constrained optimization using gravitational search algorithm

RESEARCH ARTICLE

Constrained Optimization Using Gravitational Search Algorithm

Anupam Yadav • Kusum Deep

Received: 12 January 2013 / Revised: 29 March 2013 / Accepted: 1 May 2013 / Published online: 8 October 2013

� The National Academy of Sciences, India 2013

Abstract Gravitational search algorithm is a newly born

metaheuristic which is inspired by the working force of

attraction between two masses. In this article gravitational

search algorithm is employed to solve the constrained

optimization problems. A variety of state-of-the-art

benchmark problems are taken into account to justify the

performance of the gravitational search algorithm. The

results of gravitational search algorithms are compared

with two state-of-the-art Particle Swarm Optimization

algorithms in various aspects. Pairwise one tailed t test is

applied to justify the statistical significance of the results

and time complexity analysis is also performed. Finally the

conclusions are drawn based on the experimental results.

keywords Gravitational search algorithm �Particle swarm optimization � Constrained optimization

Introduction

Constrained optimization problems are very significant to

the science and engineering community because many real

life and engineering design problems can be modeled as

constrained optimization problems. Mathematically a

constrained optimization problem can be modeled as

Maximize or Minimize f ðx1; x2; x3; . . .; xnÞ; ð1Þ

subject to inequality constraints

gjðx1; x2; x3; . . .; xnÞ 6 0; j ¼ 1; 2; 3; . . .q ð2Þ

and equality constraints

hjðx1; x2; x3; . . .; xnÞ ¼ 0; j ¼ qþ 1; qþ 2; . . .m ð3Þ

where f : Rn ! R is a real valued function where g and

h are constraints. These problems have much importance in

engineering and design for obtaining the optimal

parameters involve in the design and engineering. There

are two class of methods available for solving these

problems, the first one is deterministic techniques and the

other one is non deterministic techniques. The role of non

deterministic techniques gets much more importance due to

the limitations of deterministic techniques. The differential

free nature of non deterministic techniques stimulates its

importance in comparison to the deterministic techniques.

Nature inspired optimization techniques are recent

paradigm to the non deterministic optimization methods.

Genetic Algorithm, Differential Evolution, Particle Swarm

Optimization etc. are some examples of nature inspired

algorithms. In recent years Deep and Bansal [1] designed a

PSO for solving economic dispatch problem as a

constrained optimization problem. Deep et al. [2] applied

PSO for evaluating the hypocentral parameters of an

earthquake by utilizing some real life data. The promising

solution of these problems and the well applicability of

nature inspired algorithms on constrained optimization

problems directs to figure out the applicability of some

more nature inspired optimization techniques to the

constrained optimization problems. In the present article

a newly developed gravitational search algorithm (GSA) is

utilized to solve the constrained optimization problems.

GSA is proposed by Rashedi [3] and Rashedi et al. [4].

It is inspired by the Newton’s theory on force of attraction

A. Yadav (&) � K. Deep

Department of Mathemtics, Indian Institute of Technology

Roorkee, Roorkee, India

e-mail: [email protected]

K. Deep

e-mail: [email protected]

123

Natl. Acad. Sci. Lett. (September–October 2013) 36(5):527–534

DOI 10.1007/s40009-013-0165-8

between two masses. The proposed technique is success-

fully applied on unconstrained optimization problems [4]

as well as many real life applications [5, 6]. In this article

constrained optimization problems are tried to solve with

the help of GSA. A set of 24 state-of-the-art problems

presented in IEEE CEC 2006 [7] are solved with the help

of GSA. A parameter free constraint handling method is

employed to handle the constraints. The results of the GSA

is compared with two state-of-the-art variants of Particle

Swarm Optimization.

The organization of the paper is done in the following

way, In the Sect. 2 a brief idea of GSA is presented, Sect. 3

a brief idea of Particle Swarm Optimization is discussed. In

Sect. 4 the performance and evaluation of GSA is pre-

sented based on experiments over the benchmark functions,

finally the conclusions are drawn with future scope.

