=Paper=
{{Paper
|id=None
|storemode=property
|title=Evolutionary bacterial foraging algorithm to
solve constraint numerical optimization problems
|pdfUrl=https://ceur-ws.org/Vol-1659/paper8.pdf
|volume=Vol-1659
|authors=Betania Hernández-Ocaña,Efren Mezura-Montes,Ma. Del Pilar Pozos-Parra
|dblpUrl=https://dblp.org/rec/conf/lanmr/Hernandez-Ocana16
}}
==Evolutionary bacterial foraging algorithm to
solve constraint numerical optimization problems==
https://ceur-ws.org/Vol-1659/paper8.pdf
Evolutionary Bacterial Foraging Algorithm to
solve constraint numerical optimization problems
Betania Hernández-Ocaña1 , Efrén Mezura-Montes2 , and Ma. Del Pilar Pozos-Parra1
1
Juárez Autonomous University of Tabasco, Cunduacán, Tabasco, México
2
Department of Artificial Intelligence, University of Veracruz, Xalapa, Veracruz,
México {betania.hernandez@ujat.mx,
pilar.pozos@ujat.mx,
emezura@uv.mx}
Abstract. A version of Modified Bacterial Foraging Optimization Al-
gorithm to solve Constraints Numerical Optimization is tested. The pro-
posal uses mutation operator, skew mechanism and local search operator.
To prove the effectiveness of the mechanism and adaptations proposed,
24 well-known test problems are solved along set experiments. Perfor-
mance measures are used for validating results obtained by the proposal
and they are compared against state-of-the-art algorithms. The results
show that the proposed algorithm is able to generate feasible solutions
within of feasible region with few evaluations and improves them over the
generations. Moreover, the results are competitive against the compari-
son algorithms based on performance measures found in the literature.
Bacterial foraging optimization, mutation operator, swarm intelligence,
Constrained optimization, premature convergence.
1 Introduction
Recently nature-inspired meta-heuristic algorithms have gained popularity solving Con-
strained Numerical Optimization Problems (CNOPs). These algorithms were created in
order to solve unconstrained optimization problems, but nevertheless the main feature
of the real world problems are the constraints. In answer to this, many constraints-
handling techniques have emerged, and have been added to nature-inspired algorithms
in several proposals. In [20] the different techniques found in the specialized literature
were grouped in twelve groups, which are based on Penalty functions, Decoders, Spe-
cial Operators and Separation of objective function and constraints, Feasibility rules,
Stochastic ranking, -constrained method, Novel penalty functions, Novel special op-
erators, Multi-objective concepts, and Ensemble of constraint-handling techniques.
The nature-inspired meta-heuristic algorithms have been divided in two classes 1) the
Evolutionary Algorithms (EAs) that emulate the evolution of species and the sur-
vival of the fittest, and 2) the Swarm Intelligence Algorithms (SIAs) which have the
capability of emulate the collaborative behavior of some simple species searching for
food or shelter [8]. Some well-known EAs are Genetic Algorithms (GAs) [6], Evolution
Strategies (ES) [27], Evolutionary Programming [9], Genetic Programming (GP) [14]
and Differential Evolution (DE) [25]. Some SIAs are the Particle Swarm Optimization
(PSO) [13] and the Ant Colony Optimization (ACO) [5], these last algorithms have
gained popularity for their great performance in solving CNOP. In 2002 other SIA
58
was proposed by Passino [23] called Bacterial Foraging Optimization (BFOA), which
emulates the behavior of bacteria E.Coli in the search of nutrients in its environment.
This behavior is summarize in four process (1) chemotaxis (swim-tumble movements),
(2) swarming (communication between bacteria), (3) reproduction (cloning of the best
bacteria) and (4) elimination-dispersal (replacement of the worst bacteria). In BFOA
each bacterium tries to maximize its obtained energy per each unit of time spent
on the foraging process while avoiding noxious substances. BFOA was used initially
to solve unconstrained optimization problems, however, more recently proposals add
some constraints-handling technique to solve CNOPs, where the penalty function is
the most used [10].
