globally convergent modifications of particle swarm optimization for unconstrained optimization

In: Particle Swarm Optimization: Theory, Techniques...Editor: Andrea E. Olsson

ISBN: 978-1-61668-527-0c⃝ 2010 Nova Science Publishers, Inc.

Chapter 5

GLOBALLY CONVERGENT M ODIFICATIONS OF

PARTICLE SWARM OPTIMIZATION FOR

UNCONSTRAINED OPTIMIZATION

Emilio F. Campana1∗, Giovanni Fasano1,2†, Daniele Peri1‡1Istituto Nazionale per Studi ed Esperienze di ArchitetturaNavale (INSEAN),

via di Vallerano, 139 – Roma, ITALY2Dipartimento di Matematica Applicata, Universita Ca’Foscari Venezia,

San Giobbe, Cannaregio, 30123 Venezia, ITALY

Abstract

We focus on the solution of a class of unconstrained optimization problems, wherethe evaluation of the objective function is possibly costlyand the use of exact algo-rithms may require a too large computational burden. Several real applications, in-cluded in the latter class, claim for optimization methods where the derivatives of theobjective function are unavailable and/or the objective function must be treated as a‘black-box’. Many design optimization [15] and shape optimization [11, 26] problemsbelong to the latter class; moreover, the derivatives computed with finite differencesmay be much inaccurate. Here, expensive simulations provide information to the opti-mizer, so that each function evaluation could require up to several CPU-hours. On theother hand, for continuously differentiable functions theuse of heuristics may yieldinadequate and/or unsatisfactory results [17].We consider here the evolutionary Particle Swarm Optimization (PSO) algorithm [12].We introduce some globally convergent modifications of PSO by drawing our inspira-tion from [14], so that sequences of points are generated which admit stationary limitpoints for the objective function. The latter result is carried out for a generalized PSOscheme, where suitable ranges of the parameters are identified in order to possiblyavoid diverging trajectories for the particles [1].

Keywords : Global Optimization, Unconstrained Optimization, Particle Swarm Optimiza-tion, Derivative-free methods, Globally Convergent methods.

∗Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]

2 Emilio F. Campana, Giovanni Fasano and Daniele Peri

1. Introduction

Most of the challenging applications of optimization in applied sciences involve a remark-ably large computational cost, both in terms of machine resources and time of computation.This implies that the investigation and the use of optimization tools is a promising and ac-tive research area. The use and computability of the derivatives of a nonlinear function isjust an example of the issues in this area (see for instance [6]).In particular, engineering design encompasses real-worldproblems where both practition-ers and theoreticians are continuously asked to provide robust solutions and theoreticaladvances. Indeed, here the large number of variables often requires hundreds or thousandsof function evaluations (each evaluation possibly taking hours), in order to provide a re-liable optimal design; not to mention the difficulties to generate and use the derivatives.Hence, the use of efficient and effective derivative-free methods seems an appealing topicto investigate, in order to solve the latter problems. Furthermore, in case the formulationof the problem in hand involvesnoisy functions, the use of the derivatives imposes stronglimitations to adopt finite differences.

In this paper we focus on a modification of the PSO algorithm [12, 19], for the solutionof the unconstrainedglobal optimization problem

minx∈IRn

f(x), f :IRn → IR. (1.1)

At presentf(x) is assumed to be a nonlinear and non-convexcontinuous function. Morespecifically, the PSO algorithm attempts at detecting a global minimum of (1.1), i.e. a pointx∗ ∈ IRn such thatf(x∗) ≤ f(x), for anyx ∈ IRn.When the functionf(x) is computationally costly, exact methods may be too expensiveto solve (1.1), or they possibly do not provide a current satisfactory approximation of thesolution. In these cases the use of heuristic methods may be fruitful, whenever the compu-tational resources and/or the time allowed for the computation are severely limited. On thisguideline, PSO proved to be both effective and efficient on several practical applicationsfrom real life [19].

On the other hand, the use of heuristic methods which also generate sequences of iter-ates satisfying convergence properties, would be much attractive. Thus, we focus here onsome modifications of the PSO algorithm, where converging subsequences of iterates aregenerated. In particular, the modifications proposed guarantee that the generated sequencesof iterates converge to stationary points of the objective function (see also [9, 10, 21, 23]).We carry on our analysis in order to provide sufficient conditions, which yield the con-vergence of the resulting method. Observe that the aim of this paper is to provide robustmethods with a twofold purpose. First we want to gain advantage of PSO fast approach to aglobal solution; then, we aim at steering the solution towards a stationary point, satisfyingfirst order optimality conditions off(x).

