Multi-objective Scatter Search with External Archive for Portfolio
Optimization
Khin Lwin
1,
, Rong Qu
1
and Jianhua Zheng
2
1
ASAP Research Group, School of Computer Science, University of Nottingham, NG8 1BB, Nottingham, U.K.
2
School of Computer Science, University of Nottingham, NG8 1BB, Nottingham, U.K.
Keywords:
Evolutionary Multi-objective Portfolio Optimization, Hybrid Metaheuristic, Multi-objective Scatter Search,
Cardinality Constrained Portfolio Selection Problem, Mean-Variance Portfolio Optimization.
Abstract:
The relevant literature showed that many heuristic techniques have been investigated for constrained portfolio
optimization problem but none of these studies presents multi-objective Scatter Search approach. In this work,
we present a hybrid multi-objective population-based evolutionary algorithm based on Scatter Search with
an external archive to solve the constrained portfolio selection problem. We considered the extended mean-
variance portfolio model with three practical constraints which limit the number of assets in a portfolio, restrict
the proportions of assets held in the portfolio and pre-assign specific assets in the portfolio. The proposed
hybrid metaheuristic algorithm follows the basic structure of the Scatter Search and defines the reference
set solutions based on Pareto dominance and crowding distance. New Subset generation and combination
methods are proposed to generate efficient and diversified portfolios. Hill Climbing operation is integrated
to search for improved portfolios. The performance of the proposed multi-objective Scatter Search algorithm
is compared with the Non-dominated Sorting Genetic Algorithm (NSGA-II), Strength Pareto Evolutionary
Algorithm (SPEA-2) and Pareto Envelope-based Selection Algorithm (PESA-II). Experimental results indicate
that the proposed algorithm is a promising approach for solving the constrained portfolio selection problem.
Measurements by the performance metrics indicate that it outperforms NSGA-II, SPEA2 and PESA-II on the
solution quality within a shorter computational time.
1 INTRODUCTION
The portfolio selection problem is concerned with the
optimal allocation of a limited capital among avail-
able risky assets. The mean-variance (MV) model of
Markowitz (Markowitz, 1952) (Markowitz, 1959) is
a single period static portfolio model based on two
criteria: to maximize the reward of a portfolio (mea-
sured by the mean of return), and to minimize the risk
of the portfolio (measured by the variance of return).
It is considered to be the first mathematical model that
formalizes the concept of diversification and is widely
used in finance. It helps risk-averse investors to con-
struct desirable portfolios by taking into considera-
tion of the trade-off between risk and expected return.
With the above stated two conflicting objectives, the
portfolio selection problem could be considered as a
class of multi-objective optimization problems. As a
consequence, a single unique solution that optimizes
all the conflicting objectives hardly exists and instead
there exists a set of solutions which are optimal in
such a way that no other solutions in the search space
are superior to them when all objectives are consid-
ered simultaneously. This set of solutions is known as
non-dominated or Pareto optimal set.
Although Markowitz’s MV model is the funda-
mental theory of Modern Portfolio Theory, direct
application of this model is not of much practical
uses mainly due to the fact that it is simplified with
unrealistic assumptions. In many practical situa-
tions, constraints such as cardinality, quantity and
pre-assignment constraints are required. The cardi-
nality constraint imposes a limit on the number of as-
sets in the portfolio either to simplify the management
of the portfolio or to reduce transaction costs. The
quantity constraint restricts the proportion of each as-
set in the portfolio to lie between the lower and upper
bounds in order to avoid very small (or large) and un-
realistic holdings. The pre-assignment constraint re-
quires certain asset(s) to be included in the portfolio.
These constraints are hard in the sense that they have
to be satisfied at any time. In this work, Markowitz’s
MV model extended with cardinality, quantity and
pre-assignment constraints is studied.
When the basic model is extended with such prac-
tical constraints, the problem transforms to an NP-
Hard problem (Moral-Escudero et al., 2006) and ex-
111
Lwin K., Qu R. and Zheng J..
Multi-objective Scatter Search with External Archive for Portfolio Optimization.