Further investigations have address the fact that BFOA is particularly sensitive to the
step size parameter, a recent study of the step size was reported in [11], where differ-
ent approaches were compared and the dynamic control mechanism was found to be
slightly superior to static, random, and adaptive versions. In [21] BFOA was adapted
in the proposal called Modified Bacterial Foraging Optimization Algorithm (MBFOA)
to solve CNOPs. The proposal inherits the four processes of BFOA, however, them are
divided as independent processes that interact sequentially, therefore the parameters
that determine the number of swim, number of tumble and the swarming loop were
eliminated. Moreover, The feasible rules, proposed by Deb [3], are used as a constraint-
handling technique. Unlike of BFOA where the step size is static, in MBFOA the step
size (used in the swim movements) was adapted according the boundary of the decision
variables.
The aim of this paper is tested a version of MBFOA which used mechanisms based
on evolutionary operators, Skew mechanism for the initial swarm and Local Search
Operator called Evolutionary MBFOA (EMBFOA) in 24 well-known test problems.
The results obtained will be compared against what is currently the most successful
and well-known algorithm of the state of the art: Memetic-SAMSDE [7]. We evaluate
our results with performance measures found in the literature. An advantage of the
new proposal is that it uses the same parameter values for all test problems. It is the
first time that the mutation is used as swim operator.
The document is organized as follows: Section 2 briefly describes the EMBFOA. Section
3 presents and compares the results obtained by proposal and another state-of-the-art
algorithms. Finally, Section 4 presents the general conclusions and future works.
2 Evolutionary Modified Bacterial Foraging Algorithm
Evolutionary Modified Bacterial Foraging Algorithm is an algorithm derived from MB-
FOA in order to improve the performance in constrained spaces. Four modifications
are made: (1) a skew mechanism for the initial swarm of bacteria is applied, (2) two
swim operators are applied in the chemotaxis process to improve the exploration and
exploitation of the bacteria; one of them uses a random step size easy to implement and
other one use mutation, (3) the elimination-dispersal process is adapted to preserve the
diversity of the swarm, and (4) a local search operator is added.
In BFOA, MBFOA and EMBFOA a bacterium i represents a potential solution to
the CNOP (i.e., a n-dimensional real-value vector identified as x), and it is defined as
θi (j,G), where j is its chemotaxis loop index and G is the current cycle of the algorithm.
Skew mechanism for the initial swarm: The initial swarm of Sb bacteria is formed
from three groups. The first group is formed of bacteria randomly skewed towards the
59
lower limit of the decision variables. The second group is formed of bacteria randomly
skewed towards the upper limit of the decision variables. Finally, a group of randomly
located bacteria without skew, as in the original MBFOA, is used. The formulas to set
the limits for the first and second group per variable are presented in Equations 1 and
2.
[Li , Li + ((Ui − Li)/ss)] (1)
[Ui − ((Ui − Li)/ss), Ui ] (2)
where ss is the skew size (ss > 1), for which large values increase the skew effect
and small values decrease the skew effect. The aim of this skew in the initial swarm,
combined with the two swim operators and the random stepsize control, is to avoid the
swarm of bacteria converging prematurely (behavior observed in the original MBFOA
caused by its swarming , reproduction process and the fixed stepsize), and improve the
exploration and exploitation of the search space in the initial phase of the search.
Chemotaxis: In this process two swims are interleaved, in each cycle only one of
the exploitation swim or exploration swim is performance. The process starts with the
exploitation swim (classical swim). However, a bacterium will not necessarily interleave
exploration and exploitation swims, because if the new position of a given swim, θi (j +
1, G) has a better fitness (based on the feasibility rules) than the original position
θi (j, G), another similar swim in the same direction will be carried out in the next
cycle. Otherwise, a new tumble for the other swim will be computed. The process
stops after Nc attempts. The exploration swim uses the mutation between bacteria
and is computed as indicated in Equation 3:
θi (j + 1, G) = θi (j, G) + (β − 1)(θ1r (j, G) − θ2r (j, G)) (3)
where θ1r (j, G) and θ2r (j, G) are two different randomly selected bacteria from the
swarm. β is an user-defined parameter used in the swarming that defines the closeness
of the new position of a bacterium with respect to the position of the best bacterium,
in this operator, β − 1 is a positive control parameter for scaling the different vectors
into (0,1], i.e., it scales the area where a bacterium can move.
The exploitation swim is computed as indicated in Equation 4:
θi (j + 1, G) = θi (j, G) + C(i, G)φ(i) (4)
where φ(i) is calculated with the original BFOA Equation 5 and C(i, G) is the random
stepsize of each bacterium update in each generation with the Equation 6.