Gravitational Search Algorithm

GSA [3, 4] is a recent heuristic optimization algorithm. It is

inspired by Newton’s basic physical theory that a force of

attraction works between every particle in the universe and

this force is directly proportional to the product of their

masses and inversely proportional to the square of distance

between their positions. All these particles are named as

agents (particles). In GSA each particle is equipped with

four kind of properties: position, mass, active gravitational

mass and passive gravitational mass. The position of the

mass provides the solution of the problem. Gravitational

masses and inertial masses can be evaluated using fitness

function. Each kind of masses follows the following two

basic laws of physics:

1. Law of gravity: Each particle attracts every other

particle and the gravitational force between two

particles is directly proportional to the product of their

masses and inversely proportional to the square of the

distance between them.

2. Law of motion: The current velocity of any mass is

equal to the sum of the fractions of its previous

velocities and the variations of the velocity. Variation

in the velocity of acceleration of any mass is equal to

the force acted on the system divided by the mass of

inertia.

Inspired by the definitions above we are able to define

physics of GSA. Let the position of the ith particle at any

instant t in a D-dimensional search space be

Xtiðxt

i1; xti2; . . .; xt

iDÞ for i ¼ 1. . .ps. The force of attraction

on the ith particle to jth particle is defined as in the fol-

lowing equation

FtijD ¼ Gt �

Mtpi �Mt

aj

Rtij

� xtid � xt

jd

� �ð4Þ

where d ¼ 1; 2; . . .D;Mtpi is the passive gravitational mass

related to ith particle at time t, Mait is the active

gravitational mass related to jth particle at time t, Gt is

the gravitational constant at time t, � is a small constant and

Rijt is the Euclidian distance between the two particles i and

j given by the following equation:

Rtij ¼ kXt

i ;Xtjk2 ð5Þ

The value of gravitational constant Gt can be calculated as

Eq. 6

Gt ¼ Gt0 � exp �aiter

itermax

� �� ð6Þ

where a and Gt0 are descending coefficient and initial value

respectively, iter is the current iteration and itermax is the

maximum number of iterations.

The total force of attraction exerted by the ith particle at

time t in a D-dimensional space is given by Eq. 7

Ftid ¼

Xps

i¼1;i 6¼j

randðÞFtijd ð7Þ

where d ¼ 1; 2; . . .D and rand() is a random number in the

interval [0,1], which is added to provide the stochastic

nature to the algorithm. By using the law of motion the

acceleration of ith particle is given by the following

equation:

actid ¼

Ftid

Mtii

ð8Þ

where Mtii is the inertial mass of the ith particle. The

velocity and position of particles are calculated as follow:

Vtþ1id ¼ randðÞ � Vt

id þ actid ð9Þ

xtþ1id ¼ xt

id þ Vtþ1id ð10Þ

where rand() is uniform random variable in [0,1]. The

gravitational and inertial masses are simply calculated by

the fitness evaluations. A greater mass can be treated as

better particle and having higher force of attraction so that

they can influence other particles with high level of

attraction. The gravitational and inertial mass will be

updated with the help of following equations:

Mai ¼ Mpi ¼ Mii for i ¼ 1; 2. . .ps ð11Þ

mti ¼

fitti � worstt

bestt � worsttð12Þ

Mti ¼

mtiPps

i¼1 mti

ð13Þ

528 A. Yadav, K. Deep

123

where fitit represents the fitness value of the ith particle at time t.

and bestt & worstt may be defined as in the following equations:

bestt ¼ minðfittjÞ; j 2 f1; . . .psg ð14Þ

worstt ¼ maxðfittjÞ; j 2 f1; . . .psg ð15Þ

The exhaustive procedure of GSA is explained in Table 1

Particle Swarm Optimization

Particle swarm optimization is a nature inspired stochastic

population based optimization search technique, inspired by

the social behavior of fish and birds. PSO was first introduced

by Kennedy and Eberhart. It uses the learning, information

sharing and position updating strategy of each solution.

