dominant solution wrt.the constraint functions are favored. That is, solution x is considered dominant tosolution y ifx have the same or better constraint violation on all constraint functions,and has at least one constraint function with less constraint violation than y. Thisfacilitates a search steadily progressing towards the feasible, unconstrained part ofobjective space.

3.1.2 Many-objective optimization

Many-objective optimization is when our problem has more than 3 objectives, which

is also the case in many Grundfos related problems. The more precise we wish tomake our model, the more objectives and constraint functions may be included, anda centrifugal pump design problem may easily contain 5 - 10 objectives.

The challenge of many-objective optimization lies in the dimensionality of theobjective space. The more dimensions, the more trade-offs between the objectives canbe found. This may be illustrated by a simple counting argument; the more dimensionsof the objective space, the more edges and corners of the induced hypercube will exist,representing possible non-dominated solutions. Thus, covering a high-dimensionalPareto front can be very hard, and will require a much larger population than forproblems with 2 or 3 objectives.

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MODCO algorithms, and with this the CFDE algorithm, are potentially well suitedfor many-objective problems with high-dimensional Pareto fronts, since the more fo-

cused search does not need to put any effort into covering the full pareto-front, whichcan be very difficult in high dimensional spaces, as argued above. As higher dimen-sional problems also calls for larger population sizes, the problem with choosing fromthe result set again becomes apparent for standard MOEAs, whereas this is circum-vented using MODCO algorithms, e.g. CFDE.

3.1.3 The effect of changing MODCO parameters

Another hope from applying MODCO algorithms to Grundfos problems, is to makean investigation into different parameters settings of the MODCO algorithms. For

test problems, the number of knees and the interesting part of the objective spaceis given, but this is not the case when handling real world problems. Here, any apriori knowledge of the problem and its objective space is hard to come by due tothe complexity of the objective functions, and as discussed, should not be consideredavailable in general.

An investigation into the newly introduced MODCO parameters could reveal howthese are used by a DM, and to which extent parameter changes affect the results.The experiments in [3] demonstrated a robust behavior of the CFDE algorithm, inthat solutions converged to the same areas of the objective space under changingparameter settings. This behavior needs to be verified for real world problems, espe-

cially for constrained problems, and for problems with an unknown number of kneeregions, arbitrarily placed in the objective space. This includes an investigation ofthe correlation between the parameters N, KNC and KPD to discover any synergiesor discrepancies between these.

Further, a study on the DE parameters F and CF could also be interesting,especially in relation to the subpopulation approach. DE is intended to cause smallerand smaller changes during a run, given individuals converge towards the same areaof the decision space, e.g. mapping to a knee region or the global Pareto front. Here,we want to find out if the subpopulation approach enhances this behavior, whichresembles local search.

3.1.4 A more efficient search?

Lastly, we intent to make some investigations as to find out if the subpopulation ap-proach leads to a more efficient search. Overall, we intend to discover if all MODCOmechanisms are truly enhancing convergence, or if there are any discrepancies amongthem. This includes making experiments regarding the number of computations per-formed by MODCO algorithms vs. standard MOEAs.

It is likely that we may decrease the population size of e.g. the CFDE algorithmwithout affecting the quality of the reported solutions, since KNC


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is depending on the population size, we may decrease the number of computationscorrespondingly. The question is, how much we can decrease population size without

having the quality of results deteriorating. Further, the secondary fitness assignmentof CFDE requires less computations than the density measures of NSGA-II or SPEA2,but we still need some research on the efficiency of the alternating measures, wrt.newly developed performance indices, as discussed below.

3.2 Inventing performance indices

One related challenge when designing new algorithms, is how to judge their per-formance. As mentioned, the MODCO goals differ from the ones of more standardMOEAs, and hence new performance indices must be invented to facilitate comparison

- towards both standard MOEAs and other MODCO algorithms.Several performance indices are suggested in [4], but due to the different cardinality

of MODCO results, these need to be adapted and possibly changed in order to apply.So due to time and space limitations, these were not used in [3], where a more generalapproach was used in tests, based on the universal notion of dominance and objectivespace distance. However, they are a starting point for the future development of PIs,which are to be incorporated in the new version of [2], as discussed in Section 2.2.

The first goal of MODCO is closeness towards the true Pareto front, which is thesame goal as of standard MOEAs. That is why we may adopt several metrics fromMOO literature, but with respect to different cardinality of compared result sets. For

example, it may be possible to use the hypervolumen indicator [ 13], which calculatesthe dominated space of solutions, given a reference point. This has the advantage ofgiving an indication of to which extent a solutions set is better than another, whichis much more graded than the dominance relations used for statistics in [3]. However,the hypervolumen indicator will favor solutions sets which covers as much objectivespace as possible, and thus the lower cardinality of MODCO solutions set will bepenalized. This may be circumvented by only comparing the most similar solutionsin case of different cardinality of results, as we did in [ 3], but more investigation isneeded in order to see if such a comparison is really fair. Same approach could beused for the epsilon indicator [13], which gives the distance a population must bemoved towards an ideal point in order to dominate another population. Both of these

indicators are Pareto compliant, but will be affected by the low number of results inMODCO solution sets.

The second goal of MODCO is global distinctiveness, and we may measure thisin decision space, objective space or by using category functions [2; 4]. Again wemay be inspired by standard metrics such as the M2 indicator giving the averagedistance between solutions which are far enough away from each other in objectivespace, wrt. some niching radius corresponding to in CFDE, as noticed in [4]. Asimilar indicator M2 exist for design distinctiveness. Further, indicators for topologybased distinctiveness are suggested in [4], such as the one developed by Rasmus Ursem,to reveal topological distinctions such as hills or valleys, based on testing a solution

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position wrt. neighboring solutions. Using these different indicators, we may see towhich extent the intended distinctiveness is achieved, based on statistical analysis.