This paper is specifically concerned with theoretical properties of some modificationsof PSO. A complete numerical experience on this issue requires a wide and detailed inves-tigation, which is beyond the scope of the paper and deservesa paper on its own.

In this paper we use the subscripts to identify the particlesin a PSO scheme, while thesuperscripts indicate the iteration. We denote byI the identity matrix of suitable dimension.

Globally Convergent Modifications of Particle Swarm Optimization... 3

The symbolwk = O(zk) indicates that there exist an indexk and a constanta > 0, suchthat the real sequences{wk} and{zk} satisfy∣wk∣ ≤ a∣zk∣, for anyk > k.

In Section 2.-3. we describe a generalized PSO iteration. Then, the Sections 4.-5. intro-duce both the theory and the motivations for our modificationof PSO iteration. Finally, inSections 6.-7. we describe some new algorithms and we carry out the related convergenceanalysis.

2. A Generalized Scheme for PSO

PSO solves (1.1) by iteratively generating subsequences ofpoints inIRn, which possiblyapproach a solution. At the current step of any subsequence,the next point both depends onthe position of the current point in the subsequence, and theinformation off(x) providedby the other subsequences. In particular, we use the subscript j to indicate the subsequence,while the superscriptk indicates the iterate in the subsequence.

We preliminarily consider the following PSO iteration for any k ≥ 0 (see [1]):

vk+1j = �

[

wkvkj + cjrj ⊗ (pkj − xkj ) + cgrg ⊗ (pkg − xkj )]

,

xk+1j = xkj + vk+1

j ,

(2.1)

wherej = 1, ..., P is used to indicate thej-th particle (i.e. thej-th subsequence of points),P is a finite integer, and the vectorsvkj andxkj aren-real vectors, which respectively repre-sent thespeed(i.e. the search direction) and thepositionof thej-th particle at stepk. Withrj ⊗ (pkj − xkj ) (similarly with rg ⊗ (pkg − xkj )) we indicate that every entry of the vector(pkj − xkj ) must be multiplied by a different value of the coefficientrj. Finally, then-realvectorspkj andpkg satisfy the condition

f(pkj ) ≤ f(xℓj), for any ℓ ≤ k, pkj ∈ {xℓj}

f(pkg) ≤ f(xℓj), for any ℓ ≤ k and j = 1, . . . , P, pkg ∈ {xℓj},(2.2)

while �,wk, cj , rj , cg, rg are positive bounded coefficients. Observe thatpkj represents the‘best position’ in thej-th subsequence, whilepkg is the ‘best position’ among all the subse-quences. We recall that the choice of the coefficients is often problem dependent, thoughseveral standard values for them were proposed in the literature [3, 20, 27]. Anyway, gen-eral rules for assessing the coefficients in (2.1) are still under investigation.

Of course relations (2.1) include the case where either theinertia coefficientwk or theconstrictioncoefficient� are used. Moreover, without loss of generality we assume that rjandrg are uniformly distributed random parameters withrj ∈ [0, 1] andrg ∈ [0, 1].

One possible generalization of (2.1) is obtained by assuming that possibly the speedvk+1j depends on theP vectorspkℎ − xkj (see also [16]),ℎ = 1, . . . , P , and not only on the

vectorspkj − xkj , pkg − xkj [1]. The resulting new iteration for thej-th particle is given for


anyk = 0, 1, ... by

vk+1j = �k

j

[

wkj v

kj +

P∑

ℎ=1

cℎ,jrℎ,j(pkℎ − xkj )

]

,

xk+1j = xkj + vk+1

j .

(2.3)

Observe that in (2.3) the coefficientscℎ,j andrℎ,j both depend on the particlej and theremaining particles (ℎ). It can be readily seen [2] that for each particlej, assuming thatrℎ,j is the same for all the entries of(pkℎ − xkj ), �

kj = �j andwk

j = wj , for anyk ≥ 0, theiteration (2.3) is equivalent to thediscrete stationary (time-invariant) system

Xj(k + 1) =

⎛

⎜

⎜

⎜

⎜

⎝

�jwjI −P∑

ℎ=1

�jcℎ,jrℎ,jI

�jwjI

(

1−

P∑

ℎ=1

�jcℎ,jrℎ,j

)

I

⎞

⎟

⎟

⎟

⎟

⎠

Xj(k) +

⎛

⎜

⎜

⎜

⎜

⎝

P∑

ℎ=1

�jcℎ,jrℎ,jpkℎ

P∑

ℎ=1

�jcℎ,jrℎ,jpkℎ

⎞

⎟

⎟

⎟

⎟

⎠

,

(2.4)where

Xj(k) =

⎛

⎝

vkj

xkj

⎞

⎠ ∈ IR2n, k ≥ 0. (2.5)

