A Winning Score-based Evolutionary Process for

Multi-and Many-objective Peptide Optimization

Susanne Rosenthal

1,3

and Markus Borschbach

2,3

Rheinische Fachhochschule K

oln, University of Applied Sciences, Cologne, Germany

FHDW, University of Applied Sciences, Competence Center Optimized Systems, Bergisch Gladbach, Germany

Steinbeis Innovation Center ”Intelligent and Self-Optimizing Software Assistance Systems”, Bergisch Gladbach, Germany

Keywords:

Winning-score based Selection, Multi- and Many-objective Optimization, Biochemical Optimization,

Evolutionary Algorithm.

Abstract:

Target identiﬁcation as part of drug design is a long process with high laboratory evaluation costs since optimal

candidate leads have to be identiﬁed in an iterative process including the determination of diverse physiochem-

ical properties, which have to be optimized simultaneously. MOEAs have become an established optimiza-

tion method in in silico-aided drug design processes. Since target identiﬁcation becomes more complex, the

dimension of molecular optimization problems increases. Less work has been done so far to evolve an evo-

lutionary process efﬁciently solving both, multi- and many-objective molecular optimization problems while

considering application-speciﬁc conditions of molecule optimization. This work presents the enhancement of

a MOEA especially evolved for molecular optimization. The proposed algorithm is applicable to multi- and

many-objective molecular optimization problems identifying a selected number of qualiﬁed candidate pep-

tides within a very low number of iterations. It has a simple framework structure and optionally uses two

types of winning-score ranking method as survival selection. Default parameters are provided in the compo-

nents to enable a non-expert use. This algorithm is benchmarked to the recently proposed and promising AnD

(ANgle-based selection and shift-based Density estimation strategy) on molecular optimization problems up

to 6 objectives. Furthermore, the selection principles are exemplarily compared and discussed.

1 INTRODUCTION

Computer-assisted techniques gain importance in the

area of drug discovery and development. The suc-

cess of molecule design depends on simultaneous op-

timization on often conﬂicting biological and phys-

iochemical properties. Multi-objective Evolutionary

Algorithms (MOEAs) have proven to enhance the po-

tential of improving promising drug candidates (Ni-

colotti et al., 2011). The increase of complexity in

pharmaceutical research results in the challenge of

the design of a Many-objective Evolutionary Algo-

rithm (MaOEA) especially for molecular optimiza-

tion. To the best of our knowledge, less work has

been done so far regarding this issue. The design of

a MOEA as well as MaOEA for molecular optimiza-

tion has to take account of several application-speciﬁc

conditions: the objective of target identiﬁcation usu-

ally requires expensive and time-consuming labora-

tory work since the numerical approximation of pep-

tide properties is challenging. Therefore, the objec-

tive function evaluations have to be limited to save re-

sources. Furthermore, the algorithm provides default

parameter settings to enable the non-expert use and

does not utilize weight vectors or reference points,

which are commonly unknown in real-world appli-

cations but have an impact on the algorithm perfor-

mance.

In addressing these issues, a single-objective evo-

lutionary algorithm especially evolved for molecu-

lar optimization has been introduced in (R

ockendorf

and Borschbach, 2012), (Krause et al., 2018) pro-

viding an exponential ﬁtness improvement within the

very low number of 10 iterations and a standard

population size. This approach has been enhanced

to a MOEA with similar properties. This MOEA

is termed COmponent-Speciﬁc Evolutionary Algo-

rithm for Molecular Optimization (COSEA-MO) and

is presented in (Rosenthal and Borschbach, 2017b).

COSEA-MO identiﬁes a selected number of highly

qualiﬁed candidate peptides with a wide range of

genetic diversity within 10 iterations. Nevertheless,

Rosenthal, S. and Borschbach, M.

A Winning Score-based Evolutionary Process for Multi-and Many-objective Peptide Optimization.

DOI: 10.5220/0008065800490058

In Proceedings of the 11th International Joint Conference on Computational Intelligence (IJCCI 2019), pages 49-58

ISBN: 978-989-758-384-1

the increase of the problem dimension reveals well-

known challenges of Pareto deﬁnition-based MOEAs

in solving Many-objective Optimization Problems

(MaOPs) comprising more than three objectives: the

inability of adequately differentiate higher dimen-

sional solutions and the loss of selection pressure.

