Glowworm swarm based optimization algorithm for multimodal ... › 3e77 › 9cfdbcd1e6df... · 212 K.N. Krishnanand and D. Ghose / Glowworm swarm based optimization algorithm for

Multiagent and Grid Systems – An International Journal 2 (2006) 209–222 209IOS Press

Glowworm swarm based optimizationalgorithm for multimodal functions withcollective robotics applications

K.N. Krishnanand∗ and Debasish Ghose∗∗Guidance, Control, and Decision Systems Laboratory, Department of Aerospace Engineering, Indian Institute ofScience, Bangalore 560 012, India

Received 1 June 2005

Revised 25 August 2005, 25 March 2006

Accepted 12 April 2006

Abstract. This paper presents multimodal function optimization, using a nature-inspired glowworm swarm optimization (GSO)algorithm, with applications to collective robotics. GSO is similar to ACO and PSO but with important differences. A keyfeature of the algorithm is the use of an adaptive local-decision domain, which is used effectively to detect the multiple optimumlocations of the multimodal function. Agents in the GSO algorithm have a finite sensor range which defines a hard limit onthe local-decision domain used to compute their movements. The GSO algorithm is memoryless and the glowworms do notretain any information in their memory. Some theoretical results related to the luciferin update mechanism in order to provethe bounded nature and convergence of luciferin levels of the glowworms are provided. Simulations demonstrate the efficacy ofthe GSO algorithm in capturing multiple optima of several multimodal test functions. The algorithm can be directly used in arealistic collective robotics task of simultaneously localizing multiple sources of interest such as nuclear spills, aerosol/hazardouschemical leaks, and fire-origins in a fire calamity.

Keywords: Glowworm swarm optimization, multimodal functions, ant colony optimization, particle swarm optimization, collec-tive robotics

1. Introduction

Intelligent group behavior exhibited by biologicalswarms like ants, termites, bees, wasps, and bacte-ria is a result of actions performed by relatively sim-ple individuals that are solely based on neighbor-interactions and local information from the environ-ment inhabited by the agent-collective. The above be-havioral metaphor offers an insight into the basis todevise distributed algorithms that solve complex prob-lems related to diverse fields such as optimization [1,

∗Graduate Student.∗∗Corresponding author. Tel.: +91 80 2293 3023; Fax: +91 80

2360 0134; E-mail: [email protected].

6,23], multi-agent decision making [21], and collec-tive robotics [11,19]. Recent literature abounds withexamples of such biomimetic algorithms including antcolony optimization (ACO) techniques [6] that are ap-plied to a number of optimization problems [2,10]),bacterial chemotaxis based optimization [23], socialforaging swarms [21], and several swarm based collec-tive robotic algorithms [11,16,17,19].

1.1. Multimodal function optimization

Multimodal function optimization has been ad-dressed extensively in the recent literature [3–5,9,16,17,20,22,24–26]. Most prior work on this topic fo-cussed on developing algorithms to find either theglobal optimum or all the global optima of the given

ISSN 1574-1702/06/$17.00 © 2006 – IOS Press and the authors. All rights reserved

210 K.N. Krishnanand and D. Ghose / Glowworm swarm based optimization algorithm for multimodal functions

multimodal function, while avoiding local optima.However, there is another class of optimization prob-lems which is different from the problem of findingonly the global optimum of a multimodal function.The objective of this class of multimodal optimizationproblems is to find multiple local optima having eitherequal or unequal function values [3,4,9,16,17,20,22].In particular, the interest in this paper lies in develop-ing an algorithm which, while serving for numericalmultimodal function optimization on one hand, couldbe directly used in a realistic collective robotics task ofsimultaneously localizing multiple sources of interestlike nuclear spills, aerosol/hazardous chemical leaks,and fire-origins in a fire calamity.

1.2. The glowworm metaphor

Inspired by the concept of emergent behavior,a novelglowworm1 swarm optimization (GSO) algorithm wasdeveloped [16] that finds solutions to optimizationproblems defined on multimodal functions. In this al-gorithm, the agents are initially deployed randomly inthe objective function space. The agents carry a lu-minescence quantity called luciferin along with them.Agents are thought of as glowworms that emit a lightwhose intensity of luminescence is proportional to theassociated luciferin. Each glowworm uses the luciferinto (indirectly) communicate the function-profile infor-mation at its current location to the neighbors. Theglowworms depend on a variable local-decision do-main – that is bounded above by a circular sensorrange – to compute their movements. Each glowwormselects a neighbor that has a luciferin value more thanits own, using a probabilistic mechanism, and movestowards it. These movements enable the glowwormsto split into subgroups, exhibit a simultaneous taxis-behavior towards, and rendezvous at, the optimum lo-cations leading to the detection of multiple optima ofthe given objective function.