DOI: 10.5220/0004552501110119
In Proceedings of the 5th International Joint Conference on Computational Intelligence (ECTA-2013), pages 111-119
ISBN: 978-989-8565-77-8
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
act methods are inadequate to solve the problem in an
efficient way. In recent years, several researches have
been conducted applying metaheuristics to the portfo-
lio selection problem with real-world constraints. Al-
though the portfolio selection problem involves two
conflicting objectives, many researches in the litera-
ture have been performed with single objective ap-
proaches with aggregating function (Chang et al.,
2000) (Lwin and Qu, 2013). Recently, there are an
increasing number of multi-objective evolutionary ap-
proaches that have been studied. The main advantage
of evolutionary multi-objective portfolio optimization
is that an estimation of the efficient risk-return frontier
can be searched in a single run as opposed to the sin-
gle objective approaches with multiple runs. Several
researches have been conducted to solve the portfo-
lio optimization problem by adopting the evolutionary
multi-objective approach (Branke et al., 2009) (Skol-
padungket et al., 2007).
The aim of this paper is to apply the multi-
objective scatter search to the mean-variance portfo-
lio optimization problem. The rest of the paper is
organized as follows. Section 2 provides the mean-
variance model with cardinality, quantity and pre-
assignment constraints considered in this study. Sec-
tion 3 presents the multi-objective Scatter Search al-
gorithm with external archive and describes the de-
tails of its components. Section 4 is provided with nu-
merical results and conclusion remarks are presented
in Section 5.
2 MULTI-OBJECTIVE
CONSTRAINED PORTFOLIO
OPTIMIZATION
The portfolio optimization problem concerns the
choice of an optimal set of assets to include in the
portfolio and the distribution of investment among
them. In the multi-criteria variant of portfolio opti-
mization problem, the model is usually formalized as
follows:
min f
1
=
N
i=1
N
j=1
w
i
w
j
σ
i j
(1)
max f
2
=
N
i=1
w
i
µ
i
(2)
sub ject to
N
i=1
w
i
= 1 (3)
N
i=1
s
i
K, (4)
ε
i
s
i
w
i
δ
i
s
i
, i = 1, ..., N, (5)
s
i
z
i
, i = 1, ..., N (6)
s
i
{
0,1
}
, i = 1, ..., N (7)
z
i
{
0,1
}
, i = 1, ..., N (8)
where N is the number of available assets, µ
i
is the
expected return of asset i (i = 1...N), σ
i j
is the covari-
ance between assets i and j (i = 1...N; j = 1...N), and
w
i
(0 w
i
1) is the decision variable which repre-
sents the proportion held of asset i. Eq(3) defines the
budget constraint (all the money available should be
invested) for a feasible portfolio. K is the maximum
number of invested assets in the portfolio, s
i
denotes
whether asset i is invested or not, z
i
denotes a binary
value such that z
i
is set to one if asset i is included
in the pre-assigned set in the portfolio. If s
i
equals to
one, asset i is chosen to be invested and the proportion
of capital w
i
lies in [ε
i
, δ
i
], where 0 ε
i
δ
i
1. Oth-
erwise, asset i is not invested and w
i
equals to zero.
3 MULTI-OBJECTIVE SCATTER
SEARCH WITH EXTERNAL
ARCHIVE (MOSSwA)
In this paper, a multi-objective Scatter Search with
external archive has been developed for constrained
portfolio selection problem. Scatter Search (SS)
(Glover et al., 2000) is a population based meta-
heuristic that uses sets of solutions called the refer-
ence sets to combine its solutions to construct others.
The method generates reference sets from a popula-
tion of solutions. Then a subset of solutions is se-
lected from the reference sets. The selected solutions
are combined to generate new solutions. The gener-
ated solutions are enhanced by an improvement pro-
cedure. The result of the improved solution can mo-
tivate the updating of the reference sets. The pro-
posed multi-objective Scatter Search with external
archive (MOSSwA) consists of procedures namely,
initial population creation, reference sets formation,
subset generation, solution combination, improve so-
lution, reference set update and archive update. The
detailed procedure of the MOSSwA is provided in
Figure 1.