This work presents an enhancement of COSEA-

MO to solve multi- and many-objective molecular op-

timization problems by the traditional Winning Score

(WS) technique (Maneeratano et al., 2006) or op-

tionally a new difference-based WS selection strat-

egy. Furthermore, the selection principles of these

WS techniques are exemplarily analyzed and dis-

cussed. The performance of the enhanced versions

of COSEA-MO are compared to a recently proposed

and promising MaOEA termed AnD (ANgle-based

selection and shift-based Density estimation strategy)

(Lee et al., 2018) on a three-dimensional up to a six-

dimensional molecular optimization problem.

The outline of this work is as follows: Section 2

gives an overview of the related work, the proposed

approach with WS-based selection strategies is intro-

duced in section 3 with the discussion of the selection

principles. Section 4 presents the experiments, sec-

tion 5 concludes this work and gives an outlook on

future work.

2 RELATED WORK

MOEAs are classiﬁable into three categories ac-

cording to their selection strategies: Pareto-based,

decomposition- based and indicator-based algo-

rithms. The effectiveness of MOEAs on Many-

objective Optimization Problems (MaOPs) is signif-

icantly decreased with the problem dimension (Ishi-

huchi et al., 2011). In the case of Pareto-based

MOEAs such as NSGA-II (Deb et al., 2002) and

SPEA2 (Zitzler et al., 2002), the number of non-

dominated solutions increases signiﬁcantly with the

problem dimension since the Pareto dominance prin-

ciple as elitism strategy for survival selection is not

capable to adequately differentiate candidate solu-

tions. In the case of decomposition-based MOEAs

such as MOEA/D (Zhang and Li, 2007), it is chal-

lenging to deﬁne weight vectors or reference points

in higher dimensions. A rising dimension size leads

to a challenging consumption of computational time

in the case of indicator-based MOEAs such as SMS-

EMOA (Beume et al., 2007) and IBEA (Zitzler and

unzli, 2004).

Several enhancements have been published to im-

prove the performance of Pareto-, decomposition-

and indicator-based MOEAs on MaOPs: The intu-

itive way of improving Pareto-based MOEAs is to

ﬁnd alternative Pareto dominance deﬁnitions. The ε-

dominance principle uses a factor ε to compare the

dominance principle of individuals (Laumanns et al.,

2002). L-dominance is introduced selecting individ-

uals with objectives of similar importance regarding

the objective value improvement (Zou et al., 2008).

Fuzzy dominance methods are presented using rank-

ing schemes to select promising individuals (He et al.,

2014). Moreover, a grid-based method is published

adjusting the grid size to control the proportion of

Pareto-optimal solutions (Yang et al., 2013).

Several enhanced variants of MOEA/D for

MaOPs have been published in the past. Most re-

cent algorithms are MOEA/D-AM2M which adap-

tively allocates the search effort (Liu et al., 2017),

MOEA/D-DU which exploits the perpendicular dis-

tance from the individuals to the weight vectors (Yuan

et al., 2016) and MOEA/D-PaS which uses a Pareto

adaptive scalarization method (Wang et al., 2016).

The hypervolume is the mostly used indicator in

indicator-based MOEAs such as in IBEA and SMS-

EMOA. Its major disadvantage is the experimental in-

crease of the computational complexity with the di-

mension increase. To overcome this disadvantage,

other indicator-based methods have been introduced

recently. The Inverse Generational Distance Plus

(IGD

) indicator is used in IGD

-EMOA to address

MaOPs up to 8 objectives (Lopez and Coello, 2016).

Furthermore, the collaboration of different indicators

of low computational complexity has been proven to

be a promising solution for solving MaOPs (Lee et al.,

2018).

Beneath these approaches, the widely used

NSGA-II has been improved to NSGA-III (Deb and

Jain, 2014) for MaOPs by the use of a set of

predeﬁned well-distributed reference points. Non-

dominated solutions close to this set are prioritized.