This paper presents the details of this new approachand provides several extensions. Firstly, the local-decision domain update rule formulated in [16] is mod-ified that results in substantial performance enhance-

1The glowworm belongs to a family of beetles known as theLampyridae or fireflies and produces natural light that is used as asignal to attract a mate. Luciferin is one of the several componentsthat are involved in a chemical reaction responsible for producing thebioluminescent light. This light is also used to attract prey. Generalidea in the glowworm algorithm is similar in the sense that glowwormagents are attracted to other glowworm agents that have brighterluminescence [28].

ment in terms of the number of iterations required forconvergence of the algorithm. Secondly, the luciferinupdate rule and probability distribution function aremodified in order to address the connectivity problemscaused by the earlier formulae. A simple multimodalfunction was chosen to test the feasibility of the algo-rithm in [16]. However, in this paper, the algorithmis evaluated on a series of more complex multimodalfunctions. Next, the glowworm algorithm is appliedto optimization of a discontinuous multimodal func-tion. Working of the algorithm is also tested in higherdimensional spaces.

The paper is organized as follows. A complete de-scription of the glowworm algorithm is given in Sec-tion 2. Section 3 provides two theoretical proofs relatedto the luciferin update rule of the glowworm algorithm.Section 4 presents the simulation results. A brief men-tion of the collective robotics platform that was built totest the algorithms developed in this paper is given inSection 5. A comparison of the GSO algorithm withrelated work is carried out in Section 6. The paperconcludes with a few remarks in Section 7.

2. The Glowworm Swarm Optimization (GSO)algorithm

In the glowworm algorithm,physical entities (agents)are considered that are randomly distributed in theworkspace. The agents in the glowworm algorithmcarry a luminescence quantity called luciferin alongwith them. Agents are thought of as glowworms thatemit a light whose intensity is proportional to the asso-ciated luciferin and have a variable decision range r i

d,bounded by a circular sensor range rs (0 < ri

d � ris).

Each glowworm is attracted by the brighter glow ofother neighboring glowworms. A glowworm identi-fies another glowworm as a neighbor when it is locatedwithin its current local-decision domain. Agents inthe glowworm algorithm depend only on informationavailable in the local-decision range to make their deci-sions (Fig. 1(a)). The resulting algorithm is highly de-centralized and caters to the requirements of collectiverobotic systems.

2.1. Algorithm description

The GSO algorithm starts by placing the glowwormsrandomly in the workspace so that they are well dis-persed. Initially, all the glowworms contain an equalquantity of luciferin. Each iteration consists of aluciferin-update phase followed by a movement-phasebased on a transition rule.

K.N. Krishnanand and D. Ghose / Glowworm swarm based optimization algorithm for multimodal functions 211

rkd

r jd

rsj ij

k

local-decision domains

radial sensor range ofagent k

radial sensor rangeof agent j

krs

a

b

cd

ef

local-decision range

1

2

34

5 6

glowworm

(a) (b)

Fig. 1. a) rkd < d(i, k) = d(i, j) < rj

d< rk

s < rjs. Agent i is in the sensor range of (and is equidistant to) both j and k. But, they have

different decision-domains. Hence, only j uses the information of i. b) Emergence of a directed graph based on the relative luciferin level ofeach agent and availability of only local information. Agents are ranked according to the increasing order of their luciferin values. For instance,the agent a whose luciferin value is highest is ranked ‘1’ in the figure.

2.1.1. Luciferin-update phaseThe luciferin update depends on the function value

at the glowworm position and so, even though all glow-worms start with the same luciferin value during theinitial iteration, these values change according to thefunction values at their current positions. During theluciferin update phase, each glowworm adds, to its pre-vious luciferin level, a luciferin quantity proportional tothe measured value of the sensed profile (temperature,radiation level) at that point. In the case of a functionoptimization problem, this would be the value of theobjective function at that point. Also, a fraction of theluciferin value is subtracted to simulate the decay inluciferin with time. The luciferin update rule is givenby:

�j(t + 1) = (1 − ρ)�j(t) + γJj(t + 1) (1)

where, ρ is the luciferin decay constant (0 < ρ < 1)and γ is the luciferin enhancement constant and J j(t)represents the value of the objective function at agentj’s location at time t.

2.1.2. Movement-phaseDuring the movement-phase, each glowworm de-

cides, using a probabilistic mechanism, to move to-wards a neighbor that has a luciferin value more than itsown. That is, they are attracted to neighbors that glowbrighter. Fig. 1(b) shows the emergence of a directedgraph among a set of six glowworms based on theirrelative luciferin levels and availability of only local in-

formation. For instance, there are four glowworms (a,b, c, and d) that have relatively more luciferin than theglowworm e. Since e is located in the sensor-overlapregion of c and d, it has only two possible directionsof movement. For each glowworm i, the probability ofmoving towards a neighbor j is given by:

pj(t) =�j(t) − �i(t)∑

k∈Ni(t)�k(t) − �i(t)

(2)

where, j ∈ Ni(t), Ni(t) = {j : di,j(t) < rid(t);

�i(t) < �j(t)}, t is the time (or step) index, di,j(t)represents the euclidian distance between glowwormsi and j at time t, �j(t) represents the luciferin levelassociated with glowworm j at time t, ri

d(t) representsthe variable local-decision range associated with glow-worm i at time t, and rs represents the radial range ofthe luciferin sensor. Let the glowworm i select a glow-worm j ∈ Ni(t) with pj(t) given by Eq. (2). Then thediscrete-time model of the glowworm movements canbe stated as:

xi(t + 1) = xi(t) + s

(xj(t) − xi(t)‖xj(t) − xi(t)‖

)(3)

where s is the step-size.