Mathematically, PSO can be formulated as: Let the position of

the ith particle in a D-dimensional search space be

Xtiðxt

i1; xti2; . . .; xt

iDÞ with a flag of velocity Vti ðvt

i1; vti2; . . .; vt

idÞat any moment t, where i = 1 to ps, where ps is the swarm size.

Let Pbestit and Gbesti

t denote the latest best position of the

particle (personal best) and global best at the moment t. Initially

Pbestit and Xi

t are same. From the theory of the PSO the velocity

update equation and position update equation can be written as:

Vtþ1i ¼ c1Vt

i þ c2ðXti � Gbestt

iÞ þ c3ðXti � Pbestt

iÞ ð16Þ

Xtþ1i ¼ Xt

i þ Vtþ1i ð17Þ

The updated position of particles may be obtained by

applying Vit?1 on the position Xi

t (Eq. 17). The exhaustive

procedure of PSO is explained in Table 2

Performance and Evaluation of GSA

Test Functions

To judge the optimization ability of the GSA over con-

strained optimization problems, a set of state-of-the-art

problems from CEC 2006 [7] has been taken for the jus-

tification. A brief detail of the functions are listed in

Table 3. Few good constraint handling methods [8–11] are

available which may also utilize with the GSA. The fol-

lowing constraint handling method is applied to deal with

the constraint violations. The parameter-exempt constraint

dealing approach by computing constraint violation [7] is

used to deal with the constraints. The grade of constrained

violation [7] of single x for the jth constraint may be

computed as follows:

GjðxÞ ¼ maxfgjðxÞ; 0g1� j� q ð18Þ

GjðxÞ ¼ maxfjhjðxÞ � ej; 0gqþ 1� j�m; ð19Þwhere e is a positive tolerance esteem for equality con-

straints. The grade of constraints violation for an single

root x is G(x) =P

j=1m Gjx. To compare the results of the

GSA two popular variants of PSO is taken from the liter-

ature. There algorithms are

1. Basic PSO

2. SPSO 2011

The parameter setting of all the algorithms are assumed

same as cited in the concerned articles.

Table 1 Pseudo code of

gravitational search algorithm

Gravitational Search Algorithm 529

123

Results and Analysis

All the algorithms are coded on Matlab 2011b platform.

100 runs experiment are performed for each algorithm, a

run consists of maximum 4,000 iterations, the population

size for each algorithm is uniformly taken as 60. The

results of the performed experiments are presented in the

form of best, mean, worse and standard deviation of fitness

value along with the mean infeasibility of the particles in

the final population of a median run. It is recorded in the

results that the out of 24 problems GSA is able to solve 17

problems. GSA solved six problems for which the other

two algorithms are not able to solve. For the problems

g01, g02, g04, g06, g08, g11, g12, g13, g14, g15, g16, g18,

g19 and g24 the best recorded value is either better or

comparable to other two algorithms. The mean infeasibility

of almost all the problems by GSA is near to 0. Even for

few problems the worse recorded value is better than the

best recorded value of rest two algorithms. The results for

which fully infeasible solution is obtained is represented as

’NaN’. The results of the problems g20-g23 are not listed

in the Table 4, since none of the algorithms are able to

provide the feasible solution. Due to the very small ratio of

the feasible region and search region as discussed in the

Table 3, GSA is not able to reach towards the optimal

solution for few problems, since this very small ratio

increases the complexity of the problems and that is why

the working force of attraction among the particles become

less effective to move them towards global optima.

t Test

Pair wise one tailed t test is applied with 98� of freedom at

0.05 level of significance, over the fitness value of the

problems. This test involves the problems for which all the

algorithms are successful to give feasible result. Table 5

gives the results of the t test. The pairwise mean, standard

deviation(STD), Standard error mean, p value along with

conclusion of the test is listed. ’a-’, ’a’ and ’a?’ shows,

the Algo1 is significantly worse, alike and significantly

better to Algo2 respectively. It is observed that GSA per-

forms significantly better than other two algorithms for

more than seven problems.

