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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017 731
Opposition-Based Memetic Search for
the Maximum Diversity Problem
Yangming Zhou, Jin-Kao Hao, and Béatrice Duval
Abstract—As a usual model for a variety of practical
applications, the maximum diversity problem (MDP) is computa-
tional challenging. In this paper, we present an opposition-based
memetic algorithm (OBMA) for solving MDP, which integrates
the concept of opposition-based learning (OBL) into the well-
known memetic search framework. OBMA explores both candi-
date solutions and their opposite solutions during its initialization
and evolution processes. Combined with a powerful local opti-
mization procedure and a rank-based quality-and-distance pool
updating strategy, OBMA establishes a suitable balance between
exploration and exploitation of its search process. Computational
results on 80 popular MDP benchmark instances show that the
proposed algorithm matches the best-known solutions for most of
instances, and finds improved best solutions (new lower bounds)
for 22 instances. We provide experimental evidences to highlight
the beneficial effect of OBL for solving MDP.
Index Terms—Learning-based optimization, maximum diver-
sity, memetic search, opposition-based learning (OBL), tabu
search (TS).
I. INTRODUCTION
GIVEN a set Nof nelements where any pair of elements
are separated by a distance, the maximum diversity
problem (MDP) aims to select a subset Sof m(mis given
and m<n) elements from Nin such a way that the sum of
pairwise distances between any two elements in Sis maxi-
mized. Let N={e1,e2,...,en}be the given set of elements
and dij ∈Rbe the distance between eiand ej(dij =dji).
Formally, MDP can be formulated as the following quadratic
binary problem [28]:
max f(x)=1
2
n
i=1
n
j=1
dijxixj(1)
s.t.
n
i=1
xi=m(2)
x∈{0,1}n(3)
Manuscript received September 17, 2016; revised January 2, 2017
and February 8, 2017; accepted February 13, 2017. Date of publica-
tion February 24, 2017; date of current version September 28, 2017.
(Corresponding author: Jin-Kao Hao.)
Y. Zhou and B. Duval are with the Department of Computer
Science, LERIA, University of Angers, 49045 Angers, France (e-mail:
yangming@info.univ-angers.fr;bd@info.univ-angers.fr).
J.-K. Hao is with the Department of Computer Science, LERIA
Laboratory, University of Angers, 49045 Angers, France, and also
with the Institut Universitaire de France, 75231 Paris, France (e-mail:
hao@info.univ-angers.fr).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEVC.2017.2674800
where the binary variable xk=1(k=1,...,n) if element
ek∈Nis selected; and xk=0 otherwise. Equation (2)
ensures that a feasible solution xexactly contains m
elements.
MDP belongs to a large family of diversity or dispersion
problems whose purpose is to identify a subset Sfrom a set
Nof elements while optimizing an objective function defined
over the distances between the elements in S[39]. Over the
past three decades, MDP has been widely studied under dif-
ferent names, such as max-avg dispersion [46], edge-weighted
clique [32], dense k-subgraph [15], maximum edge-weighted
subgraph [33], and equitable dispersion [39]. In addition, MDP
also proves to be a useful model to formulate a variety of
practical applications including facility location, molecular
structure design, agricultural breeding stocks, composing jury
panels and product design [22], [35]. In terms of computational
complexity, MDP is known to be NP-hard [18].
Given the interest of MDP, a large number of solution
methods for MDP have been investigated. These methods
can be divided into two main categories: 1) exact algorithms
and 2) heuristic algorithms. In particular, exact algorithms
like [3] and [34] are usually effective on small instances
with n<150. To handle larger instances, heuristic algo-
rithms are often preferred to find suboptimal solutions in
an acceptable time frame. Existing heuristic algorithms for
MDP include construction methods [18], [22], greedy ran-
domized adaptive search procedure (GRASP) [2], [13], [47],
iterative tabu search (ITS) [38], variable neighborhood
search (VNS) [4], [8], fine-tuning iterated greedy algo-
rithm (TIG) [30], and memetic and hybrid evolutionary
algorithms (MSES [11], G_SS [17], MAMDP [53], and
TS/MA [52]). Comprehensive surveys and comparisons of
some important heuristic algorithms prior to 2012 for MDP
can be found in [4] and [35].
Recently, research into enhancing search algorithms via
machine learning techniques has gained increasing interest
in artificial intelligence and operations research. Machine
learning is one of the most promising and salient research
areas in artificial intelligence, which has experienced a rapid
development and has become a powerful tool for a wide
range of applications. Researchers have made much effort
on using machine learning techniques to design, analyze, and
select heuristics to solve large-scale combinatorial search prob-
lems [6], [26], [29], [45], [55]. Among the existing heuristics
for MDP, two methods involve hybridization of heuristics
and machine learning techniques. In [47], the proposed
GRASP_DM algorithm combines GRASP with data mining
1089-778X c
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732 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017
technique (i.e., frequent itemset mining). After each GRASP
phase, the data mining process extracts useful patterns from
recorded elite solutions to guide the following GRASP itera-
tions. These patterns correspond to items that are shared by a
significant number of elite solutions. Another learning-based
heuristic is LTS-EDA [51], which uses data mining techniques
(k-means clustering and estimation of distribution algorithms)
to extract useful information from the search history of tabu
search (TS) in order to guide the search procedure to promising
search regions. These learning-based methods have reported
competitive results when they were published.
In this paper, we propose a new learning-based optimiza-
tion method for solving MDP. The proposed “opposition-
based memetic algorithm (OBMA)” integrates the concept of
opposition-based learning (OBL) into the popular memetic
algorithm (MA) framework. OBMA brings several improve-
ments into the original MA framework. First, we employ OBL
to reinforce population initialization as well as the evolution-
ary search process, by simultaneously considering a candidate
solution and its corresponding opposite solution. Second, we
apply a TS procedure for local optimization which relies on an
improved parametric constrained neighborhood. Third, we pro-
pose a rank-based quality-and-distance (RBQD) pool updating
strategy to maintain a healthy population diversity. We identify
the main contributions of this paper as follows.
1) From an algorithmic perspective, we explore for the first
time the usefulness of OBL to enhance a popular method
(i.e., MA) for combinatorial optimization. We investigate
how OBL can be beneficially integrated into the MA
framework and show the effectiveness of the approach
within the context of solving the MDP.