2.1.3. Local-decision range update ruleWhen the glowworms depend on only local informa-

tion to decide their movements, it is expected that thenumber of peaks captured would be a strong functionof the radial sensor range. For instance, if the sensor


range of each agent covers the entire workspace, all theagents move to the global optimum point and the localoptima are ignored. Since it is assumed that a prioriinformation about the objective function (e.g., numberof maxima and minima) is not available, in order todetect multiple peaks, the sensor range must be madea varying parameter. For this purpose, each agent i isassociated with a local-decision domain whose radialrange ri

d is dynamic in nature (0 < rid � ri

s). Thelocal-decision domain update rule given in [16] resultsin an oscillatory behavior of r i

d(t). To smoothen the re-sponse, a new update rule is proposed where an explicitthreshold parameter nt is used to control the numberof neighbors at each iteration. A substantial enhance-ment in performance is noticed by using the rule givenbelow:

rid(t + 1) = min{rs, max{0, ri

d(t) + β(4)

(nt − |Ni(t)|)}}where, β is a constant parameter.

3. Luciferin convergence proof

Two theoretical proofs are provided for the luciferinupdate rule proposed in the previous section. First, itis proved that, due to luciferin decay, the maximumluciferin level �max

j is bounded asymptotically. Thisproof is similar to Stutzle and Dorigos’ [6,27] propo-sition proving the bounded nature of pheromone levelsin their ant-colony algorithm. Second, the luciferin � j

of all glowworms co-located at a peak Xi is shown toconverge to the same value �∗i .

Theorem 1. Assuming that the luciferin update rule in(1) is used, the luciferin level �j(t) for any glowwormj is bounded above asymptotically as follows:

limt→∞ �j(t) � lim

t→∞ �maxj (t) =

(γ

ρ

)Jmax (5)

where Jmax is the global maximum value of the objec-tive function.

Proof: Given that the maximum value of the objec-tive function is Jmax and the luciferin update rule inEq. (1) is used, the maximum possible quantity of lu-ciferin added to the previous luciferin level at any itera-tion t is γJmax. Therefore, at iteration 1, the maximumluciferin of any glowworm j is (1− ρ)�0 + γJmax. Atiteration 2, it is (1 − ρ)2�0 + [1 + (1 − ρ)]γJmax, andso on. Generalizing the process, at any iteration t, themaximum luciferin �max

j (t) of any glowworm j is then

Table 1Values of parameters that are kept fixed in all the simulations

ρ γ s ε β nt

0.4 0.6 0.03 0.05 0.08 5

given by:

�maxj (t) = (1 − ρ)t�0 +

t−1∑i=0

(1 − ρ)iγJmax (6)

Clearly,

�j(t) � �maxj (t) (7)

Since 0 < ρ < 1, from Eq. (6) we have that

as t → ∞, �maxj (t) →

(γ

ρ

)Jmax (8)

Using Eqs (7) and (8), we have the result in Eq. (5). �

Theorem 2. For all glowworms j co-located at peak-locations X∗

i associated with objective function valuesJ∗

i � Jmax (where i = 1, 2, . . . , m and m is the num-ber of peaks), assuming that the luciferin update rule inEq. (1) is used, �j(t) increases or decreases monotoni-

cally and asymptotically converges to �∗i =(

γρ

)J∗

i .

Proof: The stationary luciferin �∗i associated withpeak i satisfies the following condition:

�∗i = (1 − ρ)�∗i + γJ∗i ⇒ �∗i =

(γ

ρ

)J∗

i (9)

If �j(t) < �∗i for glowworm j co-located at peak-location X∗

i , then using Eq. (9) we have

J∗i (t) >

(ρ

γ

)�j(t) (10)

Now,

�j(t + 1) = (1 − ρ)�j(t) + γJ∗i > (1 − ρ)

�j(t) + γ

(ρ

γ�j(t)

)(11)

⇒ �j(t + 1) > �j(t) (12)

i.e., �j(t) increases monotonically.Similarly, if �j(t) > �∗i for glowworm j co-located

at peak-location X ∗i , then using Eqs (9) and (11), it is

easy to show that

�j(t + 1) < �j(t) (13)

i.e., �j(t) decreases monotonically.From Eqs (12) and (13), it is clear that the luciferin

�j(t) of glowworm j co-located at a peak-location X ∗i

asymptotically converges to �∗i .


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J 6

J3(x, y) J4(x, y), m =2

Fig. 2. The multimodal function profiles chosen to test the GSO algorithm.