Time Complexity

To compare the time complexity of the GSA in a median

run, its time consumption at key points are measured by in

built ’Matlab Profiler’, that determines the proportionality

of time consumption at key points. Table 6 shows the

comparative results of CPU time at key points of the

algorithm, for GSA these key process points are

Table 2 Pseudo code of

particle swarm optimization

algorithm


123

Table 3 Characteristics of test problems

Prob. D Type q (%) LI NI LE NE a

g01 13 Quadratic 0.0111 9 0 0 0 6

g02 20 Nonlinear 99.9971 0 2 0 0 1

g03 10 Polynomial 0.0000 0 0 0 1 1

g04 5 Quadratic 52.1230 0 6 0 0 2

g05 4 Cubic 0.0000 2 0 0 3 3

g06 2 Cubic 0.0066 0 2 0 0 2

g07 10 Quadratic 0.0003 3 5 0 0 6

g08 2 Nonlinear 0.8560 0 2 0 0 0

g09 7 Polynomial 0.5121 0 4 0 0 2

g10 8 Linear 0.0010 3 3 0 0 6

g11 2 Quadratic 0.0000 0 0 0 1 1

g12 3 Quadratic 4.7713 0 1 0 0 0

g13 5 Nonlinear 0.0000 0 0 0 3 3

g14 10 Nonlinear 0.0000 0 0 3 0 3

g15 3 Quadratic 0.0000 0 0 1 1 2

g16 5 Nonlinear 0.0204 4 34 0 0 4

g17 6 Nonlinear 0.0000 0 0 0 4 4

g18 9 Quadratic 0.0000 0 13 0 0 6

g19 15 Nonlinear 33.4761 0 5 0 0 0

g20 24 Linear 0.0000 0 6 2 12 16

g21 7 Linear 0.0000 0 1 0 5 6

g22 22 Linear 0.0000 0 1 8 11 19

g23 9 Linear 0.0000 0 2 3 1 6

g24 2 Linear 79.6556 0 2 0 0 2

D is the number of decision variables, q = |F|/|S| is the estimated ratio between the feasible region and the search space, LI is the number of

linear inequality constraints, NI the number of nonlinear inequality constraints, LE is the number of linear equality constraints and NE is the

number of nonlinear equality constraints. a is the number of active constraints at x

Table 4 Comparative results of objective function values for 4,000 Iterations

Problem Algorithm Best Mean Worse STDEV Mean infeas.