2) From a computational perspective, we compare the
proposed OBMA algorithm with state-of-the-art results
on several sets of 80 large size MDP benchmark
instances with 2000–5000 elements. Our results indicate
that OBMA matches most of the best-known results and
in particular finds improved best solutions (new lower
bounds) for 22 instances. These new bounds are valuable
for the assessment of new MDP algorithms. These com-
putational results demonstrate the competitiveness of
OBMA and the benefit of using OBL to enhance an MA.
The reminder of this paper is organized as follows.
After a brief introduction of OBL and memetic search in
Section II, we present the proposed OBMA in Section III.
Sections IV and Vshow computational results and com-
parisons as well as an experimental study of key issues of
the proposed algorithm. The conclusions and perspective are
provided in Section VI.
II. BACKGROUND
This section introduces the concept of OBL and the general
memetic search framework, which are then combined in the
proposed approach.
A. Opposition-Based Learning
OBL was originally proposed as a machine intelligence
scheme for reinforcement learning [49]. The main idea behind
OBL is the simultaneous consideration of a candidate solu-
tion and its corresponding opposite solution. To explain the
concept of opposition, we consider a real number x∈[a,b],
then the opposite number xis defined as x=a+b−x.
For the case of MDP, we define the concept of opposite
solution in Section III. OBL is a fast growing research field
in which a variety of new theoretical models and technical
methods have been studied to deal with complex and signifi-
cant problems [1], [40], [50], [54]. Recently, the idea of OBL
has also been used to reinforce several global optimization
methods such as differential evolution, particle swarm opti-
mization, biogeography-based optimization, artificial neural
network, and bee and ant colony optimization [5], [54].
To apply OBL to solve an optimization problem, one needs
to answer a fundamental question: given a solution from the
search space, why is it more advantageous to consider an oppo-
site solution of the current solution than a second pure random
solution? For 1-D search space, a proof and an empirical evi-
dence confirmed how much an opposite solution is better than
a uniformly generated random solution [41]. This result was
further generalized to the N-dimensional search spaces for
black-box (continuous) problems in [42].
We observe that existing studies on OBL-based optimiza-
tion concerns only global optimization with two exceptions.
In 2008, Ventresca and Tizhoosh [50] proposed a diversity
maintaining population-based incremental learning algorithm
for solving the traveling salesman problem (TSP), where
the concept of opposition was used to control the amount
of diversity within a given sample population. In 2011,
Ergezer and Simon [14] hybridized open-path opposition and
circular opposition with biogeography-based optimization for
solving the graph coloring problem and TSP. The main dif-
ficulty of these applications is how to define and evaluate
opposite solutions in a discrete space. OBL being a generally
applicable technique, its efficiency depends on the matching
degree between the definition of OBL and the solution space
of the considered problem, as well as the rationality justifying
a combination of OBL with a search algorithm [54].
B. Memetic Algorithm
The MA framework [27], [36] is a well-known hybrid
search approach combining population-based search and local
optimization. MA has been successfully applied to tackle
numerous classical NP-hard problems [9], [24], such as
graph coloring [31], graph partitioning [7], [16] and gen-
eralized quadratic multiple knapsack [10]aswellasthe
MDP [11], [53].
A typical MA algorithm (Algorithm 1) begins with a set of
random or constructed solutions (initial population). At each
generation, MA selects two or more parent solutions from the
population, and performs a recombination or crossover oper-
ation to generate one or more offspring solutions. Then a
local optimization procedure is invoked to improve the off-
spring solution(s). Finally, a population management strategy
is applied to decide if each improved offspring solution is
accepted to join the population. The process repeats until a
stopping condition is satisfied. We show below how OBL and
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 733
Algorithm 1 General MA Framework
1: Input: a problem instance and population size p
2: Output: the best solution S∗found
// Build an initial population
3: P={S1,S2,...,Sp}←PoolInitialization ()
// We suppose it is a maximization problem, record the best
solution found so far
4: S∗=arg max{f(Si)|i=1,2,...,p}
5: while a stopping condition is not reached do
6: (Si,...,Sj)←ParentsSelection (P)
// Generate an offspring solution
7: So←CrossoverOperator(Si,...,Sj)
// Improve the offspring solution
8: So←LocalImprovement(So)
// Accept or discard the improved solution
9: P←UpdatePopulation (P,So)
// Update the best solution found
10: if f(So)>f(S∗)then
11: S∗←So
12: end if
13: end while
MA can be combined to obtain a powerful search algorithm
for the highly combinatorial MDP.
III. OPPOSITION-BASED MEMETIC
ALGORITHM FOR MDP
We describe in this section the proposed OBMA for MDP.
We start with the issues of solution representation and search
space, followed by a detailed presentation of the ingredients
of the proposed approach.
A. Solution Representation and Search Space
Given an MDP instance with set N={e1,e2,...,en}
and integer m, any subset S⊂Nof size mis a feasible
solution. A candidate solution Scan then be represented by
S={eS(1),eS(2),...,eS(m)}such that S(i)is the index of ele-
ment iin Nor equivalently by a binary vector of size of n
such that exactly mvariables receive the value of 1 and the
other n−mvariables receive the value of 0. The quality of
a candidate solution Sis assessed by the objective function f
of (1).
Given an MDP instance, the search space is composed
of all the m-element subsets of N, i.e., ={S⊂N:
|S|=m}. The size of is given by n
m=[n!/(m!(n−m)!)]
and increases extremely fast with nand m.
B. Main Scheme
The proposed OBMA algorithm for MDP is based on oppo-
sition learning, which relies on the key concept of opposite
solution in the context of MDP, which is defined as follows.
Definition 1 (Opposite Solution): Given an MDP instance
with set Nand integer m,letSbe a feasible solution of
represented by its binary n-vector x, an opposite solution S
corresponds to a feasible binary vector xwhose components
match the complement1of xas closely as possible.
1Let x∈{0,1}n, its complement xis an n-vector such that x[i]=1if
x[i]=0; x[i]=0ifx[i]=1.