4. Simulation results

The glowworm algorithm is tested on the followingset of multimodal functions:

J1(x, y) = 200 − (x2 + y2 − 11)2(14)−(x + y2 − 7)2

J2(x, y) = 20 + (x2 − 10 cos(2πx)(15)

+y2 − 10 cos(2πy))

J3(x, y) = 25 − �x� − �y� (16)

J4(X) =m∑

i=1

cos2(X(i)) (17)

In J3(x, y), �p� is defined as the nearest integer lessthan p. In J4(X), X is a position vector in the m-

dimensional search space with m � 2. Functions inEqs (14)–(16) are used for tests on two dimensionalsearch spaces and to represent the measured entity pro-file. We use the function J4(X) Eq. (17) for tests onhigher dimensional search spaces (results for m = 2,3, 4, and 5 are reported). The values of various param-eters that are fixed in all simulations is shown in Ta-ble 1. Parameters that are specific to each test functionis shown in Table 2.

4.1. Performance measures

Indices such as the number of iterations for conver-gence, fraction of peaks captured, and average mini-mum distance to peak locations are used to measure theperformance of the algorithm. A glowworm is consid-


ered to be co-located at a peak when it is located withina small ε-distance from the peak and it is assumed thata peak is captured when at least three glowworms areco-located at its location. The mean minimum distanceto the peak locations (sources) dminav gives a veryuseful insight in the present context, considering themultiplicity of the sources and the need to evaluate thedeviation of the agent’s final locations from the peaklocations. In particular, dminav is given by:

dminav =1n

n∑i=1

min{di1, . . . , diQ} (18)

where dij = ‖Xi − Sj‖, i = 1, . . . , n, j = 1, . . . , Q,Xi and Sj are the locations of glowworm i and sourcej, respectively, and Q is the number of peak locations.

4.2. Tests on the J1 function

The standard Himelblau multimodal function [4]modified to convert all local minima to local max-ima is given in Eq. (14). J1(x, y) has four localmaxima of equal heights at (3, 2), (−3.779,−3.283),(−2.805, 3.131), and (3.584,−1.848) (Fig. 2(a)).

4.2.1. Constant local-decision rangeAs a first step, the radial range ri

d of each glowwormis kept constant, in order to characterize the sensitivityof the number of peaks detected to the size of the local-decision domain. As noted earlier, the local-decisionrange greatly influences the determination of variouspeaks. When the decision-range is more than 9, all theglowworms moved to a single local maximum. Fig-ure 3(a) shows the emergence of the solution as thenumber of iterations increases when r i

d = 9. A radialrange of 6.8, 4, and 3 resulted in detection of 2, 3, and4 peaks, respectively (Figs 3(b)–3(d)).

4.2.2. Variable local-decision rangeThe number of peaks captured is observed to be a

strong function of the radial sensor range. Since it isassumed that a priori information about the objectivefunction (e.g., number of maxima and minima) is notavailable, the decision-range cannot be kept constantand must be made a varying parameter. Figures 4(a)and 4(b) show the emergence of the solution when thelocal-decision range update rule in [16] and Eq. (4) areused, respectively. During these simulations, a valueof ri

d(0) = 7 was chosen for each glowworm i. Notethat all the peaks are detected within 1500 iterationswhen the earlier update rule [16] is used. However, ittakes only 300 iterations to capture all peaks when themodified decision range update rule Eq. (4) is used.

Table 2The values of n and rs used for various test functions

Function Workspace-size n rs

J1 (−6, 6) × (−6, 6) 100, 150 9, 7, 6.8, 4, 3J2 (−5, 5) × (−5, 5) 1500 2J3 (−2, 2) × (−2, 2) 1000 0.75J4 Πm

i=1(−π, π) 200–6000 1.5, 4, 5(for m = 3, 4, 5)

4.2.3. Response to presence of a forbidden regionThe obstacle situation described in Fig. 5(a) is sim-

ulated by biasing the random initial placement of theglowworms in a manner that none of the glowworms isdeployed in a circular (forbidden) region of radius 1.5unit centered at (2, 2). The forbidden region is placedsuch that the peak located at (3, 2) lies within the for-bidden region. The simulation result in Fig. 5(b) showsthat, following the biased-random deployment as de-scribed above, there is no instance when a glowwormenters the forbidden region and since one of the peaksis obscured by the forbidden region, only three peaksare detected.

4.2.4. Effect of step-size on convergenceAccording to the glowworm algorithm presented in

this paper, a glowworm with the maximum luciferin ata particular iteration remains stationary during that iter-ation. Ideally, the above property leads to a dead-locksituation when all the glowworms are located such thatthe peak-location lies outside the convex-hull formedby the glowworm positions. Since the agent move-ments are restricted to the interior region of the convex-hull, all the glowworms converge to a glowworm thatattains maximum luciferin value during its movementswithin the convex-hull. As a result, all the glowwormsget co-located away from the peak-location. However,the discrete nature of the movement update rule auto-matically takes care of this problem which could bedescribed in the following way. During the movementphase, each glowworm moves a distance of finite step-size s towards a neighbor. Hence, when a glowworm iapproaches closer to a glowworm j that is located near-est to a peak such that the inter-agent distance becomesless than s, i crosses the position of j and becomes aleader to j. In the next iteration, i remains stationaryand j crosses the position of i thus regaining its lead-ership. This process of interchanging of roles betweeni and j repeats until they reach the peak.