g01 Basic PSO -1.213E?01 -1.173E?01 -1.127E?01 2.947E-01 1.00E-02

SPSO 2011 -1.294E?01 -1.131E?01 -9.157E?00 1.115E?00 1.00E-02

GSA -1.348E?01 -1.450E?02 -1.348E?01 5.038E?01 1.67E-04

g02 Basic PSO -3.896E-01 -3.475E-01 -3.088E-01 2.347E-02 1.00E-02

SPSO 2011 -4.715E-01 -4.263E-01 -3.756E-01 2.881E-02 1.00E-02

GSA -4.768E-01 -1.063E-01 -4.711E-02 4.247E-02 1.00E-02

g03 Basic PSO -5.393E-01 -2.411E-01 -6.085E-02 1.443E-01 7.73E-04

SPSO 2011 -7.998E-01 -5.990E-01 -4.398E-01 1.091E-01 2.87E-03

GSA -2.356E-02 -1.816E-02 -1.244E-03 4.085E?02 0.00E?00

g04 Basic PSO -3.065E?04 -3.063E?04 -3.062E?04 8.971E?00 1.00E-02

SPSO 2011 -3.057E?04 -3.042E?04 -3.031E?04 8.335E?01 1.00E-02

GSA -3.065E?04 -2.782E?04 -2.332E?04 2.146E?03 2.83E-03

g05 Basic PSO NaN NaN NaN NaN 0.00E?00

SPSO 2011 NaN NaN NaN NaN 7.60E-03

GSA NaN NaN NaN NaN 0.00E?00

g06 Basic PSO -6.933E?03 -6.897E?03 -6.855E?03 2.983E?01 1.00E-02

SPSO 2011 -6.892E?03 -5.936E?03 -4.667E?03 7.435E?02 1.00E-02


123

initialization of particles, calculation of mass, calculation

of acceleration position update and fitness calculation. The

key process points of both the variants of PSO are initial-

ization of particles, initialization of velocity, velocity

update and fitness calculation. Two problems g01 and g02

are considered to compare the consuming time. It can be

observed from Table 6 that the time consumption in cal-

culation of acceleration and mass is less in comparison to

Table 4 continued

Problem Algorithm Best Mean Worse STDEV Mean infeas.