∆(i)
φ(i) = p (5)
∆T (i)∆(i)
where ∆(i) is a uniformly distributed random vector of size n with elements within the
[-1,1].
C(i, G) = R ∗ Θ(i) (6)
where Θ(i) is a randomly generated vector of size n with elements within of the range
of the each decision variable: [U pperk , Lowerk ], k = 1, ....n, and R is an user-defined
parameter to scale the stepsize, this value should be close to zero, eg. 5.00E-04. The
initial C(i, 0) is generated using Θ(i). This random stepsize allows that the bacteria
can move in different direction into of the search space and avoids the premature
convergence as is suggest in [12].
It is important to remark that the exploration swim (Equation 3) performs larger
60
movements due the mutation operator which used bacteria randomly. On the other
hand, the exploitation swim (Equation 4) generates small movements using the random
stepsize in the search process.
Swarming: In the half cycle of the chemotaxis process the swarming operator is applied
with the Equation 7, where β is an user-defined parameter positive into (1, 2]. However,
in this proposal if a solution violates the boundary of decision variables, unlike of
MBFOA, a new solution xi is generated randomly which is bounded by lower and
upper limits Li ≤ xi ≤ Ui .
θi (j + 1, G) = θi (j, G) + β(θB (G) − θi (j, G)) (7)
where θi (j + 1, G) is the new position of bacterium i, θi (j, G) is the current position of
bacterium i, θB (G) is the current position of the best bacterium in the swarm so far
at generation G, and β defines the closeness of the new position of bacterium i with
respect to the position of the best bacterium θB (G). The attractor movement applies
twice in a chemotaxis loop, while in the remaining steps the tumble-swim movement
is carried out.
Reproduction: To reduce premature convergence due to bacteria duplication, The re-
production takes place only at certain cycles of the algorithm (defined by the RepCycle
parameter) deleting the Sr worst bacteria and duplicating the remaining Sb -Sr .
Elimination-dispersal: To preserver and increase the diversity of swarm, the elimination-
dispersal process takes place at certain cycles of the search process defined by the
EliCycle user-defined parameter. The number of bacteria to eliminate-dispersal is de-
fined by the user in the parameter Se . In this proposal, the diversity of swarm is nec-
essary because the mutation is effective when there is already some level of diversity
in the existing swam. Moreover, the generation of new bacteria avoids the premature
convergence.
Local Search Operator: Sequential Quadratic Programming (SQP) [24] is incor-
porated into EMBFOA as a local search operator in order to help the algorithm to
generate better results and based on the improved behavior observed by memetic al-
gorithms [22]. SQP is also used once in the middle of the search process. However,
the user can define the frequency of usage of the local search operator by defining the
parameter local search frequency LSG . The structure of EMBFOA is showed in the
Figure 1.
Fig. 1. EMBFOA general process.
61
3 Results and analysis
In this section are presented and compared the results obtained by EMBFOA in the
CEC2006 benchmark against MBFOA and Memetic-SAMSDE which is one the best
state-of-the-art algorithms for solving this benchmark. EMBFOA was coded in Matlab
R2009b, and run on a PC with a 3.5 Core 2 Duo Processor, 4GB RAM, and Windows
7.
The 95%-confidence Wilcoxon Signed Rank Test (WSRT) [2] suggested for nature-
inspired algorithms comparison in [4] is used as the statistical test to validate the
differences in the samples of runs. The scores are based on the best fitness value of
each algorithm in each test problem.
The input parameter values used by MBFOA are: Sb =40, Nc =20, Sr =2, R=1.20E-03
and β=1.5. For EMBFOA the parameters are: Sb = 40, Nc = 20, Sr = 2, R= 1.20E-
03, β= 1.5, RepCycle= 100, EliCycle= 50, LSG = 1 and (GM AX/2) generations,
M axL S= 5,000 Fes and ss= 8; they were fine-tuned by the iRace tool [16]. The iRace
obtained five possible values for each parameter and the average of the five values was
used. In this case, the only fixed parameter were GM AX according to the termination
condition of the maximum number of evaluations (Max FEs) of CEC2006 benchmark,
and the maximum number of evaluations for the local search operator M axL S which
is 5, 000 evaluations.