The function J2(x, y) is called Rastrigin’s func-tion [23]. A set of 1500 agents is deployed in a region of


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(a) (b)

−6 −4 −2 0 2 4 6−6

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Y

(c) (d)

Fig. 3. Emergence of the solution when local-decision domain range a) rid

= 9 (one peak is detected). b) rid

= 6.8 (two peaks are detected). c)rid = 4 (three peaks are detected). d) rid = 3 (all four peaks are detected).

10×10 units centered at (0, 0) that contains a total num-ber of 100 peaks (Fig. 2(b)). Figures 6(a) to 6(d) showthe emergence of the solution and co-location of groupsof glowworms at various peaks (after 500 iterations) forthe constant and adaptive local-decision domain cases,respectively. Note that 92% of the peaks are capturedin the variable decision domain case (r i

d(0) = 2) asagainst a mere 8% in the constant decision domain case(ri

d(t) = 2, ∀ t). The above observation serves as arepresentative example to support the fact that the useof a variable decision domain, instead of a constantone, significantly improves the ability of the algorithmto capture multiple peaks.


J3(x, y) [7] is a step function (Fig. 2(c)) with 16regions at 7 different heights within a workspace of(−2, 2) × (−2, 2). The initial locations of the glow-worms, emergence of the solution, and the final solu-tion are shown in Figs 7(a), (b), and (c), respectively.Different square regions are marked according to theincreasing order of their function values. Note that re-gions marked with the same number have equal func-tion values. All the glowworms initially located in re-gion 7 remain stationary as they are located in the flatregion of global maxima. Note that glowworms in theinterior locations (except the ones that are very close


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0 500 1000 1500

0.5

1

1.5

2

Number of iterationsAver

age

min

imum

dis

tanc

e to

the

peak

loca

tions

(dm

inav

)

Earlier local−decision range update rule

Modified local−decision range update rule

(a) (b) (c)

Fig. 4. a) Emergence of solution when the local-decision range update rule in [16] is used. b) Emergence of solution when the local-decisionrange update rule Eq. (4) is used. c) Plot of dminav(t) for both the cases.

O

A

T

?

−6 −4 −2 0 2 4 6−6

−4

−2

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2

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6

X

Y

forbiddenregion

(a) (b)

Fig. 5. a) Situation where inter-agent communication helps a robot to select a feasible direction towards the source b) Response of the glowwormalgorithm to the presence of a circular forbidden region of radius 1.5 unit centered at (2,2). Emergence of the solution after 300 iterations.

to and on the edges) of all other regions move towardsand settle on the edges of the next higher peak-regions.Refer to Fig. 7(c) to observe this pattern.

4.5. Tests in higher dimensional spaces

We use the J4(X) function Eq. (17) to test the GSOalgorithm on higher dimensional spaces. The func-tion J4(X) has a total number of (2k + 1)m peaks, ofequal function value, in a search space whose range is∏m

i=1[−kπ, kπ], k = 1, 2, . . . Preliminary simulation

runs were carried for each dimension-case (m = 3, 4,5) where the number of glowworms was fixed at somevalue and the percent of peak-captures was logged forvarious values of rs. The value of rs that gave peakperformance in each case was used in the main sim-ulations to test the performance of the algorithm as afunction of number of agents. Maximum performanceis observed at rs = 1.5, 4, and 5 for m = 3, 4, and5, respectively. Figures 8(a), (b), and (c) show thenumber of glowworms co-located at each peak for thethree-dimensional case, when n = 200, 600, and 1200,


Y

(a) (b)

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1

2

3

4

5

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

(c) (d)

Fig. 6. a) Solution of the Rastrigin’s function when the local-decision domain range rid

is kept constant. b) Solution of the Rastrigin’s functionwhen the local-decision domain range rid is made adaptive. c) Co-location of groups of glowworms at various peaks in constant decision domaincase. d) Co-location of groups of glowworms at various peaks in the variable decision domain case.

respectively. Peak-captures of 26%, 85%, and 100%are obtained in the above three cases, respectively. Thenumber of peak-captures increases with increase in thevalues of n. The plot of dminav(t) for various valuesof n is shown in Fig. 8(d). Note that increase in thenumber of glowworms also increases the rate of con-vergence. Figure 9 shows the peak-capture results forthe four dimensional case. Peak-captures of 36%, 85%,and 90% are obtained when n = 1000, 3000, and 5000are used, respectively. Figure 10(a) shows the numberof glowworms co-located at each peak for the five di-mensional case. A peak-capture of 46% was observedwhen n = 6000. To characterize the dependence of

required number of glowworms on the dimensionalityof the problem, the number of glowworms nm requiredfor 46% peak-capture for m = 1, . . ., 5 are obtainedand enumerated in Table 3. The values of nm/n(m−1)

show that the number of glowworms needed to capturea given fraction of the total number of peaks does notincrease exponentially.