Algorithm 2 OBMA for MDP
1: Input:an×ndistance matrix (dij), and an integer m<n
2: Output: the best solution S∗found
// Build an initial population, Section III-C
3: P={S1,S2,...,Sp}←OppositionBasedPoolInitialize()
4: S∗←arg max{f(Si):i=1,2,...,p}
5: while a stopping condition is not reached do
6: Randomly select two parent solutions Siand Sjfrom P
// Generate an offspring solution and its opposite solution
by crossover, Section III-E
7: So,So←BackboneBasedCrossover(Si,Sj)
// Perform a double trajectory search, Section III-D
8: So←TabuSearch(So)/∗trajectory 1: search around So∗/
9: if f(So)>f(S∗)then
10: S∗=So
11: end if
// Insert or discard the improved solution, Section III-F
12: P←RankBasedPoolUpdating (P,So)
13: So←TabuSearch(So)/∗trajectory 2: search around So∗/
14: if f(So)>f(S∗)then
15: S∗=So
16: end if
// Insert or discard the improved solution, Section III-F
17: P←RankBasedPoolUpdating (P,So)
18: end while
According to this definition, if m<(n/2),Scorresponds to
any subset of elements of size of mfrom N\S.Ifm=(n/2),
the unique opposite solution is given by S=N\S.Ifm>
(n/2),Sis any subset of n−melements from N\S, completed
by other 2m−nelements from S.
We observe that a diversification framework introduced
in [21] also yields the type of opposite solution provided
by our definition and applies to constraints more general
than the constraint defined by (2). We make two related
comments. First, as we observe in Section IV,theMDP
benchmark instances in the literature correspond to the case
m≤(n/2). Second, in practice, when an opposite solution
is required while there are multiple opposite solutions, we
can just select one solution at random among the candidate
opposite solutions.
The proposed OBMA algorithm consists of four key compo-
nents: an opposition-based population initialization procedure,
a backbone-based crossover operator, an opposition-based
double trajectory search (ODTS) procedure and an RBQD pool
updating strategy. OBMA starts from a collection of diverse
elite solutions which are obtained by the opposition-based ini-
tialization procedure (Section III-C). At each generation, two
parent solutions are selected at random from the population,
and then the backbone-based crossover operator (Section III-E)
is applied to the selected parents to generate an offspring solu-
tion and a corresponding opposite solution. Subsequently, the
ODTS procedure (Section III-D) is invoked to search simulta-
neously from the offspring solution and its opposite solution.
Finally, the rank-based pool updating strategy (Section III-F)
decides whether these two improved offspring solutions should
be inserted into the population. This process repeats until a
stopping condition (e.g., a time limit) is satisfied. The general
framework of the OBMA algorithm is shown in Algorithm 2
while its components are described in the following sections.
734 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017
Algorithm 3 Opposition-Based Population Initialization
1: Input: population size p
2: Output: an initial population P={S1,S2,...,Sp}
3: count =0
4: while count <pdo
5: /∗Generate a random solution Srand its opposite solution
Sr∗/
6: Sr←TabuSearch (Sr)
7: Sr←TabuSearch (Sr)
8: /∗Identify the better solution between Srand Sr∗/
9: S←arg max{f(Sr), f(Sr)}
10: /∗Insert Sinto the population Por modify it ∗/
11: if Sis different from any solutions in the Pthen
12: Add Sinto the population Pdirectly
13: else
14: Modify Sand add it into the population P
15: count ←count +1
16: end if
17: end while
C. Opposition-Based Population Initialization
The initial population Pis composed of pdiverse and high
quality solutions. Unlike traditional population initialization,
our population initialization procedure integrates the concept
of OBL. As shown in Algorithm 3, the OBL-based initializa-
tion procedure considers not only a random candidate solution
but also a corresponding opposite solution. Specifically, we
first generate a pair of solutions, i.e., a random solution
Sr∈and a corresponding opposite solution Sraccording to
Definition 1(if multiple opposite solutions exist, one of them
is taken at random). These two solutions are then improved
by the TS procedure described in Section III-D. Finally, the
better one of the two improved solutions Sis inserted into the
population if Sis not the same as any existing individual of
the population. Otherwise, we modify Swith the swap oper-
ation (see Section III-D1) until Sbecomes different from all
individuals in Pbefore inserting it into the population. This
procedure is repeated until the population is filled up with p
solutions. With the help of this initialization procedure, the
initial solutions of Pare not only of good quality, but also of
high diversity.
D. Opposition-Based Double Trajectory Search Procedure
In the proposed OBMA algorithm, we use an ODTS pro-
cedure for local optimization. ODTS simultaneously searches
around an offspring solution Soand one opposite solution So.
The local optimization procedure used here is an improved
constrained neighborhood TS. TS is a well-known metaheuris-
tic that guides a local search heuristic to explore the solution
space beyond local optimality [20]. The original constrained
neighborhood TS algorithm was proposed in [53], which is
specifically designed for MDP by using a constrained neigh-
borhood and a dynamic tabu tenure management mechanism.
Compared with this TS algorithm, our improved TS proce-
dure (see Algorithm 4) distinguishes itself by its parametric
constrained neighborhood which allows the search process to
explore more promising candidate solutions. In the following,
we present the key ingredients of this local optimization pro-
cedure including the parametric constrained neighborhood, the
Algorithm 4 Parametric Constrained Neighborhood TS
1: Input: a starting solution S, the maximum allowed iterations
MaxIter
2: Output: the best solution S∗found
3: S∗←S
4: iter ←0
5: Initialize the tabu list
6: Calculate the gain(ei)for each element ei∈Naccording to
Eq. (4)
7: while iter <MaxIter do
8: minGain ←min{gain(ei):ei∈S}
9: Determine subset Uc
Saccording to Eq. (6)
10: maxGain ←max{gain(ei):ei∈N\S}
11: Determine subset Uc
N\Saccording to Eq. (7)
12: Choose a best eligible swap(eu,ev)(see Sect. III-D3)
13: S←S\{eu}∪{ev}
14: Update the tabu list and gain(ei)for each element ei∈N
accordingtoEq.(8)
15: if f(S)>f(S∗)then
16: S∗←S
17: end if
18: iter ←iter +1
19: end while
fast neighborhood evaluation technique and the dynamic tabu
tenure management scheme.
1) Parametric Constrained Neighborhood: In general, local
search for MDP starts from an initial solution Sand sub-
sequently swaps an element of Sand an element of N\S
according to some specific transition rule (e.g., accepting the
first or the best improving transition). Clearly, the size of this
neighborhood is bound by O(m(n−m)) and an exhaustive
exploration of all the possible swap moves is too time-
consuming for the large values of n. To reduce the size of the
swap neighborhood, we employ an extension of a candidate
list strategy sometimes called a neighborhood decomposition
strategy [19] or a successive filtration strategy [44], and which
we refer to as a constrained swap strategy [53]. As it is shown
in Section V-A, although this constrained swap strategy accel-
erates the search process, it imposes a too strong restriction
and may exclude some promising swap moves for the TS pro-
cedure. In this paper, we introduce the parametric constrained
neighborhood which adopts the idea of the constrained neigh-
borhood, but weakens the imposed constraint by introducing a
parameter ρ(ρ1) to control the size of the explored neigh-
borhood. Both constrained neighborhoods rely on the notion
of move gain of each element eiwith respect to the objective
value of the current solution Sdefined as follows:
gain(ei)=
ej∈S
dij,i=1,2,...,m.(4)
Let swap(eu,ev)denote the swap operation which
exchanges an element eu∈Sagainst an element ev∈N\S.