24 well-known CNOPs found in [15] were used in the experiments. Max Fes allowed for
each one of the 24 test problems was 240,000 evaluations. The tolerance for equality
constraints was set to ε = 1E − 04, this value is proposed in the specialized literature
of nature-inspired algorithms to solve CNOPs [1, 19, 20].
In this experiment the performance of EMBFOA is compared against MBFOA and
some EAs, such as Mememtic Self-Adaptive Multi-Strategy Differential Evolution (Memetic-
SAMSDE) which is currently one of the best state-of-the-art algorithm for solving the
set of test problems used in this paper. This algorithm also uses SQP as local search.
Unlike the EMBFOA that uses SPQ only twice, in Memetic-SAMSDE is used each 50
generations. The Adaptive Penalty Formulation with GA (APF-GA) [28], the Mod-
ified Differential Evolution (MDE) [18], the Adaptive Tradeoff Model with evolution
strategy (ATMES) [29], the Multimembered evolution strategy (SMES) [17], and the
Stochastic ranking with evolution strategy (SR+ES) [26]. Even when ATMES, SMES
and SR solved solely the first 13 test problems a computation against them is shown.
The number of evaluations computed by APF-GA and MDE was 500,000. SR+ES used
350,000 evaluations while for Memetic-SAMSDE, EMBFOA, ATMES and SMES used
240,000 evaluations. The number of independent runs used by EMBFOA, Memetic-
SAMSDE, ATMES, SMES and SR+ES was 30, and APF-GA and MDE used 25 inde-
pendent runs. In the Table 1 the results of all algorithms for the 24 test problems are
present whit the best solutions of each test problem is highlighted in bold. From the
results obtained by each algorithm, the Wilcoxon Sign-Rank test suggest that there is
significant difference between EMBFOA and MBFOA and there is no significant dif-
ference between EMBFOA, Memetic-SAMSDE and MDE. However the last algorithm
uses 500,000 evaluations which are the double evaluation used by EMBFOA. There
is a significant difference in favor of EMBFOA when it is compared against APF-GA,
ATMES, SME and SR+ES. Moreover APF-GA and SR+ES used more number of eval-
uations than EMBFOA.
In this experiment the performance of EMBFOA was compared against five state-of-
the-art nature-inspired algorithms used to solve CNOPs successfully, with the results
62
Table 1. Statistical comparison of MBFOA, EMBFOA, Memetic-SAMSDE, APF-GA,
MDE, ATMES, SMES and SR+ES. ’-’ means that the value is not proportionate by
the algorithm
Prob. f (x∗ ) Criteria MBFOA EMBFOA Memetic-SAMSDE APF-GA MDE ATMES SMES SR+ES
g01 -15 Best -9.88536 -15 -15 -15 -15 -15 -15 -15
Average -4.91338 -14.921 -15 -15 -15 -15 -15 -15
St.d 2.11E+00 2.97E-01 0 0 0 1.60E-14 0 0
g02 -0.8036191 Best -0.44923 -0.8036191 -0.8036191 -0.803601 -0.8036191 -0.803339 -0.803601 -0.803515
Average -0.33277 -0.679341365 -0.8036191 -0.803518 -0.78616 -0.790148 -0.785238 -0.781975
St.d 4.70E-02 6.23E-02 0 1.00E-04 1.20E-02 1.30E-02 1.67E-02 2.00E-02
g03 -1.0005001 Best -1.0004 -1.001 -1.0005 -1.001 -1.0005 -1 -1 -1
Average -1.0004 -0.993 -1.0005 -1.001 -1.0005 -1 -1 -1
St.d 2.30E-05 4.16E-02 0 0 0 5.90E-05 0.0000209 0.0000209
g04 -30665.5387 Best -30665.1 -30665.539 -30665.539 -30665.539 -30665.5386 -30665.539 -30665.539 -30665.539
Average -30663.3 -30665.539 -30665.539 -30665.539 -30665.5386 -30665.539 -30665.539 -30665.539
St.d 4.19E+00 1.17E-11 0 1.00E-04 1.00E-04 7.40E-12 0 2.00E-05
g05 5126.49671 Best - 5126.497 5126.497 5126.497 5126.497 5126.498 5126.599 5126.497
Average - 5126.497 5126.497 5127.5423 5126.497 5127.648 5174.492 5128.881
St.d - 2.78E-08 0 1.43E+00 0 1.80E+00 5.01E+01 3.50E+00