5. Glowworms

Four small-wheeled robots christened Glowworms(named after the glowworm algorithm) were built


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50

60

70

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Num

ber o

f glo

ww

orm

s co

loca

ted

at e

ach

peak

0 50 100 150 200 250 3000

0.5

1

1.5

Number of iterations

Aver

age

min

imum

dis

tanc

e to

pea

ks (d

min

av(t)

)

n = 1200

n = 600

n = 200

(a) (b)

Fig. 8. a) Plot of number of glowworms co-located at each peak when m = 3, n = 600. b) Plots of average minimum distance to peak locationsdminav(t).

Table 3Number of glowworms required for 46% peak-capture of J6 functionfor m = 1, . . . , 5

m Number of peaks nm nm/n(m−1)

1 3 62 9 41 6.833 27 300 7.314 81 1500 55 243 6000 4

to conduct our experiments [19]. Each Glowworm(Fig. 10(b)) has been designed to provide features of ba-sic mobility on plain/smooth surfaces,obstacle sensing,relative localization/identification of neighbors, andinfrared-based luciferin glow/reception. A circular ar-ray of sixteen infrared transmitters placed radially out-ward is used as the glowworm’s beacon to obtain a nearcircular emission pattern around the robot. The glow

consists of an infrared light modulated by an 8-bit serialbinary signal that is proportional to the Glowworm’sluciferin value at the current sensing-decision-actioncycle (Refer to [19] for a description of the Glowwormhardware modules). A real-robot-implementation ofthe GSO algorithm was carried out in [19], using a setof four glowworms, in the context of a sound sourcelocalization task.

6. Comparison of GSO with related work

The GSO technique is a population-based algorithmand falls under the category of swarm intelligencemethods. The algorithm is in the same spirit as theACO and PSO techniques but is different in many as-pects that help in achieving simultaneous detection of


0 10 20 30 40 50 60 70 80 900

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Peak Number.

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0 100 200 300 400 5000

0.2

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Number of iterationsAver

age

min

imum

dis

tanc

e to

pea

k lo

catio

ns (d

min

av(t)

)

n = 1000

n = 2000

n = 3000

n = 4000

rs = 4

n = 5000

(a) (b)

Fig. 9. a) Plot of number of peak-captures for m = 4, n = 3000. b) Plots of average minimum distance to peak locations dminav(t).

multiple local optima of multimodal functions. Thisis a problem not directly addressed by ACO or PSOtechniques. Generally, ACO and PSO techniques areused for locating global optima. However, the objec-tive in this paper is to locate as many of the peaks aspossible. This requirement is the main motivation forformulating the GSO technique.

6.1. ACO vs. GSO

Generally, ACO techniques are used and found to beeffective in a discrete setting where gradient-based al-gorithms do not work too well. However, in the presentcase, GSO is applied to the continuous domain becausegradient-based algorithms do not produce satisfactoryresults when multiple peaks are sought. Note that con-tinuous multimodal test functions (Fig. 2) were con-sidered in all the simulation examples. The GSO isloosely based on Bilchev and Parmee’s approach [1,2], which is a special variant of ACO to solve continu-ous optimization problems, but with several significantmodifications. A brief description of this ACO variantis given in order to facilitate easier comparison with theGSO technique.

In Bilchev and Parmee’s approach [1], a finite setof regions are randomly placed in the search space.Each path between the nest location and a region iis associated with a virtual pheromone τi(t) at eachiteration t. Initially, τi(t = 0) = τ0 for all agents. Theprobability that an agent selects region i is given by:

pi(t) =ταi (t)ηβ

i (t)∑Nj=1 τα

j (t)ηβj (t)

(19)

where, ηi(t) reflects the local-desirability of a portionof the solution, α and β represent relative weights, andN is the number of regions. The agent then moves tothe selected region’s center, measures the value of theobjective function at that point, moves a short distancein a random direction, shifts the region’s center to thenew point if it finds an improvement in the solution, andthen comes back to the nest. The pheromone updateassociated with the region is given by

τi(t + 1)={(1 − ρ)τi(t) + γ∆J, if ∆J > 0(1 − ρ)τi(t), Otherwise

(20)

where, ∆J is the improvement made in the solutionand γ is a proportionality constant.

This process is repeated with a new probability dis-tribution according to Eq. (19). With the increase innumber of iterations, the pheromone concentration as-sociated with inferior regions decay (and may disap-pear eventually) and good regions get reinforced withtime, finally converging to the solution.