GSA -6.934E?03 2.264E?05 7.859E?05 2.082E?05 1.67E-04

g07 Basic PSO 8.204E?01 1.215E?02 1.732E?02 2.928E?01 9.79E-03

SPSO 2011 4.101E?01 1.080E?02 3.207E?02 9.673E?01 1.00E-02

GSA 4.286E?01 2.151E?02 4.652E?02 9.207E?01 1.67E-04

g08 Basic PSO -9.583E-02 -9.583E-02 -9.583E-02 3.440E-14 1.00E-02

SPSO 2011 -9.583E-02 -9.583E-02 -9.583E-02 1.600E-17 1.00E-02

GSA -9.583E-02 -9.440E?00 1.401E-01 1.704E?01 0.00E?00

g09 Basic PSO 7.112E?02 7.309E?02 7.470E?02 1.159E?01 1.00E-02

SPSO 2011 6.809E?02 6.892E?02 7.324E?02 1.515E?01 1.00E-02

GSA 7.043E?02 1.355E?06 9.020E?06 2.170E?06 1.67E-04

g10 Basic PSO 1.079E?04 1.190E?04 1.332E?04 8.307E?02 8.75E-03

SPSO 2011 7.702E?03 1.156E?04 1.606E?04 2.948E?03 1.00E-02


g11 Basic PSO 7.509E-01 7.692E-01 8.265E-01 2.183E-02 1.70E-03

SPSO 2011 7.503E-01 7.747E-01 8.805E-01 4.423E-02 6.81E-03

GSA 7.452E-01 8.570E-01 1.253E?00 1.022E?00 1.67E-04

g12 Basic PSO -1.000E?00 -1.000E?00 -1.000E?00 1.150E-05 1.00E-02

SPSO 2011 -1.000E?00 -1.000E?00 -1.000E?00 0.000E?00 1.00E-02

GSA -1.000E?00 -1.000E?00 -1.000E?00 0.000E?00 1.00E-02



GSA 9.978E-01 1.394E?06 7.659E?07 9.903E?06 1.67E-04



GSA -4.330E?01 -1.039E?03 -4.330E?01 2.620E?02 1.67E-04


SPSO 2011 9.618E?02 9.641E?02 9.699E?02 2.950E?00 1.00E-02

GSA 9.610E?02 8.124E?02 9.881E?02 1.039E?02 1.67E-04

g16 Basic PSO -1.894E?00 -1.886E?00 -1.880E?00 3.794E-03 1.00E-02

SPSO 2011 -1.800E?00 -1.633E?00 -1.457E?00 1.115E-01 1.00E-02

GSA -1.821E?00 3.379E-01 3.012E?00 1.096E?00 1.67E-04






GSA -5.621E-01 1.452E?00 1.451E?02 5.861E?01 1.67E-04

g19 Basic PSO 1.163E?02 1.497E?02 1.785E?02 1.600E?01 1.00E-02

SPSO 2011 9.809E?01 1.319E?02 2.210E?02 3.238E?01 1.00E-02

GSA 5.170E?01 1.793E?03 3.433E?03 9.090E?02 2.83E-03

g24 Basic PSO -5.508E?00 -5.508E?00 -5.508E?00 5.270E-06 1.00E-02

SPSO 2011 -5.508E?00 -5.504E?00 -5.489E?00 6.173E-03 1.00E-02

GSA -5.508E?00 -3.757E?00 -3.628E-01 1.484E?00 5.00E-03


123

the velocity initialization and velocity updation, so the total

consuming time for GSA to solve the problem g01 and g02

is less while others take little more time. This is a great

advantage of GSA, because while dealing with large scale

optimization problems the CPU time consumed by GSA

will be considerably less.

Conclusion

In the current article a new GSA is employed to solve the

constraint optimization problems. A set of stat-of-the-art

problems are solved using GSA which are taken from CEC

2006. 100 runs experiment is performed for maximum 4000

iterations. The results of GSA are compared with Basic PSO

and SPSO 2011 in the form of best, mean, worse and

standard deviation of the objective function values. The

statistical significance of the results are justified with the

one tailed t test. It is concluded that the performance of the

GSA is good for most of the problems and for few problems

its results are comparable with Basic PSO as well as SPSO

2011. While doing experimental analysis it is observed that

a more compatible constraint handling technique to GSA

may provide some better results to the existing.

Table 5 Pairwise t test results over objective function values with 95 % confidence interval at 0.05 level of significance

Problem Algo1 vs Algo2 Mean STD Std. error mean p value Conclusion

g01 GSA-Basic PSO 0.44 0.68 0.14 0.004 a

GSA-SPSO 2011 2.44 0.68 0.14 0.000 a?

g02 GSA-SPSO 2011 0.09 0.03 0.01 0.000 a?

GSA-SPSO 2011 0.42 0.03 0.01 0.000 a?

g04 GSA-Basic PSO 694.99 364.18 72.84 0.000 a?

GSA-SPSO 2011 781.66 157.88 31.58 0.000 a?

g07 GSA-Basic PSO 320.95 283.78 56.76 0.000 a?

GSA-SPSO 2011 353.07 283.49 56.70 0.000 a?

g09 GSA-Basic PSO 10.65 6.85 1.37 0.000 a?

GSA-SPSO 2011 24.92 6.30 1.26 0.000 a?

g12 GSA-Basic PSO 0.00 0.00 0.00 0.020 a-

GSA-SPSO 2011 0.00 0.00 0.00 0.021 a-

g18 GSA-Basic PSO -0.20 0.06 0.01 0.000 a?

GSA-SPSO 2011 0.24 0.07 0.01 0.000 a?

g19 GSA-Basic PSO 481.71 111.77 22.35 0.000 a?

GSA-SPSO 2011 526.02 110.26 22.05 0.000 a?

g24 GSA-SPSO 2011 0.00 0.00 0.00 0.000 a?

GSA-SPSO 2011 0.00 0.00 0.00 0.000 a?

Table 6 Comparision of CPU time at some key processing points of the algorithms

g01 Initializate X Initialization V V calculation Fitness Total time (s)

Basic PSO 0.01 0.01 0.2 3 3.22

SPSO 2011 0.1 0.02 0.02 3.1 3.24

Initializate X Mass Acceleration Fitness

GSA 0.01 0.001 0.003 2.9 2.914

g02 Initializate X Initialization V V calculation Fitness

Basic PSO 0.02 0.02 0.35 5 5.39

SPSO 2011 0.02 0.03 0.034 6 6.084

Initializate X Mass Acceleration Fitness

GSA 0.02 0.002 0.023 4 4.045


123

Acknowledgments The first author is thankful to Council of Sci-

entific and Industrial Research, HRDG Group for the financial support

with grant No. 9834-11-44. I would also like to thanks to the anon-

ymous reviewers for their valuable suggestion and comments for

revising this article.

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constrained optimization using gravitational search algorithm

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