g06 -6961.81388 Best -6960.83 -6961.813875 -6961.813875 -6961.814 -6961.814 -6961.814 -6961.814 -6961.814
Average -6950.8 -6961.813875 -6961.813875 -6961.814 -6961.814 -6961.814 -6961.284 -6875.94
St.d 1.21E+01 4.63E-12 0 0 0 4.60E-12 1.85E+00 1.60E+02
g07 24.3062091 Best 24.72605 24.3062 24.3062 24.3062 24.3062 24.306 24.327 24.307
Average 25.91026 24.3062 24.3062 24.3062 24.3062 24.316 24.475 24.374
St.d 8.36E-01 1.20E-07 0 0 0 1.10E-02 1.32E-01 6.60E-02
g08 -0.095825 Best -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825
Average -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825
St.d 1.60E-12 1.79E-17 0 0 0 2.80E-16 0 0
g09 680.630057 Best 680.6534 680.63 680.63 680.63 680.63 680.63 680.632 680.63
Average 680.706 680.63 680.63 680.63 680.63 680.639 680.643 680.656
St.d 4.25E-02 5.24E-13 0 0 0 1.00E-02 1.55E-02 3.40E-02
g10 7049.24802 Best 7082.964 7049.24802 7049.24802 7049.24802 7049.24802 7052.253 7051.903 7051.903
Average 7356.791 7049.24802 7049.24802 7077.6821 7049.24802 7250.437 7253.047 7253.047
St.d 4.89E+02 1.27E-04 0 5.12E+01 0 1.20E+02 1.36E+02 1.36E+02
g11 0.7499 Best 0.7499 0.7499 0.7499 0.7499 0.7499 0.75 0.75 0.75
Average 0.7499 0.7499 0.7499 0.7499 0.7499 0.75 0.75 0.75
St.d 2.12E-06 1.86E-07 0 0 0 3.40E-04 1.52E-04 8.00E-05
g12 -1 Best -0.9999 -1 -1 -1 -1 -1 -1 -1
Average -0.9992 -1 -1 -1 -1 -1 -1 -1
St.d 1.95E-03 0 0 0 0 1.00E-03 0 0
g13 0.053941 Best - 0.053942 0.053942 0.053942 0.053942 0.05395 0.053986 0.067543
Average - 0.358400084 0.053942 0.053942 0.053942 0.053959 0.166385 0.067543
St.d - 2.81E-01 0 0 0 1.30E-05 1.77E-01 3.10E-02
g14 -47.764888 Best -42.5345 -47.764888 -47.764888 -47.76479 -47.764887 - - -
Average -38.6844 -47.764888 -47.764888 -47.76479 -47.764874 - - -
St.d 2.52E+00 1.26E-13 0 1.00E-04 1.40E-05 - - -
g15 961.715022 Best 961.7153 961.71502 961.71502 961.71502 961.71502 - - -
Average 961.7177 961.71502 961.71502 961.71502 961.71502 - - -
St.d 1.57E-03 2.87E-05 0 0 0 - - -
g16 -1.905155 Best -1.90335 -1.905155 -1.905155 -1.905155 -1.905155 - - -
Average -1.88754 -1.905155 -1.905155 -1.905155 -1.905155 - - -
St.d 5.64E-02 2.06E-15 0 0 0 - - -
g17 8853.53967 Best - 8912.914588 8853.5397 8853.5398 8853.5397 - - -
Average - 8927.107321 8823.5397 8888.4876 8853.5397 - - -
St.d - 2.68E+00 0 2.90E+01 0 - - -
g18 -0.866025 Best -0.85966 -0.866025 -0.866025 -0.866025 -0.866025 - - -
Average -0.73024 -0.866025 -0.866025 -0.865925 -0.866025 - - -
St.d 1.18E-01 5.08E-05 0 1.00E-04 0 - - -
g19 32.655592 Best 49.47301 32.655592 32.655593 32.655593 32.64827 - - -
Average 117.2929 32.65581447 32.655593 32.655593 33.34125 - - -
St.d 7.33E+01 6.40E-04 0 0 8.47E-01 - - -
g20 0.2049794 Best - - - - - - - -
Average - - - - - - - -
St.d - - - - - - - -
g21 193.72451 Best - 193.72451 193.72451 196.63301 193.72451 - - -
Average - 245.010861 193.72451 199.51581 193.72451 - - -
St.d - 5.27E+01 0 2.36E+00 0 - - -
g22 236.430976 Best - - 236.370313 - - - - -
Average - - 245.738829 - - - - -
St.d - - 9.05E+00 - - - - -
g23 -400.0551 Best - -400.0544473 -400.0551 -399.7624 -400.0551 - - -
Average - -377.0888655 -400.0551 -394.7627 -400.0551 - - -
St.d - 6.61E+01 0 3.87E+00 0 - - -
g24 -5.50801327 Best -5.508013 -5.508013 -5.508013 -5.508013 -5.508013 - - -
Average -5.50768 -5.508013 -5.508013 -5.508013 -5.508013 - - -
St.d 2.83E-04 1.81E-15 0 0 0 - - -
of the comparison it is prove that EMBFOA is a competitive algorithm. Moreover,
the superiority is evident of EMBFOA over MBFOA to find the feasible region and
improve the feasible solutions found.