The third goal of MODCO is to discover knee regions, and so we need indicatorsto reveal this, as knee regions are not in general known a priori, as was the case withthe knee problems used for illustration in [3]. One such indicator is suggested in [4],based on knowledge of an ideal corner of the hypercube spanned by the objectivefunctions. However, the utility function of Brance used in the CFDE algorithm willgive a similar indication of knee regions, and this is very simple to both understand,adjust and use.

3.3 Inventing alternate MODCO algorithms

In order to investigate the performance of MODCO algorithms, it would be niceto have some alternatives to compete against each other. So a part of my futureresearch will be devoted to inventing alternatives to the CFDE algorithm, to facilitatecomparison.

The first step along this path is to create CFDESP2, the CFDE algorithm usingSPEA2 ranking and truncation, as described in Section 2.1. The SPEA2 paretostrength ranking is more graded than the non-dominated sorting of NSGA-II whennot all individuals of the population are non-dominated, which may be an advantagefor complex, constrained problems. Thus we get both CFDENSII and CFDESP2 tocompare against each other, just as the DEMO versions. These should incorporate the

GDE3 approach described in Section 3.1.1, in order to apply to constrained problems.Second step will be incorporating other forms of distinctiveness into the CFDEalgorithms, such that we may measure this in decision space instead of objective space,as well as using category functions, as mentioned in [2] and described further in [4]. Inshort, category functions map solutions in decision space to easily identified categories,e.g. motors of different standard sizes or electronics components of various standards.These a priori known categories are interesting distinctions wrt. solutions to theDM. However, note that the category mapping is not explicit preference functions.This way we investigate how the alternate definitions on distinctiveness will affectthe search and the results, to which extent they make sense to the DM, and how theMODCO parameters are used in the different cases.

Final step will be inventing MODCO algorithms not based on the CFDE algo-rithm, but with the same parameters to adjust. This should lead to other approachesthan the one based on a fixed user defined number of subpopulations, which at thatpoint should be thoroughly researched. One may imagine MODCO algorithms, whereonly upper and lower bounds on KNC are to be set, where after KNC and KPD will beadjusted dynamically during the run, while automatically optimizing distinctivenessand searching for knee regions. This could entail some mechanisms for dynamicallycreating subpopulations from one overall population, when knee regions or other at-tractive areas of the objective space are found, e.g. wrt. the optimization of globaldistinctiveness.

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3.4 Relevant conferences and journals

Here, we will provide an overview of the conferences and journals, to which we maysubmit future publications. We will first list conferences and then journals, as wellas give an early indication of what work may be done by these deadlines. The listfurther shows approximate dates for the conferences.

Evolutionary Multiobjective Optimization (EMO), April, 2009. IEEE Congress on Evolutionary Computation (CEC), May 2010. Genetic and Evolutionary Computations Conference (GECCO), July 2010.

These conferences are all focused on evolutionary computation, and are all wellsuited for papers on new algorithms or new applications. EMO is more specialized, inthat it is more focused on multiobjective optimization, and thus is very relevant to thisauthor. So the acceptance of [3] for this conference is a good opportunity to discuss mywork here. However, as CEC and GECCO are much further away in the future (2010),relevant submission will here include reports on testing new MODCO algorithms onGrundfos problems including new performance indices, as will be developed by theend of 2009, the approximate deadline for submissions to these conferences.

Relevant journals include:

Applied Soft Computing - www.softcomputing.org/

IEEE Transactions on Evolutionary Computations -http://ieee-cis.org/pubs/tec/

Applied Soft Computing is a journal in connection to online conferences held by theworld federation of soft computing (WFSC), and are focused on industrial applicationsof soft computing, such as evolutionary algorithms. The journal is connected to themajor publisher Elsevier, which hosts many different forms of scientific papers andjournals. The IEEE transactions on Evolutionary Computations is also interested inboth methods and applications. For these journals, a thorough application of theinitial CFDE algorithm versions (CFDENSII and CFDESP2) on Grundfos problemscould be very relevant, in that it is both a novel approach and further it is tested onan industrial engineering application, which is more interesting than test problems.This project would include a thorough rewrite of [2] and [4], to produce a single,introductory paper on the abstract class of MODCO algorithms and how to measureperformance. This work is to be done during the spring and summer of this year,2009.

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List of Figures

1 Mapping from decision space to objective space - assuming maximization 32 Multi-objective optimization process . . . . . . . . . . . . . . . . . . . 4

3 Multi-Objective Evolutionary Algorithm main loop . . . . . . . . . . . 7

4 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Population after migration . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 ZDT1 plot - 20 runs - 5 clusters . . . . . . . . . . . . . . . . . . . . . . 23

8 ZDT3 plot - 20 runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 DEB3DK plot - investigating user defined performance distinctiveness . 24

10 DO2DK plot - 20 runs - 5 clusters - 4 knees . . . . . . . . . . . . . . . 25

11 DEB2DK plot - 20 runs - 4 clusters - 4 knees . . . . . . . . . . . . . . . 25

List of Algorithms

1 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Cluster-Forming Differential Evolution . . . . . . . . . . . . . . . . . . 18

References

[1] Justesen, P.D. and Ursem, R.K.: Mult

Documents

2009 - Multi-Objective Optimization Using Evolutionary Algorithms