An appropriate design of this set is challenging, es-

pecially in the case of real-word applications. A re-

cent promising MaOEA has been proposed termed

AnD (Lee et al., 2018). It has a simple framework

structure and selects promising individuals from the

union of parent and child population for the next it-

eration with a diversity-ﬁrst-and-convergence-second

principle. AnD combines the well-known vector an-

gle and shift-based density estimation in the selection

process. Angle-based selection is used to identify two

individuals with minimal angle. This is motived by

the idea that these individuals represent the search in

the same direction and waste computational resources

if both individuals survive. The individual with lower

shift-based density estimation is deleted in order to

ensure convergence. AnD is compared to seven state-

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

of-the-art MaOEA on a variety of benchmark prob-

lems with 5, 10 and 15 objectives and reveals highly

competitive performance (Lee et al., 2018). AnD

is chosen for experimental comparison in this work

as it is the only algorithm apart from COSEA-MO

that has a simple framework structure, provides op-

timized default parameters for the non-expert use and

is independent of weight vectors or reference points,

which usually have a strong impact on the perfor-

mance and are usually unknown in real-world appli-

cations. Moreover, the framework structure of AnD is

similar to those of COSEA-MO.

The traditional WS technique has been introduced

with the Compressed–objective Genetic Algorithm

(COGA) (Maneeratano et al., 2006). Two conﬂict-

ing preference objectives, WS and a vicinity method,

are used to assign different preference levels to non-

dominated solutions to bound the increasing set of

non-dominated solutions in MaOPs. A rank is as-

signed to each non-dominated solution according to

the preference objectives to select high preferred non-

dominated solutions in survival selection and the trun-

cation method to maintain the archive size. COGA

has been enhanced to the Improved Compressed–

objective Genetic Algorithm (COGA-II) (Boolong

et al., 2010). A WS-based ranking mechanism is

applied instead of the two preference objectives of

COGA. The WS value of a non-dominated solu-

tion is determined by the weighted sum of competi-

tive scores from all objectives to the remaining non-

dominated solutions.

3 PROPOSED APPROACH

This section proposes an enhanced version of

COSEA-MO to solve MaMOPs. Its characteristics

are a simple framework structure, optimized default

parameter settings for the non-expert use, determinis-

tic dynamic variation operators and WS-based selec-

tion mechanism. Two alternative WS-based mecha-

nism rank the population and select individuals for the

next generation according to the scores. Score values

are assigned to each individual based on the number

of superior or inferior objectives in the case of the tra-

ditional WS and additionally based on the quantity

of superiority or inferiority in the case of Difference-

based Winning Score (dWS) to the remaining individ-

uals in the population. The framework of COSEA-

MO with WS-based selection is referred to as WS-

COSEA-MO, the version with dWS is termed dWS-

COSEA-MO. The framework of both is given in Al-

gorithm 1.

Algorithm 1: Framework of (d)WS-COSEA-MO.

Input: Population P

, population size N,

Archive A

= {}, number of optimal

solutions m, total number of

generations T

Output: Next generation P

t+1

and archive

update A

t+1

1: Random initialization of P

;

2: while t < T do

← RandomMatingAndVariation(P

);

← P

∪ Q

;

t+1

← (d)WS-Selection(U

);

t+1

← add m ﬁttest individuals of P

t+1

acc. to (d)W S;

t ← t + 1;

end

Firstly, the start population P

of size N is ran-

domly initialized. The individuals represent peptides

in form of character strings. During the evolution pro-

cess, an offspring generation Q

of size N is deter-

mined by randomly selecting three parents of P

for

variation. The speciﬁc number of parents is motived

to ensure a high genetic diversity of the genetic mate-

rial. The variation operators are motivated by a suit-

able balance of global and local search. Determin-

istic dynamic variation operators are suitable opera-

tors to achieve this purpose. A linear dynamic re-

combination operator and an adapted version of the

deterministic dynamic mutation operator of B

ack and

Sch

utz (B

ack and Sch

utz, 1996) is used to generate

offspring (RandomMatingAndVariation). The varia-

tion rates are adapted dynamically by predeﬁned de-

creasing functions with the iteration progress: the re-

combination operator varies the number of recombi-

nation points by a linearly decreasing function

(t) =

−

l/4

·t,

where l is the peptide length, T the total number of

the generations and t the index of the current gener-

ation. The adapted mutation operator determines the

mutation probabilities via

= (a +

l − 2

T − 1

−1

with a = 5. The mutation rates of the traditional op-

erator are reduced by a higher value for a. After that,

and Q

are combined to a population U

of size 2N.