OnceaswapSswap(eu,ev)
−−−−−−→Sis made, it provides a new solu-
tion S=S\{eu}∪{ev}and the move gain uv of this swap
can be calculated according to the following formula:
uv =fS−f(S)=gain(ev)−gain(eu)−duv.(5)
Equation (5) suggests that in order to maximize the move
gain, it is a good strategy to consider swap moves that replaces
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 735
in the current solution San element euwith a small gain by
an element evout of Swith a large gain. In other words,
the search process can only consider swap moves that involve
an element eu∗from Swith the minimal gain value and an
element ev∗in N\Swith a maximal gain value. However, the
move gain also depends on the distance du∗v∗between eu∗
and ev∗. These remarks lead to the following definition for the
parametric constrained neighborhood.
For a current solution S, let minGain =min{gain(ei):ei∈
S}and maxGain =max{gain(ei):ei∈N\S}and dmax =
max{dij :1≤i<j≤n}. The parametric constrained
neighborhood relies on the two following sets:
Uc
S=ei∈S:gain
(ei)≤minGain +ρ
2dmax(6)
and
Uc
N\S=ei∈N\S:gain
(ei)maxGain −ρ
2dmax.(7)
Therefore, a constrained neighbor solution Scan be
obtained from Sby swapping one element eu∈Uc
Sand another
element ev∈Uc
N\S. Clearly, the evaluation of all constrained
neighboring solutions can be achieved in O(|Uc
S|×|Uc
N\S|).
Conveniently, we can adjust the value of parameter ρ(ρ1)
to control the size of the constrained neighborhood.
One notices that the neighborhood of [53] is a special
case of the above neighborhood when ρ=2. In general,
a larger ρvalue leads to a less constrained neighborhood
compared to the neighborhood of [53], allowing thus addi-
tional promising candidate solutions to be considered by the
TS procedure. Section V-A confirms the effectiveness of this
parametric constrained neighborhood.
2) Fast Neighborhood Evaluation Technique: Once a
swap(eu,ev)move is performed, we need to update the gains
gain(ei)affected by the move. To rapidly determine the gain of
each element ei, we resort to the fast neighborhood evaluation
technique used in [2], [4], and [53]
gain(ei)=⎧
⎪
⎨
⎪
⎩
gain(ei)+div if ei=eu
gain(ei)−diu if ei=ev
gain(ei)+div −diu if ei= euand ei= ev.
(8)
Updating the gains of nelements requires O(n)time.
Therefore, the time to update the parametric constrained neigh-
borhood at each iteration is bounded by O(n)+O(|Uc
S|×
|Uc
N\S|).
3) Dynamic Tabu Tenure Management Scheme: Starting
with a solution S, TS iteratively visits a series of neighboring
solutions generated by the swap operator. At each iteration,
a
best swap
(i.e., with the maximum move gain uv)is
chosen among the eligible swap moves to transform the cur-
rent solution even if the resulting solution is worse than the
current solution. To prevent the search from cycling among
visited solutions, TS typically incorporates a short-term history
memory H, known as the tabu list [20].
Initially, all elements are eligible for a swap operation. Once
aswap(eu,ev)is performed, we record it in the tabu list Hto
mark element euas tabu, meaning that element euis forbidden
to join again solution Sduring the next Tuiterations (Tuis
called the tabu tenure). Similarly, element evis also marked
as tabu for the next Tviterations and thus cannot be removed
from Sduring this period. The tabu status of an element is
disabled if the swap operation with this element leads to a
solution better than any already visited solution (this rule is
called the aspiration criterion in TS). An eligible swap move
involves only elements that are not forbidden by the tabu list
or satisfy the aspiration criterion.
It is important to determine a suitable tabu tenure for the
elements of a swap. Yet, there does not exist a universally
applicable tabu tenure management scheme. In this paper, we
adopt a dynamic tabu list management technique which was
proposed in [16] and proved to work well for MDP [53].
The tabu tenure Txof an element extaking part in a swap
operation is determined according to a periodic step function
T(iter), where iter is the number of iterations. Specifically,
T(iter)takes the value of α(a parameter set to 15 in this
paper), 2 ×α,4×α, and 8 ×αaccording to the value of
iter, and each T(iter)value is kept for 100 consecutive iter-
ations (see [16], [53] for more details). Following [53], we
set Tu=T(iter)for the element eudropped from the solu-
tion and Tv=0.7∗T(iter)for the element evadded to
the solution.
To implement the tabu list, we use an integer vector Hof
size nwhose components are initially set to 0 (i.e., H[i]=
0,∀i∈[1,...,n]). After each swap(eu,ev)operation, we set
H[u](resp.H[v]) to iter +Tu(resp. iter +Tv), where iter is
the current number of iterations and Tu(resp. Tv)isthetabu
tenure explained above. With this implementation, it is very
easy to know whether an element eiis forbidden by the tabu
list as follows. If iter ≤H[i], then eiis forbidden by the tabu
list; otherwise, eiis not forbidden by the tabu list.
E. Backbone-Based Crossover Operator
The crossover operator plays a critical role in memetic
search and defines the way information is transmitted from
parents to offspring [24]. A meaningful crossover operator
should preserve good properties of parent solutions through
the recombination process. In our case, we adopt a backbone-
based crossover operator which generates an offspring solution
in the same way as in [53] while introducing additionally an
opposite solution. For MDP, the backbone is a good property
that is to be transmitted from parents to offspring, as shown
in Definition 2. Specially, the backbone-based crossover oper-
ator not only produces an offspring solution, but also creates
a corresponding opposite solution.
Definition 2 (Backbone [24], [53]): Let Suand Svbe two
solutions of MDP, the backbone of Suand Svis defined as the
set of common elements shared by these two solutions, i.e.,
Su∩Sv.
Given a population P={S1,S2,...,Sp}of pindividuals, an
offspring solution is constructed in two phases. The first phase
randomly selects two parents Suand Svin Pand identifies the
backbone which is used to form the partial offspring So, i.e.,
So=Su∩Sv.If|So|<m, then the second phase successively
extends Sowith m−|So|other elements in a greedy way.