3.1 Solution Representation
In our solution representation, two vectors of size N
are used to define a portfolio p: a binary vector s
i
,
i = 1,...,N denoting whether asset i is included in the
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
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Let
NP = number of individuals in the population
M = number of portfolio(s) in the archive
X = number of portfolio(s) in the reference set
A = the archive maintaining the M
non-dominated portfolio(s) found so far
re f 1 = the reference set maintaining the X
non-dominated portfolio(s)
re f 2 = the reference set maintaining the X
least crowded portfolio(s)
Sub = the subset that is used for generating
new trial solutions
t pop = list of portfolios in the trial population
r[x,y] = uniform random real-value between x
and y inclusive
portfolio, and a real-value vector w
i
, i = 1,...,N rep-
resenting the proportions of the capital invested in the
assets.
3.2 Initial Population Generation
Method
To generate a trial initial population, maximum K
different indexes (including all assets in the pre-
assignment subset) are randomly selected and propor-
tions are assigned to those selected assets randomly.
If the generated portfolio violates the budget and/or
quantity constraints, such solution is corrected by the
constraint handling techniques provided in Section
3.9. Hence, all generated solutions in the trial pop-
ulation are feasible.
3.3 Reference Set Formation
The reference sets are initialized from an initial pop-
ulation composed of random solutions and they are
updated by the improved and diverse solutions. Two
reference sets are maintained: ref1 and ref2. The
ref1 and ref2 sets maintain the best solutions and the
least crowded solutions from the archive (A) and the
trial population(t pop) by Pareto optimal (Fonseca and
Fleming, 1995) and the crowding distance (Deb et al.,
2002) concepts respectively.
A solution a is efficient (i.e., Pareto optimal) if
there does not exist any solution b such that b domi-
nates a. Solution a is considered to dominate solution
b if and only if:
f
1
(a) f
1
(b) AND f
2
(a) > f
2
(b)
OR
f
2
(a) f
2
(b) AND f
1
(a) < f
1
(b)
Pseudocode: MOSSwA
BEGIN
INITIALIZATION:
for each portfolio p
j
, j := 1 to NP do
t pop randomly generate an individual with pre-
assigned asset(s) included; (if infeasible)
repair by constraint handling techniques
(see Section-3.9)
end for
Repeat until certain number of generations
ARCHIVE:
A update archive with M non-dominated
portfolio(s) found so far
REF SET:
re f 1 select the X non-dominated portfolio(s) from
archive A
re f 2 select the X non-redundant and least crowded
portfolio(s) from the remaining set of
portfolios in archive A and from t pop
(see Section-3.8)
for j := 1 to NP do
SUBSET:
Sub select random |Sub| portfolio(s) from
re f 1 and/or re f 2
COMBINE:
for i := 1 to |Sub| 1 do
randomly select three portfolios from Sub
p
i
generate a new portfolio with combination
method (see Section-3.5); (if infeasible)
repair by constraint handling techniques
end for
IMPROVE:
p
new
perform local search on p
i
for certain number of moves
if p
i
dominates p
new
t pop p
i
else if p
new
dominates p
i
t pop p
new
else
t pop p
new
; t pop p
i
end if
end for
END
Figure 1: Pseudocode of the proposed MOSSwA.
The crowding distance is used to measure the ap-
proximate density of solutions toward each other in a
population. The crowding distance of a portfolio p is
calculated by taking the average distance of the two
portfolios taken one from either side of p (Deb et al.,
2002).
Multi-objectiveScatterSearchwithExternalArchiveforPortfolioOptimization
113
3.4 Subset Generation Method
This method selects a subset of solutions from the ref-
erence set to create a subset (Sub). This generated
subset (Sub) will later be used in solution combina-
tion method to produce new solutions. Our approach
generates the subset by three different variants as fol-
lows:
1. randomly select X solutions from re f 1.
2. randomly select X solutions from re f 2.