4 Conclusion
In this paper, a Swarm Intelligence algorithm called Evolutionary Modified Bacterial
Foraging Optimization Algorithm (EMBFOA) was proposed to solve constrained nu-
merical optimization problems. This proposal is an improved version of the algorithm
Modified Bacterial Foraging Optimization. The performance of EMBFOA was tested
in 24 well-known test problems using a set of performance measures found in the lit-
erature and the results obtained were compared against state-of-the-art algorithms.
According to the results, EMBFOA is better than the original algorithm MBFOA and
competitive against state-of-the-art algorithms. To the best of the authors knowledge,
63
this is the first attempt to design a BFOA-based algorithm that uses mutation as swim
operator. As future work, EMBFOA will be tested in more CNOPs and a study on the
impact of the population size and the way it is generated over the performance of the
algorithm, will be performed.
References
1. Carlos A. Coello Coello. Theoretical and Numerical Constraint Handling Tech-
niques used with Evolutionary Algorithms: A Survey of the State of the Art.
Computer Methods in Applied Mechanics and Engineering, 191(11-12):1245–1287,
January 2002.
2. G.W. Corde and D.I. Foreman. Nonparametric Statistics for Non-Statisticians: A
Step-by-Step Approach. John Wiley, Hoboken, NJ, 2009.
3. Kalyanmoy Deb. An Efficient Constraint Handling Method for Genetic Algorithms.
Computer Methods in Applied Mechanics and Engineering, 186(2/4):311–338, 2000.
4. J. Derrac, S. Garcı́a, D. Molina, and F. Herrera. A practical tutorial on the use of
nonparametric statistical tests as a methodology for comparing evolutionary and
swarm intelligence algorithms. Swarm and Evolutionary Computation, 1(1):3–18,
2011.
5. M. Dorigo, V. Maniezzo, and A. Colorni. The Ant System: Optimization by
a Colony of Cooperating Agents. IEEE Transactions of Systems, Man and
Cybernetics-Part B, 26(1):29–41, 1996.
6. A.E. Eiben and J. E. Smith. Introduction to Evolutionary Computing. Natural
Computing Series. Springer Verlag, 2003.
7. Saber M. Elsayed, Ruhul A. Sarker, and Daryl L. Essam. On an evolutionary
approach for constrained optimization problem solving. Applied soft computing,
12(0):3208–3227, 2012.
8. Andries P. Engelbrecht. Computational Intelligence. An Introduction. John Wiley
& Sons, 2nd edition, 2007.
9. Lawrence J. Fogel. Intelligence Through Simulated Evolution. Forty years of Evo-
lutionary Programming. John Wiley & Sons, New York, 1999.
10. Betania Hernández-Ocaña, Efrén Mezura-Montes, and Pilar Pozos-Parra. A re-
view of the bacterial foraging algorithm in constrained numerical optimization.
In Proccedings of the Congress on Evolutionary Computation (CEC’2013), pages
2695–2702. IEEE, 2013.
11. Betania Hernández-Ocaña, Pilar Pozos-Parra, and Efrén Mezura-Montes. Stepsize
control on the modified bacterial foraging algorithm for constrained numerical
optimization. In Proceedings of the 2014 Conference on Genetic and Evolutionary
Computation, GECCO ’14, pages 25–32, New York, NY, USA, 2014. ACM.
