1. Variant of Differential Evolution Algorithm
Richa Shukla, Bramah Hazela, Shashwat Shukla, Ravi Prakash*,
Krishn Kumar Mishra*
Computer Science and Engineering Department
Amity University , Lucknow India, 226028 and
Computer Science and Engineering Department
MNNIT Allahabad, Allahabad India, 211004*
Email: richamity22@gmail.com, bhazela@lko.amity.edu, sshukla1@lko.amity.edu,
raviprakashguddu@gmail.com, kkm@mnnit.ac.in
Abstract. Differential Evolution is a nature inspired optimization tech-
nique. It has been achieved best solutions on large area of test suits.
DE algorithm is efficient in programming and it has broad applicability
in engineering. This paper presents modified mutation vector genera-
tion strategy of basic DE for solving stagnation problem. A new vari-
ant of differential evolution that is DE New has been proposed and the
performance of DE New is tested in COmparing Continuous Optimis-
ers(COCO) framework on 24 benchmark functions and found DE New
has better exploration performance in search space in comparison to
GA, DE- PSO, DE-AUTO on Black-Box Optimization Benchmarking
(BBOB) 2015 devised by COCO.
Keywords DE New, Exploitation, Robot Kidnapping, Differential Evolution,
mean-cost.
1 Introduction
Optimization problem can be solved with the help of different types of algo-
rithms[1][12]. In this scope reputation of Evolutionary Algorithm(EA) is un-
beatable[2][3][4]. DE is an EA that is used for solving many- modal optimization
problem in continuous search space[5][7]. DE algorithm is efficient and based
on natural selection with globally strong optimization technique in continu-
ous search space[8]. DE logic works with the help of DE operators (mutation,
crossover and selection) and associated parameters[9][10][11]. Myths related to
GA and DE creates due to similarity of the operators. In GA, crossover is the
main operator but in DE mutation operator plays leader role for solving global
optimization problem[13][14]. DE is better in compare to GA because it dose not
use binary encoding[15][4] and probability density function[16]. DE variants are
very helpful for enhancing the vector position, resolve the stagnation problem
and increase the convergence speed. DE basically works on global search space.
So for better exploration of solution, here DE New navigates to investigate the
local optimum problem for solving blocked solutions. Further content of this

2. paper has been ordered as below section 2 defines Differential Evolution, sec-
tion 3 states existing and modified algorithm, section 4 presents proposed work,
practical approach of result analysis has been elaborated in section 5 and finally
conclusions are drawn in section 6.
2 Differential Evolution
Differential Evolution is a randomized, population based, meta-heuristic, nature
inspired[17] and global optimization technique[18] given by Storn and Price in
1995[6]. Mutation, crossover and selection are the DE operators[19] but DE algo-
rithm is fully depend on mutation operator. The role of mutation operator is to
generate a mutant vector and the role of crossover is to develop trial vector and
the aim of selection operator is to select best between trial vector and mutant
vector for each generation.
Best example of DE is that it uses in mobile robot systems[20] for determining
the position and orientations of the robot. The way of initialization of the robot
position and track the position is the concept of Robot kidnapping(particle kid-
napping)[21][22][23]. Robot kidnapping understands the position and movement
of robot path without telling the new position[18].
Brief description of DE operators are given below:
1.Mutation: In this operator generate a mutant vector. Select three vectors
randomly from the search space and multiply the difference of two vectors with
a constant value ’F’(range of F is in between 0-2) and add with third vector that
produces a mutant vector mi,g , this process is called mutation. mutant vector
mi,g for each generation is stated as:
xi,g=xp+F(xq-xr) (1)
Notations:
1. i=1,2,......population size.
2. p, q, r are the three random vectors of the search space for ’g’ genera-
tion[24][27].
2.Crossover: After generating of mutant vector, create trial vector with the
help of mutant operator by recombining of two different vectors.
ti,g+1=t1i,g,t2i,g,t3i,g...........................tni,g (2)
The overall equation of trial vector for crossover operator is divided in two
phases , compare randomly generated number with crossover constant value
If random number generated is ≤ CR then we will use mutant vector mi,g
for each generation otherwise choose target vector without altering parent vec-
tor[25][26][28].
3.Selection: Selection of new member is totally depend on selection operator.
Comparison between trial vector and mutant vector, must select best between
these two vector ti,g and xi,g is the aim of selection operator.