3. randomly select arbitrary R solutions from re f 1
and X R solutions from re f 2 where R X.
3.5 Solution Combination Method
This method uses the generated subset Sub (see Sec-
tion 3.4) and combines solutions from Sub and returns
one or more trial solutions. In finance literature, it is
considered to be a fundamental premise to utilize as-
sets that have low correlation with each other. Hence
the assets which are less correlated to each other are
preferable to the heavily correlated assets. It is also
commonly believed that it is beneficial to reduce the
portfolio’s standard deviation of return. Intuitively,
investors prefer higher return assets with less risk.
By taking into accounts of the above stated intu-
itive appeal, in this work, we propose three problem-
specific combination mechanisms for the constrained
portfolio selection problem as follows: Three portfo-
lios (p1, p2 and p3) from subset Sub are selected to
use in the combination mechanism. The assets se-
lected in these three portfolios are observed and anal-
ysed. The set potential is constructed by inserting all
the indexes of assets which are selected in at least two
out of the three portfolios. A new portfolio is then
constructed by selecting random n indexes of assets
in potential set where n K. Like the DE offspring
generation scheme investigated in (Robi
ˇ
c and Filipi
ˇ
c,
2005), the proportion w of those selected assets for
the new solution are assigned as follows:
w
i
:= w3
i
+ r[0, 1] × (w1
i
w2
i
)
The remaining assets of the new portfolio are selected
by one of the following heuristic methods in selection
of asset:
HigherExpRet: K - n assets which have the highest
expected return values are selected in the new so-
lution and the proportions of those selected assets
are randomly assigned.
LessSTD: K - n assets which have the least standard
deviation values are selected in the new solution
and the proportions of those selected assets are
randomly assigned.
LeastCov: K - n assets which have the lowest corre-
lation values are selected in the new solution and
the proportions of those selected assets are ran-
domly assigned.
It may be noted that the combination mechanisms
construct solutions that may or may not be feasible.
Hence, the repair mechanism is employed if the newly
generated solution violates the constraints. (see Sec-
tion 3.9)
3.6 Improvement Method
The solutions obtained by the combination method
(see Section 3.5) are improved by a local search tech-
nique (i.e., hill climbing). It simply moves the current
solution to a better candidate solution by using muta-
tion operation. This is done by randomly altering a
proportion of an asset or a selection of the asset in the
portfolio until no improved solutions can be found in
a certain number of moves. Then the trial population
is updated by adding those non-dominated solutions.
3.7 Maintaining the External Archive
The main objective of the external archive (A) is
to keep all the non-dominated solutions encountered
along the search process. This approach is adopted
in order to save and update all well spread non-
dominated solutions generated by the algorithm dur-
ing the search.
In each generation, the archive (A) is updated with
the non-dominated solutions from the trial population.
The computational time of maintaining the archive in-
creases with the archive size. Hence, the size of the
archive is restricted to a prespecified value. When the
external archive has reached its maximum capacity
(M), the crowding distances of the solutions are cal-
culated to delete the most crowded archive members.
3.8 Reference Set Update Method
Once the archive (A) has been updated with non-
dominated solutions obtained from the trial popu-
lation (t pop), the reference sets are updated. For
our proposed algorithm for the portfolio optimization
problem, the reference set update method has been
devised as follows.
The non-dominated solutions obtained from the
archive (A) are sorted by using the Pareto dominance.
The X best non-dominated solutions obtained from
the archive (A) are updated to ref1. Similarly, the so-
lutions in the trial population are sorted by the crowd-
ing distance. The X least crowded solutions obtained
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
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from the sorted archive (A) and trial (t pop) solutions
are updated to ref2.
3.9 Constraint Handling
During the population sampling, each constructed in-
dividual must be repaired if the representative port-
folio does not satisfy the constraints of the problem.
The budget constraint in Eq.(3) is satisfied by firstly
normalizing the weights: w
i
= w
i
/
N
j=1
w
j
over those
assets selected. Moreover, the bounding constraint
in Eq.(5) requires the proportion of asset i to be in
the range [ε
i
, δ
i
]. If the proportion of asset after the
normalization violates the quantity constraints, then it
employs repair mechanism described in (Chang et al.,
2000).