Finally, the WS- or dWS-based selection mechanism

is performed to select N individuals of U

for the next

generation P

t+1

((d)WS-Selection) and the archive is

updated by adding the m-optimal individuals of P

t+1

according to the scoring points. Optimal solutions de-

tected in previous generations are not added twice into

the archive.

A Winning Score-based Evolutionary Process for Multi-and Many-objective Peptide Optimization

The motivation and the decisive characteristics of

WS- and dWS-based selection are described and dis-

cussed in the sequel.

3.1 WS-based Selection Mechanisms

In (Benedetti et al., 2006), three reasons are summa-

rized about the unsatisfactory of Pareto dominance

deﬁnition in the case of a large number of objectives:

• The number of improved objective function val-

ues are not taken into account.

• The (normalized) relevance of improvement is not

taken into account.

• No preference among the objectives is considered.

In the present biochemical optimization problems, all

objectives are equally important and therefore, the last

issue is negligible. The traditional WS method meets

the ﬁrst issue in an intuitive way, it describes the dif-

ference between the number of superior and inferior

objectives between two individuals: let sup

i j

be the

number of objectives in a solution i that is superior

to the corresponding objectives in a solution j while

in f

i j

is the number of objectives in i that is inferior to

j. The WS-values of the i-th solution in a population

of size N is given by (Maneeratano et al., 2006):

W S(i) =

∑

j=1

i j

with w

i j

= sup

i j

− in f

i j

Obviously, it is w

i j

= −w j i and w

= 0. This assign-

ment ensures that solutions with high WS-values are

close to the true Pareto front.

We assume the following expression of a MaOP:

minimize F(x) = ( f

(x), f

(x),..., f

(x)) with x ∈ Ω,

where x is the decision variable in the search space

of all feasible peptides (Ω), F(x) the objective vector

and K the number of objectives. To address the sec-

ond issue additionally to the ﬁrst one, dW S is used:

dW S(x

) =

∑

j=1

∑

k=1

i jk

with

i jk











( f

) − f

))

, if f

) ≺ f

0, if f

) = f

−( f

) − f

))

, if f

)  f

where x

and x

are two individuals of the population,

N is the population size and K the number of objec-

tives. dWS has especially been evolved to rank the

solutions according to their exact objective value dif-

ferences. With this approach, the amount of objective

improvement and worsening is considered, not only

the number of superior and inferior objectives. The

quadrate of the differences ensures that higher differ-

ences have a stronger impact on the scores.

The aim of WS- or dWS-based selection is to

ﬁnd N approximately optimal individuals from U

for

the next generation. Furthermore, the update of the

archive with the m ﬁttest individuals is also based

on the ranking according to the scores. Therefore,

a score is assigned to each individual x

of the cur-

rent population relative to the remaining population

members x

. W S

or respectively dW S(x

) of an in-

dividual x

reﬂect the quality of x

relative to the re-

maining members of the current population. A high

positive score value indicates superior quality of this

individual compared to the others, whereas high neg-

ative values indicates a low qualiﬁed solution. In con-

trast to COGA and COGA2, the scores are standalone

selection criteria and are assigned to every member of

the population with the aim of ranking, they are not

only used to differentiate a pair of non-dominated so-

lutions.

After this scoring point assignment to each indivi-

dual, the population is ranked according to the scores.

The selection process of the N individuals from

performs as described in Algorithm 2. R

is the

ranked set of U

and R

(i) is the i-th front set of R

Algorithm 2: Selection process.