3. 3 Existing and Modified DE Algorithms
Algorithm 1 Basic Differential Evolution Algorithm
1: for Each particle i do
2: for each vectors xi,g of population N do
3: MUTATION
4: select three random vectors from search space for mutation
5: xi,g=xp+F(xq-xr)
6: CROSS-OVER
7: if random number generated is ≤ CR then select mi,g
8: else select target vector
9: end if
10: SELECTION
11: evaluate fitness of xi,g and ti,g+1
12: choose best fitness as solution and discard previous
13: End if termination conditions met
14: end for
15: end for
Algorithm 2 Proposed Differential Evolution Algorithm
1: for Each particle i do
2: for each vectors xi,g of population N do
3: MUTATION
4: select xp, xq, xr
5: Fix first vector xp
6: where xq > mean-cost and xr< mean-cost
7: xi,g=xp+F(xq-xr)
8: CROSS-OVER
9: if random number generated is ≤ CR then
10: select mi,g for each generation
11: else select target vector
12: end if
13: SELECTION
14: evaluate fitness of xi,g and ti,g+1
15: choose best fitness as solution and discard previous
16: End if termination conditions met
17: end for
18: end for
4 Proposed Work
In DE algorithm mutation and associated parameters maintain the performance
of search space. There are many solutions in search space that are blocked,

4. stagnated, trapped and unable to give better solution for multi-modal prob-
lems in global search space. So for solving this problem DE NEW algorithm has
been proposed. In this algorithm, strategy of basic mutation operator has been
changed and a new variant of DE algorithm (DE NEW) has been proposed in
this paper. The logic of proposed DE NEW algorithm is to select three vectors
from the search space xp, xq, xr. Calculate the mean-cost and on the basis of
mean-cost, search space has been divided into two parts. Select xpvector ran-
domly from the total population size and stored in index number c, it will be the
fixed vector in each iterations. Then select next two vectors xq and xr. Choose
xq randomly from the above mean cost and stored in index number a and select
xr randomly from the below mean cost and stored in index number b.The new
strategy of mutation operator has been used for providing better exploration of
solution and greater diversity of solutions in search space in compare to GA, DE-
PSO, DE-AUTO on Black-Box Optimization Benchmarking (BBOB)[29] 2015
devised by COCO.
Fig. 1: DE NEW
5 Result Analysis
The three popular algorithm’s have been compared with tested performance of
DE New. In below table the rank has been alloted for proposed variant and
noiseless results have been attached. Its performance decreased might be due
to lower value of randomized F, Cr etc. But in this paper, DE New has been
succeeded to give better results for each modal in 24 benchmark functions in
2D, 3D, 5D, 10D, 20D and 40D.

5. Fig. 2: Comparison Results

6. Table 1: Ranking of Proposed Variant
DIMENSIONS hcond lcond mult2 multi separable
2D 2nd 2nd 2nd 1st 1st
3D 2nd 2nd 3rd 2nd 1st
5D 2nd 2nd 2nd 2nd 2nd
10D 2nd 2nd 2nd 2nd 2nd
20D 2nd 2nd 2nd 1st 2nd
40D 2nd 2nd 4th 1st 2nd
6 Conclusion
In this paper, proposed variant of DE algorithm has been based on modified mu-
tation strategy. It gives best performance for unimodal and multi-modal prob-
lems and increase the convergence rate and resolve the stagnation problems.
In DE New unlike basic DE and other algorithms, it can get best optimization
performance. DE New is able to give satisfactory performance for DE-AUTO
and best performance for DE-PSO, GA in 2D, 3D, 5D, 10D, 20D and 40D on
various benchmark functions, however there are a lot of ways for improving it.
In future work, DE algorithm can be used in cost estimation of the softwares
and implemented in many applications. Also merge two or more Evolutionary
Algorithm for achieving desired results.
References
[1] Z. W. Geem, J. H. Kim, and G. Loganathan, “A new heuristic optimization
algorithm: harmony search,” Simulation, vol. 76, no. 2, pp. 60–68, 2001.
[2] R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic
for global optimization over continuous spaces,” Journal of global optimization,
vol. 11, no. 4, pp. 341–359, 1997.
[3] S. Tiwari, K. Mishra, and A. K. Misra, “Test case generation for modified code
using a variant of particle swarm optimization (pso) algorithm,” in Information
Technology: New Generations (ITNG), 2013 Tenth International Conference on.
IEEE, 2013, pp. 363–368.
[4] K. Mishra, S. Tiwari, and A. Misra, “Combining non revisiting genetic al-
gorithm and neural network to generate test cases for white box testing,” in
Practical Applications of Intelligent Systems. Springer, 2011, pp. 373–380.
[5] E. Mezura-Montes, J. Vel azquez-Reyes, and C. A. Coello Coello, “A com-
parative study of differential evolution variants for global optimization,” in Pro-
ceedings of the 8th annual conference on Genetic and evolutionary computation.
ACM, 2006, pp. 485–492.
[6] R. Storn and K. Price, Differential evolution-a simple and efficient adaptive
scheme for global optimization over continuous spaces. ICSI Berkeley, 1995, vol.
3.