736 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017
Specifically, we alternatively consider each parent and select
an unassigned element with maximum gain with respect to So
until Soreaches the size of m. Once the offspring solution So
is obtained, we generate its corresponding opposite solution
Soaccording to Definition 1. Consequently, we obtain two
different and distant offspring solutions Soand Sowhich are
further improved by the TS procedure of Section III-D.
F. Rank-Based Pool Updating Strategy
To maintain a healthy diversity of the population, we use
a rank-based pool updating strategy to decide whether the
improved solutions (Soand So) should be inserted into the
population or discarded. This pool updating strategy simul-
taneously considers the solution quality and the distance
between individuals in the population to guarantee the pop-
ulation diversity. Similar quality-and-distance pool updating
strategies have been used in MAs in [10], [31], [48], and [53].
For two solutions Suand Sv, we use the well-known set-
theoretic partition distance [23] to measure their distance
DSu,Sv=m−SimSu,Sv(9)
where Sim(Su,Sv)=|Su∩Sv|denotes the number of common
elements shared by Suand Sv.
Given a population P={S1,S2,...,Sp}and one solution
Siin P, the average distance between Siand the remaining
individuals in Pis computed by [10]
ADSi,P=1
p
Sj∈P,j=i
DSi,Sj.(10)
To update the population with an improved offspring solu-
tion (Soor So), let us consider the case of So(the same
procedure is applied to So). We first tentatively insert Sointo
the population P, i.e., P←P∪{So}. Then all individuals in
Pare assessed based on the following function:
ScoreSi,P=β∗RFfSi+(1−β)∗RFADSi,P
(11)
where RF(f(Si)) and RF(AD(Si,P)) represent, respectively,
the rank of solution Siwith respect to its objective value
and the average distance to the population. Specifically, RF(·)
ranks the solutions of Pin decreasing order according to their
objective values or their average distances to the population. In
case of ties, the solution with the smallest index is preferred.
βis the weighting coefficient between the objective value of
the solution and its average distance to the population, which
is empirically set to β=0.6.
Based on this scoring function, we identify the worst solu-
tion Swwith the largest score value from the population P.If
the worst solution Swis not the improved offspring So, then
the population is updated by replacing Swby So; otherwise,
Sois simply discarded.
G. Computational Complexity of OBMA
To analyze the computational complexity of the proposed
OBMA algorithm, we consider the main steps in one genera-
tion in the main loop of Algorithm 2.
AsshowninAlgorithm2, each generation of the OBMA
algorithm is composed of four components: parents selection,
backbone-based crossover, TS, and rank-based pool updating
strategy. The step of parents selection is bounded by O(1).
The backbone-based crossover operation can be achieved in
O(nm2). The computational complexity of the parametric con-
strained neighborhood search procedure is O((n+|Uc
S|×
|Uc
N\S|)MaxIter), where |Uc
S|is the number of elements that
can be swapped out from S,|Uc
N\S|is the number of elements
in N\Sthat can be swapped into S, and MaxIter is the allow-
able maximum number of iterations in TS. The computational
complexity for the pool updating strategy is O(p(m2+p)),
where pis the population size. To summarize, the total com-
putational complexity of the proposed OBMA within one
generation is O(nm2+(n+|Uc
S|×|Uc
N\S|)MaxIter).
IV. COMPUTATIONAL RESULTS
This section presents computational experiments to test the
efficiency of our OBMA algorithm. We aim to: 1) demon-
strate the added value of OBMA (with OBL) compared to the
memetic search framework (without OBL) and 2) evaluate the
performance of OBMA with respect to the best-known results
ever reported by state-of-the-art algorithms in the literature.
A. Benchmark Instances
Our computational assessment were based on 80 large
instances with 2000–5000 elements which are classified into
the following sets.
Set I contains three data sets: 1) MDG-a (also known as
Type1_22); 2) MDG-b; and 3) MDG-c. They are available at
http://www.optsicom.es/mdp/.
1) MDG-a: This data set consists of 20 instances with n=
2000 and m=200. The distance dij between any two
elements iand jis an integer number which is randomly
selected between 0 and 10 from a uniform distribution.
2) MDG-b: This data set includes 20 instances with n=
2000 and m=200. The distance dij between any
two elements iand jis a real number which is ran-
domly selected between 0 and 1000 from a uniform
distribution.
3) MDG-c: This data set is composed of 20 instances with
n=2000 and m=300, 400, 500, and 600. The distance
dij between any two elements iand jis an integer number
which is randomly selected between 0 and 1000 from a
uniform distribution.
Set II (b2500) contains ten instances with n=2500
and m=1000, where the distance dij between any two
elements eiand ejis an integer randomly generated from
[−100,100]. This data set was originally derived from
the unconstrained binary quadratic programming problem
by ignoring the diagonal elements and is part of ORLIB:
http://people.brunel.ac.uk/~mastjjb/jeb/orlib/files/.
Set III (p3000 and p5000) contains five very large instances
with n=3000 and m=1500, and five instances with
n=5000 and m=2500, where dij are integers generated
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 737
TAB L E I
EIGHTY LARGE BENCHMARK INSTANCES USED IN THE EXPERIMENTS
TAB L E II
PARAMETER SETTING OF THE OBMA ALGORITHM
from a [0,100] uniform distribution. The sources of the gen-
erator and input files to replicate this data set can be found at:
http://www.proin.ktu.lt/~gintaras/mdgp.html.
B. Experimental Settings
Our algorithm2was implemented in C++, and com-
plied using GNU gcc 4.1.2 with “−O3” option on an
Intel E5-2670 with 2.5 GHz and 2-GB RAM under Linux.
Without using any compiler flag, running the DIMACS
machine benchmark program dfmax3on our machine requires
0.19, 1.17, and 4.54 s to solve graphs r300.5, r400.5, and
r500.5, respectively. To obtain our experimental results, each
instance was solved according to the settings (including time
limit and number of runs) provided in Tables Iand II. Notice
that, like most reference algorithms of Section IV-D,weuseda
cutoff time limit (instead of fitness evaluations) as the stopping
condition. This choice is suitable in the context of MDP given
that its fitness evaluation is computationally cheap enough,
contrary to expensive-to-evaluate problems like many engi-
neering optimization problems where using fitness evaluations
is a standard practice [25].
C. Benefit of OBL for Memetic Search
To verify the benefit of OBL for memetic search, we com-
pare OBMA with its alternative algorithm OBMA0without
OBL. To obtain OBMA0, two modifications have been made
on OBMA.