The virtual pheromones associated with the variousregions in the above variant of ACO technique is equiv-alent to the luciferin quantities carried by the glow-worms in the GSO technique. However, the crucial dif-ference lies in the manner in which the stigmergic com-munication is used to make agent-movement decisions.In the ACO technique, each agent at the nest selectsa region based on a probability distribution Eq. (19)which is a function of the pheromone levels associated


Table 4ACO versus GSO

Standard ACO GSO

1 Effective in discrete setting [6] Applied to continuous domain2 Global optimum or multiple global optima of equal value [24,26] Multiple optima of equal or unequal values

Special variant of ACO [1]1 Cannot be applied when ants (agents) have limited sensing range Useful for applications where robots have limited sensor range2 Global information used Local information used3 Pheromones associated with paths from nest to regions Luciferin carried by and associated with glowworms4 Pheromone information used to select regions Luciferin information used to select neighbors5 Shifting of selected region’s center in a random direction Deterministic movements towards selected neighbor

0 50 100 150 200 2500

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peak

Luciferin glow in a circular region

Distance sensor

Luciferin receptor

Conic reflector

Processingelectronics

Sweep-platform

Safe-wanderingplatform

(a) (b)

Fig. 10. a) Number of peak-captures when m = 5 and n = 6000 are used. b) The wheeled mobile robot prototype, Glowworm, built to conductthe multiple sound source localization experiment.

with all the N regions. In contrast, each glowworm inthe GSO technique uses the luciferin information avail-able only in its current local neighborhood to selecta neighbor with higher luciferin value. The last col-umn in Table 2 shows different values of rs that wereused in the simulations in order to limit the maximumsize of local-domain of all the glowworms. While theselected region’s center is shifted to a new point in arandom direction in the ACO variant, each glowwormdeterministically moves a step distance towards the se-lected neighbor. The main differences between GSOand ACO are summarized in Table 4.

6.2. PSO vs. GSO

Particle swarm optimization is a population-basedsearch algorithm introduced by Kennedy and Eber-hart [13] and is initialized with a population of randomsolutions, called particles. Each particle moves in thesearch space with a velocity that is dynamically ad-justed according to the historical behaviors of the parti-

cle itself and its neighbors. The particles exhibit a taxisbehavior toward favorable regions over the course ofthe search process. Let Xi and Vi be the n-dimensionalposition and velocity vectors of the ith particle, respec-tively. In the global variant of PSO, the position andvelocity updates of the ith particle are given by:

Vi(t + 1) = Vi(t) + c1r1(Pi(t) − Xi(t))(21)

+c2r2(Pg(t) − Xi(t))

Xi(t + 1) = Xi(t) + Vi(t + 1) (22)

where, i = 1, 2, . . . , N , is the particle’s index, N isthe particle population-size, c1, c2 are positive con-stants, referred to as the cognitive and social param-eters, respectively, r1 and r2 are random numbersthat are uniformly distributed within the interval [0, 1],t = 1, 2, . . ., indicates the iteration number, Pi(t) is thebest previous position encountered by the i th particle upto time t, and g is the index of the particle that attainedthe best previous position among all the individuals ofthe swarm. Hence, the PSO algorithm uses a memory


Table 5PSO versus GSO

PSO GSO

1 Uses a memory element Memoryless2 Direction of movement based on previous best positions Agent movement along line-of-sight with a neighbor3 Dynamic neighborhood based on k nearest neighbors Local decision domain based on varying range4 Neighborhood range covers the entire search space Maximum range hard limited by finite sensor range5 Limited to numerical optimization models Effective detection of multiple peaks/sources in addition to numerical opti-

mization tasks

element in the velocity update mechanism of the parti-cles. However, the GSO algorithm is memoryless andthe glowworms do not retain any information in thememory. While the directions of particle movements inPSO are adjusted according to its own and global bestprevious positions, movement directions are alignedalong the line-of-sight between neighbors in the GSOalgorithm. In the local variant of PSO, Pg is replacedby the best previous position encountered by particlesin a local neighborhood. Impact of various neighbor-hood structures such as circle, star, wheel, pyramid, andvon Neumann2 square topology [14,15] and dynami-cally changing neighborhoods [12] on the performanceof the PSO technique has been investigated. The vari-able local decision domain is one of the crucial aspectsin which GSO is different from PSO which is usedeffectively to locate the multiple peaks. In PSO, thedynamic neighborhood is achieved by evaluating thefirst k nearest neighbors. Such a neighborhood topol-ogy is limited to computational models only and is notapplicable in a realistic scenario where multiple sourcelocations are sought and the neighborhood size is de-fined by the limited sensor range of the mobile agents.To address this problem, in GSO, the requirement of kneighbors is used only as an implicit parameter to con-trol the range of the variable decision domain and themaximum range is made relatively small when com-pared to the total size of the search space. Parsopoulosand Vrahatis [24] consider a multimodal function withmultiple global optima (of equal function value) andcombine the constriction-factor based PSO variant witha repulsion technique in order to sequentially detect allthe optimizers. However, GSO is tested on a similarfunction (Fig. 2(a)) to simultaneously capture all thepeaks (Fig. 4). In general, GSO follows a simultaneousapproach to capture the multiple local optima of anymultimodal function. The main differences betweenGSO and PSO are summarized in Table 5.