4 EXPERIMENTAL RESULTS
In this work, we compared the performance of the
proposed algorithm with the Non-dominated Sort-
ing Genetic Algorithm (NSGA-II) (Deb et al., 2002),
the Strength Pareto Evolutionary Algorithm (SPEA2)
(Zitzler et al., 2001) and Pareto Envelope-based Se-
lection Algorithm (PESA-II) (Corne et al., 2001).
Twenty independent runs were performed for all
experiments and the same random seed is assigned to
each set of the sum so that the algorithms start with
the same initial population. In order to ensure a fair
comparison, we have used the same population size
and archive size (if applicable) for all the algorithms
tested in this work. We have chosen to run all the al-
gorithms for the same stopping criteria (i.e. the same
number of evaluations) to generate the Pareto front.
Each algorithm also uses the same encodings (see
Section 3.1) and repair mechanism (see Section 3.9)
when a newly constructed portfolio violates the con-
sidered constraints. NSGA-II uses binary tournament
selection based on the crowding distance. NSGA-II,
SPEA2 and PESA-II use simulated binary crossover
and polynomial mutation evolutionary operators.
Before the experiments were performed, param-
eters are tuned for all algorithms using the small-
est problem instance, i.e. Hang Seng. The param-
eter value of the compared algorithms are provided
in Table 1. For constraint values, we use K = 10,
ε
i
= 0.01(i = 1,...,N), δ
i
= 1.0(i = 1,...,N) and z
i
=
1(i = 30).
Table 1: Parameter setting of for four algorithms.
Parameters MODEwAwL NSGA-II SPEA2 PESA-II
Number of Population (NP) 100 100 100 100
Number of Generation 10,000N 10,000N 10,000N 10,000N
Scaling Factor (F) 0.3 - - -
Crossover Probability (CR) 0.9 0.9 0.9 0.9
Crossover Distribution Index - 20 20 20
Mutation Probability - 1/N 1/N 1/N
Mutation Distribution Index - 20 20 20
Tournament Round - - 1 -
Number of Bisection - - - 5
Archive Size (M) 100 - 100 100
4.1 Dataset
A test data for the portfolio optimization problems
from the OR-library (Beasley, 1990) is used to
evaluate the performance of the algorithms described
above. These datasets contain the estimated returns
and the covariance matrix of ve different stock
market indices: Hang Seng in Hong Kong, DAX
100 in Germany, FTSE 100 in UK, S&P 100 in
USA and Nikkei 225 in Japan. For each set of
the test data, the number of assets N is 31, 85, 89,
98 and 225, respectively. In the current literature
of portfolio selection problem, this set of dataset
has been widely tested, and is recognized as the
benchmark to evaluate computational algorithms.
All information of the dataset itself and their best
known solutions can be accessed online at the URL
http://people.brunel.ac.uk/
mastjjb/jeb/orlib/portinfo
.html.
4.2 Performance Metric
To evaluate the performance of the proposed algo-
rithm for the constrained portfolio optimization prob-
lem, we use three criteria: Inverted Generational Dis-
tance, Generational Distance and Diversity metric
().
4.2.1 Inverted Generational Distance (IGD)
The Inverted Generational Distance (Sierra and
Coello, 2005) uses the true Pareto front as a refer-
ence and measures the distance from the true Pareto
front to the Pareto front obtained by an algorithm. It
is mathematically defined as:
IGD =
s
Q
i=1
d
2
i
Q
where Q is the number of solutions in the true Pareto
front and d
i
is the Euclidean distance between each
of the solution and the nearest member from the set
of non-dominated solutions found by the algorithm.
Multi-objectiveScatterSearchwithExternalArchiveforPortfolioOptimization
115
Figure 2: Performance Comparisons of the algorithms in term of GD, IGD and metrics for Hang Seng.
Figure 3: Performance Comparisons of the algorithms in term of GD, IGD and metrics for DAX 100.