Input: Ranked population R

with |R

| = 2N,

population size N

Output: Next generation P

t+1

1: while |P

t+1

| + |R

(i)| ≤ N do

t+1

← P

t+1

∪ R

(i);

i++;

end

2: while |P

t+1

| < N do

binary tournament selection

} ∈ R

\ P

t+1

;

if VolumeDominance(x

) <

VolumeDominance(x

) then

t+1

← P

t+1

∪ {x

};

end

else

t+1

← P

t+1

∪ {x

};

end

The population of the next iteration is ﬁlled by each

rank subsequently until the population size exceeds

N. In the case that adding the individual set of rank

(i) exceeds N, the remaining individuals for P

t+1

are selected by binary tournament selection of two in-

dividuals from the remaining ranks R

\ P

t+1

accord-

ing to the better Volume Dominance (VD) value. VD

is simply the spanned space of an individual to the

zero point (VolumeDominance). In the case of a min-

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

Figure 1: Selection principle of AnD, NSGA-II and (d)WS-

COSEA-MO.

imization problem, a lower VD value reveals a higher

solution quality than a lower one. Since all individ-

uals from the previous ranks are selected, elitism is

ensured.

3.2 Discussion of the Selection

Principles

WS and dWS as selection criteria have been chosen

under the subsequent considerations:

• the individuals are ranked according to their qual-

ity relative to each other,

• the ranking process is also applicable in higher

problem dimensions without loss of effectiveness,

• Solutions with lower function values in one ob-

jective but highly qualiﬁed function values in the

other objectives are termed boundary solutions

and are of importance for the spread of the true

Pareto front, they receive an evolutionary advan-

tage,

• individuals positioned in a crowded area achieve

similar ranks.

The potential of these targeted characteristics is

demonstrated in the following examples:

Firstly, a two-dimensional example is used to il-

lustrate the selection principle of WS-COSEA-MO,

dWS-COSEA-MO, AnD as well as NSGA-III. This

example is taken from (Lee et al., 2018). Six in-

dividuals are given, A(0,0.9), B(0.7,1), C(1,0.3),

D(0.7,0.15), E(0.9, 0.05) and F(0, 1). Four promis-

ing individuals haven to be selected into the next

generation. Figure 1 illustrates the selection princi-

ples: Since NSGA-III prefers non-dominated solu-

tions, therefore A, D, E and F are selected. In the case

of AnD, C and E are removed as C and D as well as

E and F provide minimal vector angles, but the shift-

based density estimation of C and E are worse. In

the case of WS-COSEA-MO and dWS-COSEA-MO,

A, D, E and F are selected into the next generation.

The difference between WS-COSEA-MO and dWS-

COSEA-MO is the ranking of the solutions A and D.

In the case of WS-COSEA-MO, D has a higher win-

ning score than A, this is vice versa in the case of

dWS-COSEA-MO. Summarizing, the selection prin-

ciples of the winnings score–based algorithms are

comparable to the one of NSGA-III. Point B is only

selected by AnD. This point is potentially important

to maintain the diversity in the population, but the ex-

periments in this work show that this selection mech-

anism lacks of a suitable convergence within a very

low number of generations.

Furthermore, a three-dimensional example is used

to illustrate the characteristics of WS- and dWS-based

selection according to the characteristics mentioned

above: Eleven points are given, A(0.15,0.1, 0.08),

B(0.2,0.3,0.15), C (0.02, 0.3,0.7),

D(0.25,0.75,0.32), E(0.32, 0.27,0.81),

F(0.3,0.4, 0.25), G(0.28,0.35,0.31),

H(0.32,0.43,0.28), I(0.15,0.1,0.78),

K(0.17, 0.68,0.15) and L(0.18, 0.17,0.73). The win-

ning scores as well as the difference-based winnings

scores are determined and given in Table 1. Point A

has the highest score value in both cases followed by

point B. Points C, I, K and L are boundary points and

achieve an evolutionary advantage by good score val-

ues in the case of WS. In the case of dWS, the scores

of these points are positioned in the lower half of the

ranking. F an G are positioned very close to each

other and achieve very similar scores in both cases.

D and E are worse individuals and have the lowest

scores in both cases. Summarizing, the evolutionary

advantage of the boundary points in the case of WS is

the main difference between the winning score alter-

natives.