2Named after its use in cellular automata pioneered by John vonNeumann; the neighborhood structure comprises of a square andindividuals are indexed according to a rectangular matrix so that eachone is connected to the individuals above, below, and to each side ofit, wrapping the edges.

7. Concluding remarks

A novel glowworm swarm algorithm that can beused to find solutions to multimodal function optimiza-tion problems is presented. The algorithm developedin this paper could be applied to a class of problemsrelated to collective robotics. The glowworm algo-rithm is evaluated on a series of standard multimodaltest functions from the literature. Tests on the stair-case function show that the algorithm can also handlediscontinuities in the chosen objective function. Theglowworm metaphor approach to multimodal optimiza-tion problems raises several intriguing questions aboutthe relation of parameter values and tuning of the al-gorithm based on different applications. Future workinvolves an analysis of the effect of various parame-ters on algorithmic performance, a theoretical analysisof the glowworm algorithm, and a thorough qualita-tive/quantitative comparison with other nature-inspiredoptimization algorithms. Some work in this direction,to establish the theoretical foundations of GSO, hasbeen carried out and will appear elsewhere [18].

References

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[2] E. Bonabeau, M. Dorigo and G. Theraulaz, Swarm Intelli-gence: From Natural to Artificial Systems, Oxford UniversityPress, 1999, 183–203.

[3] A. Corona, M. Marchesi, C. Martini and S. Ridella, Mini-mizing multimodal functions of continuous variables with thesimulated annealing algorithm, ACM Transactions on Mathe-matical Software 13 (1987), 262–280.

[4] K. Deb, Optimization for Engineering Design: Algorithmsand Examples, Prentice-Hall of India, Private Limited, NewDelhi, 1995, 326.

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[12] X. Hu and R. Eberhart, Multiobjective Optimization Using Dy-namic Neighborhood Particle Swarm Optimization, Proceed-ings of IEEE Congress on Evolutionary Computation, May2002, 1677–1681.

[13] J. Kennedy and R. Eberhart, Particle Swarm Optimization,Proceedings of IEEE International Conference on Neural Net-works, 1995, 1942–1948.

[14] J. Kennedy and R. Mendes, Neighborhood Topologies inFully-Informed and Best-of-Neighborhood Particle Swarms,Proceedings of IEEE International Workshop on Soft Com-puting in Industrial Applications, June 2003, 45–50.

[15] J. Kennedy, Small Worlds and Mega-Minds: Effects of Neigh-borhood Topology on Particle Swarm Performance, Proceed-ings of IEEE Congress on Evolutionary Computation, July1999, 1931–1938.

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[18] K.N. Krishnanand and D. Ghose, Theoretical Foundations forMultiple Rendezvous of Glowworm-inspired Mobile Agentswith Variable Local-Decision Domains, Proceedings of Amer-ican Control Conference, Minneapolis, Minnesota, June 2006,(to appear).

[19] K.N. Krishnanand and D. Ghose, Glowworm-Inspired RobotSwarm for Simultaneous Taxis Towards Multiple RadiationSources, Proceedings of IEEE International Conference onRobotics and Automation, Orlando, Florida, May 2006 (toappear).

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[21] Y. Liu and K.M. Passino, Stable social foraging swarms in anoisy environment, IEEE Transactions on Automtic Control49 (2004), 30–43.

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Authors’ Bios

K.N. Krishnanand received his B.E. (Hons.) de-gree in Electrical and Electronics from Birla Insti-tute of Technology and Science, Pilani, India andM.Sc. (Engg.) degree from Indian Institute of Sci-ence, Bangalore, India, in 1998 and 2004, respectively.He worked as a research scientist at the Institute ofRobotics and Intelligent Systems (IRIS), Bangalore,India, during 1998–2001. He is currently pursuing hisPh.D. study in the area of Collective Robotics at theDepartment of Aerospace Engineering, Indian Instiuteof Science, Bangalore, India. His research interests in-clude use of emergent behavior in multi-robot systems,design of mobile platforms for collective robotics ex-periments, swarm-based optimization, and agreementproblems in multi-agent networks.

Debasish Ghose obtained a BSc(Engg) degree fromthe National Institute of Technology (formerly the Re-gional Engineering College), Rourkela, India, in 1982,and an ME and a PhD degree, from the Indian Instituteof Science, Bangalore, in 1984 and 1990, respectively.He is presently a Professor of Aerospace Engineeringat the Indian Institute of Science. His research interestsare in guidance and control of aerospace vehicles, col-lective robotics, multiple agent decision-making anddistributed decision-making systems, and schedulingproblems in distributed computing systems. He is anauthor of the book Scheduling Divisible Loads in Par-allel and Distributed Systems published by the IEEEComputer Society Press. He has held visiting positionsat several other universities such as the University ofCalifornia at Los Angeles and the Kangwon NationalUniversity in South Korea.

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