Figure 4: Performance Comparisons of the algorithms in term of GD, IGD and metrics for FTSE 100.
Figure 5: Performance Comparisons of the algorithms in term of GD, IGD and metrics for S&P 100.
This metric measures both the diversity and the con-
vergence of an obtained non-dominated solution set.
The smaller the value of this metric, the closer the ob-
tained front is to the true Pareto front.
4.2.2 Generational Distance (GD)
The Generational Distance (Van Veldhuizen and La-
mont, 1998) is a variant of the IGD. It measures
how far the solutions of the computed Pareto front
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
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Figure 6: Performance Comparisons of the algorithms in term of GD, IGD and metrics for Nikkei.
Figure 8: Comparison of Obtained Efficient Frontier of all the algorithms for Constrained Portfolio Optimization Probelm.
Figure 7: Running Time Measures of the algorithms for
Constrained Portfolio Optimization Problem.
obtained by an algorithm are from those at the true
Pareto front. The smaller value indicates that all the
generated solutions are on the true Pareto front.
4.2.3 Spread ()
The Diversity metric () (Deb et al., 2002) mea-
sures the extent of spread achieved among the non-
dominated solutions as follows:
=
d
f
+ d
l
+
|Q|−1
i=1
|d
i
¯
d|
d
f
+ d
l
+ (|Q| 1)
¯
d
where d
i
is the Euclidean distance in the objective
space between consecutive solutions in the obtained
non-dominated front Q, and
¯
d is the average of these
distances. The parameters d
f
and d
l
are the Euclidean
distance between the extreme solutions of the Pareto
front P
and the boundary solutions of the obtained
front Q. The lower value of the spread () indicates a
better diversity.
4.2.4 Comparisons of the Algorithms
The experimental results of GD, IGD and of the
four algorithms performed on five datasets from OR-
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117
Table 2: Student t-Test Results of Different Algorithms on five problem instances from OR-Library.
Algorithm1 Algorithm2 Hang Seng DAX 100 FTSE 100 S & P 100 Nikkei
MOSSwA NSGA-II + + + + +
MOSSwA SPEA2 + + + + +
MOSSwA PESA-II + + + + +
NSGA-II SPEA2 +
NSGA-II PESA-II + + + + +
SPEA2 PESA-II + + + + +
library (see Section 4.1) are shown in Figure 2, Fig-
ure 3, Figure 4, Figure 5 and Figure 6. The running
time of the algorithms are shown in Figure 7. The re-
sults showed that our proposed algorithm (MOSSwA)
is not only superior in performance measures but
also is efficient in computational time compared with
NSGA-II, SPEA2 and PESA2.
For illustrative purpose, four obtained efficient
frontiers of the algorithms from a single run for ve
problem instances along with the true unconstrained
efficient frontier (UCEF) are provided in Figure 8.
To further support our observation that MOSSwA
outperforms others, we compare the IGD values of
the five algorithms by using Student’s t-test (Walpole
et al., 1998). The statistical results obtained by a
two-tailed t-test with 38 degrees of freedom at a 0.05
level of significance are given in Table 2. The re-
sult of Algorithm 1 Algorithm 2 is shown as ”+” ,
”, or when Algorithm 1 is significantly better
than, significantly worse than, or statistically equiva-
lent to Algorithm 2, respectively. Results show that
MOSSwA outperforms other algorithms in all prob-
lem instances. We therefore can conclude that the
proposed MOSSwA has the best optimization per-
formance for the portfolio optimization problem with
considered constraints.