4 EXPERIMENTAL SETUP

The performance of the proposed WS-COSEA-MO

and dWS-COSEA-MO are compared to the re-

cently published AnD on four differently dimensional

molecular optimization problems according to the

convergence behavior and diversity. All experiments

are implemented in the open source jMetal library 4.5.

(Nebro and Durillo, 2018) and uses the open source

BioJava framework 4.2.0 (Prlic et al., 2018). Each

experiment is run 20 times on each molecular opti-

mization problem with 10 iterations and a population

size of 100. The individuals are 20-mer peptides com-

posed of the 20 canonical amino acids. Short peptides

of length 20 are of speciﬁc interest because of their fa-

A Winning Score-based Evolutionary Process for Multi-and Many-objective Peptide Optimization

Table 1: Winning Score values of WS-COSEA-MO and dWS-COSEA-MO.

A B C D E F G H I K L

WS 26 8 7 -14 -15 -6 -6 -13 8 2 2

dWS 3.3 1.8 -0.9 -1.7 -2.3 0.6 0.8 0.4 -1.0 -0.03 -0.9

vorable properties as drugs.

4.1 Molecular Optimization Problems

Four molecular optimization problems with 3 to 6 ob-

jective functions predicting physiochemical proper-

ties are used as experimental studies. Table 2 presents

the composed physiochemical optimization problems

with the used abbreviations: Needleman Wunsch Al-

gorithm (NMW), Molecular Weight (MW), Average

Hydrophilicity (Hydro), Instability Index (InstInd),

Isoelectric Point (pI) and Aliphatic Index (aI). These

molecular functions are provided by the BioJava li-

brary (Prlic et al., 2018). The physiochemical func-

tions are shortly described in the following, a de-

scription of the functions is given here (Prlic et al.,

2018): NMW is a well known and used method for

the global sequence alignment of a solution to a pre-

deﬁned reference individual. This algorithm refers to

the common hypothesis that a high similarity between

molecules refers to similar molecular properties.

MW is an important peptide property as a min-

imized MW ensures a good cell permeability. MW

of a peptide sequence a of length l is calculated

summarizing the mass of each amino acid (a

) plus a

water molecule:

MW (a) =

∑

i=1

mass(a

) + 17.0073(OH) +

1.0079(H), where O (oxygen) and H (hydrogen) are

the elements of the periodic system.

A common challenge of drug peptides is the sol-

ubility in aqueous solutions, especially peptides with

stretches of hydrophobic amino acids. Therefore, Hy-

dro is calculated by the hydrophilicity scale of Hopp

and Woods (Hopp and Woods, 1983) with a window

size equal to the peptide length l. An average hy-

drophilicity value is assigned to each candidate pep-

tide a using the scales for each amino acid a

Hydro(a) =

· (

∑

i=1

hydro(a

)).

The use of molecules as therapeutic agents is po-

tentially restricted by their instability and their poten-

tial degradation by enzymes in systemic application.

The stability is addressed by the InstInd as stability is

a very important feature of drug components. InstInd

is determined by the Dipeptide Instability Weight Val-

ues (DIWV) of each two consecutive amino acids in

the peptide sequence. DIWV are provided by the

GRP-Matrix (Guruprasad et al., 1990). These values

are summarized and the ﬁnal sum is normalized by

the peptide length l:

InstInd(a) =

l−1

∑

i=1

DIWV (a

i+1

pI of a peptide is characterized as the pH-value at

which a peptide has a net charge of zero. A peptide

has its lowest solubility at its pI. Therefore, the charge

of a peptide inﬂuence the solubility in aqueous solu-

tions. The pI value is calculated as follows: Firstly,

the net charge for pH = 7.0 is determined. If this

charge is positive, the pH at 7 +3.5 is calculated; oth-

erwise the pH at 7 − 3.5 is determined. This process

is repeated until the modules of the charge is less or

equal 0.0001.

aI of a peptide is characterized as the relative vol-

ume occupied by aliphatic side chains consisting of

the amino acids alanine (Ala), valine (Val), isoleucine

(Ile) and leucine (Leu). aI is regarded as a positive

factor for the increase of thermostability. aI is calcu-

lated according to the formula:

aI = X(Ala) + d · X (Val) + e · (X(Ile) + X(Leu)),

where X(Ala), X(Val), X (Ile) und X (Leu) are mole

percent of the amino acids. The coefﬁcients d and

e are the relative volume at the valine side chain

(d = 2.9) and Lei, Ile side chains (e = 3.9) to the side

chain Ala.