1) For the population initialization phase, we randomly
generate two initial solutions at a time (instead of a
random solution and an opposite solution).
2) For the crossover phase, we perform twice the crossover
operation to generate two offspring solutions (instead of
one offspring solution and an opposite solution).
2The best solution certificates and our program will be made available at
http://www.info.univ-angers.fr/pub/hao/OBMA.html.
3dfmax: ftp://dimacs.rutgers.edu/pub/dsj/clique.
To make a fair comparison between OBMA and OBMA0,we
ran both algorithms under the same conditions, as shown in
Tables Iand II. The comparative results for the five data sets
are summarized in Tables III–VII.
In these tables, columns 1 and 2, respectively, show
for each instance its name (Instance) and the current
best objective value (fprev )jointly reported in recent stud-
ies [17], [35], [51], [53]. Columns 3–7 report the results of
the OBMA0algorithm: the difference between fprev and the
best objective value fbest (i.e., fbest =fprev −fbest ), the dif-
ference between fprev and average objective value favg (i.e.,
favg =fprev −favg), the standard deviation of objective val-
ues (σ ), the average CPU time to attain the best objective
values (tbest)and the success rate (#succ)over 30 or 15
independent runs. Columns 8–12 present the same informa-
tion of the OBMA algorithm. The best values among the
results of the two compared algorithms are indicated in bold.
At the last row, we also provide the average number of
instances for which one algorithm outperforms the other algo-
rithm. 0.5 is assigned to each compared algorithm in case
of ties.
To analyze these results, we resort to a widely used sta-
tistical methodology known as
two-tailed sign test
[12]. This
test is a popular way to compare the overall performance of
algorithms by counting the number of winning instances of
each compared algorithm and thus to identify the overall win-
ner algorithm. The test makes the null hypothesis that the
compared algorithms are equivalent. The null hypothesis is
accepted if each algorithm wins on approximately X/2 out
of Xinstances. Otherwise, the test rejects the null hypothesis,
suggesting a difference between the compared algorithms. The
critical values (CVs) for the two-tailed sign test at a signifi-
cance level of 0.05 are, respectively, CV20
0.05 =15 for X=20
instances and CV10
0.05 =9forX=10 instances. In other
words, algorithm A is significantly better than algorithm B if
A performs better than B for at least CVX
0.05 instances for a
data set of Xinstances.
From Table III which shows the results of OBMA0and
OBMA for the 20 MDG-a instances, we first observe that
both algorithms attain the best-known results reported in the
literature. However, OBMA performs better than OBMA0in
terms of the average objective value and success rate, and wins
14.5 and 13.5 instances, respectively. We also observe that the
standard deviation of the best objective values is significantly
smaller for OBMA, and OBMA wins 14.5 instances, which is
very close to the critical value (CV20
0.05 =15). Finally, com-
pared to OBMA0, OBMA needs less average CPU time to find
the best-known solutions for all instances except MDG-a_26
and wins 13.5 instances in terms of the success rate.
Table IV shows the results of OBMA0and OBMA for the
20 MDG-b instances. The best-known objective values (fprev)
of this data set were obtained by a scatter search algorithm
(G_SS) [17] with a time limit of 2 h on an Intel Core 2 Quad
CPU 8300 with 6 GB of RAM running Ubuntu 9.04 [35]. This
table indicates that both OBMA and OBMA0find improved
best-known solutions for 14 out of 20 instances and attain
the best objective values for the remaining six instances. On
the other hand, compared to the OBMA0algorithm, OBMA
738 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017
TABLE III
COMPARISON OF THE RESULTS OBTAINED BY OBMA0AND OBMA ON THE DATA SET MDG-A
TAB L E IV
COMPARISON OF THE RESULTS OBTAINED BY OBMA0AND OBMA ON THE DATA SET MDG-B
obtains a better average objective value and higher success rate
for 13.5 and 13 instances. It is worth noting that OBMA has
a steady performance, and achieves these results with a 100%
success rate on almost all instances except for MDG-b_24,
MDG-b_32, and MDG-b_33. To summarize, OBMA performs
better than OBMA0for this data set, but the differences are
not very significant at a significance level of 0.05.
Table Vpresents the results of OBMA0and OBMA for the
20 instances of the MDG-c instances. All best-known results
(fprev) were achieved by ITS [38] or VNS [8] under a time
limit of 2 h on an Intel Core 2 Quad CPU 8300 with 6 GB
of RAM running Ubuntu 9.04 [35]. We observe that both
OBMA and OBMA0obtain improved best-known solutions
for eight instances and match the best-known solutions for
four instances. In fact, OBMA improves all best-known solu-
tions obtained by VNS, but it fails to attain eight best-known
solutions found by ITS. Compared to OBMA0, OBMA obtains
two improved best solutions for MDG-c_17 and MDG-c_19.
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 739
TAB L E V
COMPARISON OF THE RESULTS OBTAINED BY OBMA0AND OBMA ON THE DATA SET MDG-C
TAB L E VI
COMPARISON OF THE RESULTS OBTAINED BY OBMA0AND OBMA ON THE DATA SET B2500
Moreover, OBMA performs significantly better than OBMA0
in terms of the average best solution (19 >CV20
0.05 =15),
success rate (15 >=CV20
0.05 =15)and standard deviation
(19 >CV20
0.05 =15)at a significance level of 0.05.
Table VI reports the results of OBMA0and OBMA for the
10 instances of the b2500 data set. From this table, we observe
that both algorithms reach the best-known values for all the
instances. Meanwhile, the average value of best objective val-
ues of OBMA is better than that of OBMA0, and the difference
of this measure between these two algorithms is weakly signif-
icant (8.5<CV10
0.05 =9). Even though there is no significant
difference on the success rate, OBMA obtains a higher suc-
cess rate for 8.5 instances, while the reverse is true only for
1.5 instances. In addition, OBMA achieves these results more
steadily than OBMA0, wining 8.5 out of 10 instances in terms
of the standard deviation.
Table VII displays the results of OBMA0and OBMA for the
ten largest instances (p3000 and p5000 instances). For these
very large instances, OBMA matches all the best-known objec-
tive values without exception while OBMA0fails to do so
for four instances. In addition, OBMA performs significantly
better than OBMA0, and wins 10, 9.5 instances in terms of
the average best objective value and success rate, respectively.
The performance of OBMA is also more stable than OBMA0
by wining 8 out of 10 instances in term of the standard
deviation.