5 CONCLUSIONS
In this work, we presented a multi-objective evolu-
tionary algorithm based on Scatter Search with exter-
nal archive (MOSSwA) for solving the mean-variance
portfolio selection problem with cardinality, quantity
and pre-assignment constraints. Experimental results
indicate that the proposed adapted multi-objective
Scatter Search algorithm (MOSSwA) outperforms the
Non-dominated Sorting Genetic Algorithm (NSGA-
II), Strength Pareto Evolutionary Algorithm (SPEA-
2) and Pareto Envelope-based Selection Algorithm
(PESA-II) in all performance measures. The pro-
posed hybrid metaheuristic algorithm follows the ba-
sic structure of the Scatter Search and defines the
reference set solutions with the Pareto dominance
and crowding distance concepts. New subset gener-
ation and combination methods are proposed to con-
tribute to the literature in order to generate efficient
and diversified portfolios. When cardinality and pre-
assignment constraints are considered, the proposed
three combination mechanisms enhance the solution
quality significantly in terms of both the computa-
tional time and all the measures of solution quality.
REFERENCES
Beasley, J. (1990). Or-library: Distributing test problems by
electronic mail. Journal of the Operational Research
Society, 41(11):1069–1072.
Branke, J., Scheckenbach, B., Stein, M., Deb, K., and
Schmeck, H. (2009). Portfolio optimization with
an envelope-based multi-objective evolutionary algo-
rithm. European Journal of Operational Research,
199(3):684–693.
Chang, T., Meade, N., Beasley, J., and Sharaiha, Y. (2000).
Heuristics for cardinality constrained portfolio op-
timisation. Computers and Operations Research,
27(13):1271–1302.
Corne, D. W., Jerram, N. R., Knowles, J. D., Oates, M. J.,
et al. (2001). Pesa-ii: Region-based selection in evo-
lutionary multiobjective optimization. In Proceedings
of the Genetic and Evolutionary Computation Confer-
ence (GECCO2001. Citeseer.
Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002).
A fast and elitist multiobjective genetic algorithm:
Nsga-ii. Evolutionary Computation, IEEE Transac-
tions on, 6(2):182–197.
Fonseca, C. and Fleming, P. (1995). An overview of evo-
lutionary algorithms in multiobjective optimization.
Evolutionary computation, 3(1):1–16.
Glover, F., Laguna, M., and Mart
´
ı, R. (2000). Fundamen-
tals of scatter search and path relinking. Control and
cybernetics, 39(3):653–684.
Lwin, K. and Qu, R. (2013). A hybrid algorithm for
constrained portfolio selection problems. Applied
Intelligence,DOI:10.1007/s10489-012-0411-7.
Markowitz, H. (1952). Portfolio selection. The Journal of
Finance, 7(1):pp. 77–91.
Markowitz, H. (1959). Portfolio selection: Efficient diver-
sification of investments. John Wiley and Sons, New
York.
Moral-Escudero, R., Ruiz-Torrubiano, R., and Suarez, A.
(2006). Selection of optimal investment portfolios
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
118
with cardinality constraints. In Evolutionary Com-
putation, 2006. CEC 2006. IEEE Congress on, pages
2382–2388. IEEE.
Robi
ˇ
c, T. and Filipi
ˇ
c, B. (2005). Demo: Differential
evolution for multiobjective optimization. In Evolu-
tionary Multi-Criterion Optimization, pages 520–533.
Springer.
Sierra, M. R. and Coello, C. A. C. (2005). Improving
pso-based multi-objective optimization using crowd-
ing, mutation and epsilon-dominance. In EMO’05,
pages 505–519.
Skolpadungket, P., Dahal, K., and Harnpornchai, N. (2007).
Portfolio optimization using multi-obj ective genetic
algorithms. In Evolutionary Computation, 2007. CEC
2007. IEEE Congress on, pages 516–523. IEEE.
Van Veldhuizen, D. A. and Lamont, G. B. (1998). Multiob-
jective evolutionary algorithm research: A history and
analysis. Technical report, Citeseer.
Walpole, R. E., Myers, R. H., Myers, S. L., and Ye, K.
(1998). Probability and statistics for engineers and
scientists, volume 8. Prentice Hall Upper Saddle
River, NJ:.
Zitzler, E., Laumanns, M., Thiele, L., Zitzler, E., Zitzler, E.,
Thiele, L., and Thiele, L. (2001). Spea2: Improving
the strength pareto evolutionary algorithm.
Multi-objectiveScatterSearchwithExternalArchiveforPortfolioOptimization
119