These six objective functions comparatively act

to reﬂect the similarity of a particular peptide and a

pre-deﬁned reference peptide: f (CandidatePept.) :=

| f (CandidatePept.) − f (ReferencePept.)|. Therefore,

the four objective functions have to be minimized

and the optimization problems are minimization prob-

lems. Furthermore, the objective values are normal-

ized by the theoretical maximal value of each objec-

tive:

) =

)

Max

for the k-objectives.

4.2 Performance Metrics

Two statistical metrics are chosen to evaluate the con-

vergence and diversity performance. These metrics

are applied on 10% approximately optimal individ-

uals in each iteration for all algorithms. These op-

timal individuals are determined by WS in all test

cases. The value of 10% optimized individuals is a pa-

rameter motivated by the number of peptides selected

for subsequent laboratory analysis and therefore mo-

tivated by the practical application. Furthermore, the

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

Table 2: Physiochemical functions of the different optimization problems.

dim. abbr. objective functions

3D 3D-MOP NMW, MW, Hydro

4D 4D-MaOP NMW, MW, Hydro, InstInd

5D 5D-MaOP NMW, MW, Hydro, InstInd, pI

6D 6D-MaOP NMW, MW, Hydro, InstInd, pI, aI

Figure 2: 3D-MOP: WS-COSEA-MO. Figure 3: 4D-MaOP: WS-COSEA-MO.

Figure 4: 3D-MOP: dWS-COSEA-MO. Figure 5: 4D-MaOP: dWS-COSEA-MO.

Figure 6: 3D-MOP: AnD. Figure 7: 4D-MaOP: AnD.

metrics are applied on the archives of dWS-COSEA-

MO and WS-COSEA-MO.

The Average Cuboid Volume (ACV) is used to

measure the convergence behavior (Rosenthal and

Borschbach, 2017a). ACV calculates the averaged

spanned space of each solution to an ideal reference

point, which is usually known in real-world applica-

tions. The ACV indicator is given by

ACV =

∑

i=1

(

∏

j=1

i j

− r

)), (1)

A Winning Score-based Evolutionary Process for Multi-and Many-objective Peptide Optimization

Figure 8: 5D-MaOP: WD-COSEA-MO. Figure 9: 6D-MaOP: WS-COSEA-MO.

Figure 10: 5D-MaOP: dWS-COSEA-MO. Figure 11: 6D-MaOP: dWS-COSEA-MO.

Figure 12: 5D-MaOP: AnD. Figure 13: 6D-MaOP: AnD.

where n is the number of individuals that are evalu-

ated, k the number of objectives and r

the ideal point.

The lower the ACV values, the better the convergence

behavior since the molecular optimization problems

have to be minimized. ACV as a simple statisti-

cal measure is preferred over traditional convergence

metrics since it is independent of Pareto optimal so-

lution sets which are usually unknown in real-world

applications, of low computation cost, independent of

the problem dimension and relative to the number of

solutions allowing a comparison of differently sized

archive sets.

A simple statistical evaluation method is used to

compare the diversity performance. The diversity is

determined by the standard deviation of the solution

set to the gravity point of this set.