12. Alireza Kasaiezadeh, Amir Khajepour, and Steven L. Waslander. Spiral bacterial
foraging optimization method: Algorithm, evaluation and convergence analysis.
Engineering Optimization, 46(4):439–464, 2014.
13. James Kennedy and Russell C. Eberhart. Swarm Intelligence. Morgan Kaufmann,
UK, 2001.
14. John R. Koza, Martin A. Keane, Matthew J. Streeter, William Mydlowec, Jessen
Yu, and Guido Lanza. Genetic Programming IV: Routine Human-Competitive
Machine Intelligence. Kluwer Academic Publishers, Hingham, MA, USA, 2003.
64
15. J.J. Liang, Thomas Philip Runarsson, Efrn Mezura-Montes, Maurice Clerc, P.N.
Suganthan, Carlos A. Coello Coello, and K. Deb. Problem definitions and eval-
uation criteria for the cec 2006 special session on constrained real-parameter op-
timization. Technical report, School of EEE Nanyang Technological University,
Singapore, September 2006.
16. Manuel López-Ibáñez, Jéremie Dubois-Lacoste, Thomas Sttzle, and Mauro Bi-
rattari. The irace package, iterated race for automatic algorithm configuration.
Technical Report TR/IRIDIA/2011-004, IRIDIA, Université Libre de Bruxelles,
Belgium, 2011.
17. E. Mezura-Montes and Carlos A. Coello Coello. A simple multimembered evolution
strategy to solve constrained optimization problems. Evolutionary Computation,
IEEE Transactions on, 9(1):1–17, Feb 2005.
18. E. Mezura-Montes, J. Velazquez-Reyes, and C.A. Coello Coello. Modified differ-
ential evolution for constrained optimization. In Evolutionary Computation, 2006.
CEC 2006. IEEE Congress on, pages 25–32, 2006.
19. Efrén Mezura-Montes, editor. Constraint-Handling in Evolutionary Optimization,
volume 198 of Studies in Computational Intelligence. Springer-Verlag, 2009.
20. Efrén Mezura-Montes and Carlos A. Coello Coello. Constraint-handling in nature-
inspired numerical optimization: Past, present and future. Swarm and Evolutionary
Computation, 1(4):173–194, 2011.
21. Efrén Mezura-Montes and Betania Hernández-Ocaña. Modified bacterial foraging
optimization for engineering design. In Cihan H. Dagli and et al., editors, Proceed-
ings of the Artificial Neural Networks in Enginnering Conference (ANNIE’2009),
volume 19 of Intelligent Engineering Systems Through Artificial Neural Networks,
pages 357–364, St. Louis, MO, USA, November 2009. ASME Press.
22. Ferrante Neri and Carlos Cotta. Memetic algorithms and memetic computing
optimization: A literature review. Swarm and Evolutionary Computation, 2:1–14,
2012.
23. Kevin M. Passino. Biomimicry of bacterial foraging for distributed optimization
and control. IEEE Control Systems Magazine, 22(3):52–67, 2002.
24. M. J. D. Powell. Algorithms for Nonlinear Constraints that use Lagrangian Func-
tions. Mathematical Programming, 14:224–248, 1978.
25. K. Price, R. Storn, and J. Lampinen. Differential Evolution: A Practical Approach
to Global Optimization. Natural Computing Series. Springer-Verlag, 2005.
26. T.P. Runarsson and Xin Yao. Stochastic ranking for constrained evolutionary
optimization. Evolutionary Computation, IEEE Transactions on, 4(3):284–294,
Sep 2000.
27. Hans-Paul Schwefel, editor. Evolution and Optimization Seeking. Wiley, New York,
1995.
28. B. Tessema and G.G. Yen. An adaptive penalty formulation for constrained evo-
lutionary optimization. Systems, Man and Cybernetics, Part A: Systems and Hu-
mans, IEEE Transactions on, 39(3):565–578, May 2009.
29. Yong Wang, Zixing Cai, Yuren Zhou, and Wei Zeng. An adaptive tradeoff model for
constrained evolutionary optimization. Evolutionary Computation, IEEE Trans-
actions on, 12(1):80–92, Feb 2008.
65