7. [7] D. Zaharie, “Critical values for the control parameters of differential evolu-
tion algorithms,” in Proceedings of MENDEL, vol. 2, 2002, p. 6267.
[8] Y. Ao and H. Chi, “Experimental study on differential evolution strategies,”
in Intelligent Systems, 2009. GCIS’09. WRI Global Congress on, vol. 2. IEEE,
2009, pp. 19–24.
[9] M. Daoudi, S. Hamena, Z. Benmounah, and M. Batouche, “Parallel diffrential
evolution clustering algorithm based on mapreduce,” in Soft Computing and Pat-
tern Recognition (SoCPaR), 2014 6th International Conference of. IEEE, 2014,
pp. 337–341.
[10] M. F. Tasgetiren, O. Bulut, Q.-K. Pan, and P. N. Suganthan, “A differen-
tial evolution algorithm for the median cycle problem,” in Differential Evolution
(SDE), 2011 IEEE Symposium on. IEEE, 2011, pp. 1–7.
[11] F. Neri and V. Tirronen, “Recent advances in differential evolution: a survey
and experimental analysis,” Artificial Intelligence Review, vol. 33, no. 1-2, pp.
61–106, 2010.
[12] S. Das and P. N. Suganthan, “Differential evolution: a survey of the stateof-
the-art,” Evolutionary Computation, IEEE Transactions on, vol. 15, no. 1, pp.
4–31, 2011.
[13] R. Joshi and A. C. Sanderson, “Minimal representation multisensor fusion
using differential evolution,” Systems, Man and Cybernetics, Part A: Systems
and Humans, IEEE Transactions on, vol. 29, no. 1, pp. 63– 76, 1999.
[14] J. Holland, “Adaption in natural and artificial systems,” Ann Arbor, MI:
University of Michigan Press, 1975.
[15] D. E. Goldberg et al., Genetic algorithms in search optimization and ma-
chine learning. Addison-wesley Reading Menlo Park, 1989, vol. 412.
[16] H.-P. Schwefel, “Evolution and optimization seeking,”John Wiley Sons,
1995.
[17] X.-S. Yang, Nature-inspired metaheuristic algorithms. Luniver press, 2010.
[18] A. R. Vahdat, N. NourAshrafoddin, and S. S. Ghidary, “Mobile robot global
localization using differential evolution and particle swarm optimization,” in
Evolutionary Computation, 2007. CEC 2007. IEEE Congress on. IEEE, 2007,
pp. 1527–1534.
[19] D. Kumar and K. K. Mishra, “Incorporating logic in artificial bee colony
(abc) algorithm to solve first order logic problems: The logical abc,” in Knowl-
edge and Smart Technology (KST), 2015 7th International Conference on. IEEE,
2015, pp. 65–70.
[20] C. Gonz alez, D. Blanco, and L. Moreno, “Optimum robot manipulator path
generation using differential evolution,” in Evolutionary Computation, 2009.
CEC’09. IEEE Congress on. IEEE, 2009, pp. 3322–3329.
[21] J. Gu and G. Gu, “Differential evolution with a local search operator,” in In-
formatics in Control, Automation and Robotics (CAR), 2010 2nd International
Asia Conference on, vol. 2. IEEE, 2010, pp. 480–483.
[22] J. M. Hereford, M. Siebold, and S. Nichols, “Using the particle swarm opti-
mization algorithm for robotic search applications,” in Swarm Intelligence Sym-
posium, 2007. SIS 2007. IEEE. IEEE, 2007, pp. 53–59.

8. [23] V. Pano and P. R. Ouyang, “Comparative study of ga, pso, and de for tuning
position domain pid controller,” in Robotics and Biomimetics (ROBIO), 2014
IEEE International Conference on. IEEE, 2014, pp. 1254–1259.
[24] M. G. Epitropakis, D. K. Tasoulis, N. G. Pavlidis, V. P. Plagianakos, and
M. N. Vrahatis, “Enhancing differential evolution utilizing proximitybased mu-
tation operators,” Evolutionary Computation, IEEE Transactions on, vol. 15,
no. 1, pp. 99–119, 2011.
[25] C. Sun, H. Zhou, and L. Chen, “Improved differential evolution algorithms,”
in Computer Science and Automation Engineering (CSAE), 2012 IEEE Interna-
tional Conference on, vol. 3. IEEE, 2012, pp. 142– 145.
[26] A. Tripathi, N. Saxena, K. K. Mishra, and A. K. Misra, “An environmen-
tal adaption method with real parameter encoding for dynamic environment,”
Journal of Intelligent Fuzzy Systems, no. Preprint, pp. 1–13.
[27] P. Melin, F. Olivas, O. Castillo, F. Valdez, J. Soria, and M. Valdez, “Op-
timal design of fuzzy classification systems using pso with dynamic parameter
adaptation through fuzzy logic,” Expert Systems with Applications, vol. 40, no.
8, pp. 3196–3206, 2013.
[28] Y.-W. Chen, C.-L. Lin, and A. Mimori, “Multimodal medical image regis-
tration using particle swarm optimization,” in Intelligent Systems Design and
Applications, 2008. ISDA’08. Eighth International Conference on, vol.3. IEEE,
2008, pp. 127–131.
[29] Brockho. (n.d.). Cec-bbob-2015 [COmparing Continuous Optimisers: COCO].
Retrieved June 11, 2016, from http://coco.gforge.inria.fr/