Finally, Table VIII provides a summary of the comparative
results for the five data sets between OBMA (OBL enhanced
MA) and OBMA0(MA without OBL). As we observe
from the table, OBMA achieves a better performance than
OBMA0, i.e., achieving improved solutions for six instances
and matching the best solutions on the remaining 75 instances.
740 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017
TAB L E VII
COMPARISON OF THE RESULTS OBTAINED BY OBMA0AND OBMA ON THE DATA SETS P3000 AND P5000
TABLE VIII
SUMMARY OF WIN STATISTICAL RESULTS
(OBMA0|OBMA) ON ALL DATA SETS
In addition, OBMA also achieves a better performance in
terms of the average best value, the success rate and the stan-
dard deviation, wining OBMA0on most benchmark instances.
Therefore, we conclude that OBL can beneficially enhance the
popular memetic search framework to achieve an improved
performance.
D. Comparison With State-of-the-Art Algorithms
We turn now our attention to a comparison of our
OBMA algorithm with state-of-the-art algorithms, includ-
ing iterated TS (ITS) [38], scatter search (G_SS) [17],
VNS [8], TIG [30], TS with estimation of distribution algo-
rithm (LTS-EDA) [51], and MA (MAMDP) [53]. We omit the
TS/MA (TS/MA) [52] and the memetic self-adaptive evolution
strategies (MSES) [11] since TS/MA performs quite similar to
MAMDP of [53] while MSES does not report detailed results.
Among these reference algorithm, only the program of the
MA (MAMDP) [53] is available. For our comparative study,
we report the results of the MAMDP algorithm by running its
code on our platform with its default parameter values reported
in [53]. For the other reference algorithms, we use their results
presented in the corresponding references. The detailed com-
parative results in terms of fbest and favg are reported in
Tables IX and X.
Table IX presents the comparative results on the 40
instances of the data sets MDG-a, b2500, p3000, and p5000
for which the detailed results of reference algorithms are avail-
able. At the last row of the table, we also indicate the number
of wining instances relative to our OBMA algorithm both in
terms of the best objective value and average objective value
(recall that a tied result counts 0.5 for each algorithm). From
this table, we observe that OBMA dominates all the reference
algorithms. Importantly, OBMA is the only algorithm which
obtains the best-known values and the largest average objective
values for all 40 instances.
Table Xdisplays the comparative results on the data sets
MDG-b and MDG-c. The best-known objective values fprev
for the MDG-b instances are obtained by G_SS [17] while the
fprev values of the MDG-c instances are obtained by ITS and
VNS [35], both with a time limit of 2 h. No result is available
for the TIG and LTS-EDA algorithms for these data sets. The
results of our OBMA algorithm (and MAMDP) are obtained
with a time limit of 10 min. Table Xindicates that both OBMA
and MAMDP improve the best-known results for the major-
ity of the 40 instances. Moreover, compared to MAMDP, our
OBMA algorithm obtains an improved best objective value
for one MDG-b instance and three MDG-c instances, while
matching the best objective values for the remaining instances.
Finally, OBMA dominates MAMDP in terms of the average
objective value, wining 18 out of the 20 MDG-b instances and
all 20 MDG-c instances.
To summarize, compared to the state-of-the-art results, our
OBMA algorithm finds improved best-known solutions (new
lower bounds) for 22 out of the 80 benchmark instances,
matches the best-known solutions for 50 instances, but fails
to attain the best-known results for eight instances. Such a
performance indicates that the proposed algorithm competes
favorably with state-of-the-art MDP algorithms and enriches
the existing solution arsenal for solving MDP.
V. EXPERIMENTAL ANALYSIS
In this section, we perform additional experiments to gain
some understanding of the proposed algorithm including the
parametric constrained neighborhood, the RBQD pool man-
agement and the benefit of OBL for population diversity.
A. Study of the Parametric Constrained Neighborhood
Our TS procedure relies on the parametric constrained
neighborhood whose size is controlled by the parameter ρ.
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 741
TAB L E IX
COMPARISON OF OBMA WITH OTHER ALGORITHMS ON THE DATA SETS MDG-A,B2500, P3000, AND P5000
To highlight the effect of this parameter and determine a
proper value, we ran the TS procedure to solve the first
ten instances of MDG-a (i.e., MDG-a_21–MDG-a_30) with
ρ∈[1,10]. Each instance was independently solved until
the number of iterations reached MaxIter. Fig. 1shows
the average objective values achieved (left) and the aver-
age CPU times consumed (right) by TS on these ten
instances.
As we see from Fig. 1(left), the average objective value
has a drastic rise when we increase ρfrom 1 to 3. Then,
it slowly increases if we continue to increase ρto 10.
On Fig. 1(right), the average CPU time of TS needed
to finish MaxIter iterations continuously increases when ρ
increases from 1 to 10. As ρincreases, the size of the
constrained neighborhood also increases, thus the algorithm
needs more time to examine the candidate solutions. To
make a compromise between neighborhood size and solu-
tion quality, we set the scale coefficient ρto4inour
experiments.
B. Effectiveness of the Pool Updating Strategy
To validate the effectiveness of the RBQD pool
updating strategy, we compare it with the general
quality-and-distance (GQD) pool updating strategy used
in [53]. GQD evaluates each individual by a weighted sum
of the quality and the distance to the population. In this
experiment, we compared the performance of the OBMA
algorithm under these two pool updating strategies (the two
OBMA variants are called OBMARBQD and OBMAGQD).
The experiment was performed on the largest data set, i.e.,
p3000 and p5000. We performed 20 runs of each algorithm
to solve each instance, and recorded the best objective value
(fbest), the difference between the average objective value and
the best objective value (favg), the standard deviation of
objective value over each run (σ), the average time of one
run (tavg), the average time over the runs which attained fbest
(tbest), and the success rate (#succ).
Table XI shows the comparison of the results obtained
by OBMA under the RBQD strategy (OBMARBQD) and the
742 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 5, OCTOBER 2017
TAB L E X
COMPARISON OF OBMA WITH MAMDP [53]ON THE DATA SETS MDG-B AND MDG-C,
THE BEST-KNOWN RESULTS ARE OBTAINED BY G_SS [17], ITS, AND VNS [35]
TAB L E XI
COMPARISON OF THE RESULTS OBTAINED BY OBMA UNDER THE RBQD POOL UPDATING STRATEGY
(OBMARBQD)AND THE GQD POOL UPDATING STRATEGY (OBMAGQD)
GQD strategy (OBMAGQD). From the table, we observe that
OBMARBQD achieves the same best objective values for all
tested instances compared with OBMAGQD. However, for the
five metrics, OBMARBQD performs better than OBMAGQD for
much more instances, and, respectively, winning 8, 8, 6, 6,
and 8 out of 10 tested instances. These results confirm the
effectiveness of our proposed RBQD pool updating strategy.