4.3 Experimental Results

The performance results of WS-COSEA-MO, dWS-

COSEA-MO and AnD on 3D-MOP are depicted in

Figure 2, 4 and 6, the results of 4D-MaOP are shown

in Figure 3, 5 and 7, the results of 5D-MaOP are pre-

sented in Figure 8, 10 and 12 and those of 6D-MaOP

in Figure 9, 11 and 13. Generally, WS-COSEA-MO

and dWS-COSEA-MO provide a continuous con-

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

vergence improvement within 10 iterations, whereas

AnD does not provide any convergence behavior in

this low number of iterations in these molecular op-

timization problems except for a slight improvement

in 6D-MaOP, though these ACV values are worse

compared to those of the winning score-based algo-

rithms. Especially in the case of WS-COSEA-MO,

an exponential convergence improvement is observ-

able. As a consequence of the missing convergence,

AnD provides the highest diversity values in all test

cases. Moreover, AnD has the highest standard devia-

tion values of ACV and diversity indicating an highly

varying performance. Comparing dWS-COSEA-MO

and WS-COSEA-MO, it is observable that the tradi-

tional winning score-based algorithm provides a bet-

ter convergence behavior and the diversity is of a

higher level as well in all test cases.

The archive sizes of WS-COSEA-MO and dWS-

COSEA-MO after 10 iterations in each test case have

also been examined. Generally, the archive sizes

of both algorithms are in the same range of a 95%

conﬁdence interval between 32 and 47. The mean

of the archive sizes of both algorithms is the same,

avg = 39. Figure 14 depicts the archive performance

of WS-COSEA-MO and dWS-COSEA-MO after 10

iterations. As is has been expected from the previ-

ous results, WS-COSEA-MO provides higher quali-

ﬁed solutions than dWS-COSEA-MO, whose diver-

sity is better compared to dWS-COSEA-MO as well.

Since AnD does not converge within this low

number of 10 iterations, further experiments have

been applied with a higher iteration number of 100. In

the case of 3D-MOP to 5D-MaOP, no convergence is

observable wihin the 100 iterations, whereas conver-

gence behavior is observable in the case of 6D-MaOP.

As AnD has not been applied to optimization prob-

lems with less than 5 objectives so far, this allows the

hypothesis that AnD is only suitable for optimization

problems with a higher dimension number.

5 CONCLUSION

This work presents an enhancement of COSEA-MO

that is especially evolved for multi-objective molec-

ular optimization addressing the application-speciﬁc

condition of identifying highly qualiﬁed candidate

peptides while limiting the number of objective func-

tion evaluations to save resources. COSEA-MO is

enhanced by WS-based selection mechanism. Two

types of winning scores are used to and differen-

tiate the individuals of a population effectively in

multi- and many-objective molecular optimization

problems. The performance of WS-COSEA-MO and

Figure 14: (d)WS-COSEA-MO: Performance of archives

after 10 iterations.

dWS-COSEA-MO is compared to the recently pro-

posed and promising AnD in terms of convergence,

diversity and exemplary analysis of the selection prin-

ciples. AnD is chosen for benchmarking as it has a

similar properties compared to COSEA-MO and is

the only MaOEA apart from COSEA-MO that has a

simple framework structure, provides optimized de-

fault parameters for the non-expert use and is inde-

pendent of weight vectors or reference points, which

usually have a strong impact on the performance and

are usually unknown in real-world applications. WS-

COSEA-MO reveals superior performance in terms

of convergence and diversity in all test cases. It

complies the problem-speciﬁc requirement of allocat-

ing an evolutionary advantage to boundary solutions.

WS-COSEA-MO provides exponential convergence

improvement within the very low number of 10 iter-

ations. Otherwise, AnD does not reveal any conver-

gence behavior within this low number of iterations,

since diversity is the preference objective of this evo-

lutionary process at the cost of convergence.

In future research, the selection process is ana-

lyzed regarding to the important biochemical objec-

tive of genetic dissimilarity among the candidate pep-

tides. The improvement of dWS as part of the selec-

tion mechanism is in the focus for the speciﬁcation of

the search process. Furthermore, the prioritization of

different objectives in the evolutionary process is in

the focus and suitable method for this purpose will be

addressed. Moreover, since descriptor-based ﬁtness

functions are missing for diverse but important molec-

ular properties and the evaluation time of these molec-

ular properties have to be reduced, the proposed ap-

proach will be revised by surrogate-assisted principles

(Diaz-Manriquez et al., 2016) as pre-screening tech-

niques in advance of the expensive laboratory analysis

for further improvement.

A Winning Score-based Evolutionary Process for Multi-and Many-objective Peptide Optimization

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