C. Opposition-Based Learning Over Population Diversity
In this section, we further verify the benefit brought by
OBL in maintaining the population diversity of the OBMA
algorithm. To assess the diversity of a population, a suitable
metric is necessary. In this experiment, we resort to
minimum
distance
and
average distance
of individuals in the population
to measure the population diversity. The minimum distance is
defined as the minimum distance between any two individ-
uals in the population, i.e., MD =mini=j∈{1,2,...,p}D(Si,Sj).
Correspondingly, the AD is the average distance between all
individuals in the population, as defined by (10).
Using the data sets MDG-a and b2500, we compared the
diversity of the population with or without OBL. The popula-
tion initialization (PI0) procedure without OBL first generates
two random solutions, which are then, respectively, improved
by the TS procedure. The best of two improved solutions is
inserted into the population if it does not duplicate any exist-
ing individual in the population. We repeat this process until
pdifferent solutions are generated. In contrast, the population
initialization with OBL (PIOBL) is the procedure described in
Section III-C, which considers both a random solution and its
corresponding opposite solution. We solved each instance 20
times and recorded the minimum distance and average distance
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 743
Fig. 1. Average objective values and average CPU times spent on ten
MDG-a instances obtained by executing TS with different values of the scale
coefficient ρ.
Fig. 2. Comparative results of the populations built by population initializa-
tion with OBL (PIOBL) or without OBL (PI0).
of each population initialization procedure on each instance.
The comparative results of the population constructed with or
without OBL are shown in Fig. 2, where the x-axis shows the
instances in each benchmark and y-axis indicates the average
distance and minimum distance.
From Fig. 2, we observe that the population built by PIOBL
has a relatively larger average distance and minimum distance.
This is particularly true for all instances of the MDG-a data set
except for MDG-a_31. Also, the population produced by PIOBL
has a larger minimum distance than that of PI0for 18 out of
20 instances of the MDG-a data set. Equal or better results are
found for the b2500 data set, since the population generated
by PIOBL dominates the population produced by PI0in terms
of the average and minimum distances. This experiment shows
that OBL helps the OBMA algorithm to start its search with
a population of high diversity, which is maintained by the
RBQD strategy during the search.
VI. CONCLUSION
We have proposed an OBMA which uses OBL to
improve an MA for solving MDP. The OBMA algorithm
employs OBL to reinforce population diversity and improve
evolutionary search. OBMA distinguishes itself from existing
MAs by three aspects.
1) A double trajectory search procedure which simultane-
ously both a candidate solution and a corresponding
opposite solution.
2) A parametric constrained neighborhood for effective
local optimization.
3) An RBQD pool updating strategy.
Extensive comparative experiments on 80 large benchmark
instances (with 2000–5000 items) from the literature have
demonstrated the competitiveness of the OBMA algorithm.
OBMA matches the best-known results for most of instances
and in particular finds improved best results (new lower
bounds) for 22 instances which are useful for the assess-
ment of other MDP algorithms. Our experimental analysis has
also confirmed that integrating OBL into the memetic search
framework does improve the search efficiency of the classical
memetic search. It would be interesting to study the behavior
of the OBMA algorithm when it is applied to
As future work, several potential research lines can be fol-
lowed. First, to further improve OBMA, it is worth studying
alternative strategies for tuning tabu tenure, generating initial
solutions, and managing population diversity. Second, it would
be interesting to study the behavior of the OBMA algorithm on
much larger instances (e.g., with tens of thousands items) and
investigate whether techniques developed for large scale con-
tinuous optimization [37], [43] could be helpful in this setting.
Third, OBL being a general technique, it is worth studying its
usefulness within other heuristic algorithms. Finally, it would
be interesting to investigate the opposition-based optimiza-
tion approach for solving additional combinatorial problems
including those with other diversity and dispersion criteria.
ACKNOWLEDGMENT
The authors would like to thank the anonymous referees
for their insightful comments and suggestions which helped
us to significantly improve this paper. They would also like
to thank Dr. Q. Wu for kindly sharing the source code of the
MAMDP algorithm described in [53] and Y. Zhou from the
China Scholarship Council (2014–2018).
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Yangming Zhou received the B.Sc. degree from
Donghua University, Shanghai, China, in 2011,
and the M.Sc. degree from Zhejiang University,
Hangzhou, China, in 2014. He is currently pursu-
ing the Ph.D. degree with the LERIA Laboratory,
University of Angers, Angers, France.
His current research interests include evolution-
ary computation, memetic algorithms, metaheuris-
tics, and machine learning for solving combinatorial
optimization problems.
ZHOU et al.: OPPOSITION-BASED MEMETIC SEARCH FOR MDP 745
Jin-Kao Hao received the B.Sc. degree from the
School of Computer Science, National University of
Defense Technology, Changsha, China, in 1982, the
M.Sc. degree from the National Institute of Applied
Sciences, Lyon, France, in 1987, the Ph.D. degree
from the University of Franche-Comté, Besançon,
France, in 1991, and the Professorship Diploma
HDR (Habilitation à Diriger des Recherches) degree
from the University of Science and Technology of
Montpellier, Montpellier, France, in 1998.
He is a Distinguished Professor (Professeur des
Universités de classe exceptionnelle) of Computer Science with the University
of Angers, Angers, France. He has authored or co-authored over 230 peer-
reviewed publications and co-edited nine books in Springer’s LNCS series. His
current research interests include design of effective algorithms and intelligent
computational methods for solving large-scale combinatorial search problems,
bioinformatics, data science, telecommunication networks, and transportation.
Dr. Hao is a Senior Fellow of the Institut Universitaire de France. He is on
the editorial board of seven international journals. He has served as an Invited
Member of over 200 Program Committees of International Conferences.
Béatrice Duval received the master’s degree from
the Ecole Normale Supérieur de Fontenay-aux-
Roses, Montrouge, France, in 1984, and the Ph.D.
degree in machine learning from the University of
Paris-Sud, Orsay, France, in 1991.
She is currently an Associate Professor of
Computer Science with the University of Angers,
Angers, France. Her current research interests
include data mining, machine learning, applications
in bioinformatics with hybrid methods combining
machine learning techniques, and metaheuristics.