We observe that the sequence{Xj(k)} identifies the trajectory of thej-th particle in thestate spaceIR2n, and it can be split into thefree responseXjℒ(k) and theforced responseXjℱ (k) (see also [18]). In other words, for anyk ≥ 0, Xj(k) may be rewritten accordingwith

Xj(k) = Xjℒ(k) +Xjℱ (k), (2.6)

where

Xjℒ(k) = Φj(k)Xj(0), Xjℱ (k) =k−1∑

�=0

Hj(k − �)Uj(�), (2.7)

and (after a few calculations -see also [1])

Φj(k) =

⎛

⎜

⎜

⎜

⎜

⎝

�jwjI −P∑

ℎ=1

�jcℎ,jrℎ,jI

�jwjI

(

1−

P∑

ℎ=1

�jcℎ,jrℎ,j

)

I

⎞

⎟

⎟

⎟

⎟

⎠

k

, (2.8)


Hj(k − �) =

⎛

⎜

⎜

⎜

⎜

⎝

�jwjI −P∑

ℎ=1

�jcℎ,jrℎ,jI

�jwjI

(

1−

P∑

ℎ=1

�jcℎ,jrℎ,j

)

I

⎞

⎟

⎟

⎟

⎟

⎠

k−�−1

, (2.9)

Uj(�) =

⎛

⎜

⎜

⎜

⎜

⎝

P∑

ℎ=1

�jcℎ,jrℎ,jp�ℎ

P∑

ℎ=1

�jcℎ,jrℎ,jp�ℎ

⎞

⎟

⎟

⎟

⎟

⎠

. (2.10)

We remark that unlikeXjℱ(k), the free responseXjℒ(k) in (2.6)-(2.7) only depends on theinitial point Xj(0), and not on the vectorp�ℎ, � ≥ 0. The latter observation will be largelyused to carry out our results. In particular, the next section will be devoted to report somerelevant analysis on the free responseXjℒ(k) (see also [3]).

3. Issues on the Parameters Assessment in PSO

It is well known (see for instance [18]) that if thej-th trajectory{Xj(k)} in (2.6) is non-diverging, it satisfies the condition

limk→∞

Xj(k) = limk→∞

Xjℱ (k), j = 1, . . . , P ;

i.e. the free responseXjℒ(k) is bounded away from zero only for finite values of the indexk. Moreover, from (2.8) we haveΦj(k) = Φj(1)

k, for anyk ≥ 0, and it was proved [2]that the2n eigenvalues of the unsymmetric matrixΦj(1) are real. In particular, by settingfor the sake of simplicity in (2.8)

aj = �jwj , !j =

P∑

ℎ=1

�jcℎ,jrℎ,j, (3.1)

we can prove [1] that the matrixΦj(1) has at most the two distinct eigenvalues�j1 and�j2

with

�j1 =1− !j + aj −

[

(1− !j + aj)2 − 4aj

]1/2

2

�j2 =1− !j + aj +

[

(1− !j + aj)2 − 4aj

]1/2

2.

(3.2)

In addition, each of them has algebraic multiplicityn. A necessary (but in general notsufficient) condition for thej-th trajectory{Xj(k)} to be non-diverging, is provided by thefollowing result (see also [18]), which imposes someconditions on the coefficients of PSOiteration.

Proposition 1 Consider the PSO iteration (2.3). For anyj ∈ {1, . . . , P} and anyk ≥ 0,let rj,ℎ be the same for all the entries of the vector(pkℎ − xkj ), with�k

j = �j andwkj = wj.


Suppose that for any particlej ∈ {1, . . . , P} the eigenvalues�j1 and�j2 in (3.2) satisfythe conditions

∣�j1∣ < 1

∣�j2∣ < 1.(3.3)

Then, for anyj the sequence{Xjℒ(k)} satisfieslimk→∞Xjℒ(k) = 0. The condition (3.3)is also a necessary condition for the trajectory{Xj(k)} to be non-diverging. □

We highlight that most of the typical settings for PSO parameters proposed in the literature(see e.g. [3, 27]), satisfy the condition (3.3). In the lightof the results in Proposition 1, acouple of issues still arise, which deserve further consideration.

1. The hypotheses in Proposition 1 neither ensure that for a fixedj the sequence{Xj(k)}is converging, nor they guarantee that{Xj(k)} admits limit points. I.e., for a fixedjthere may be diverging subsequences of{Xj(k)} even if (3.3) holds.

2. Suppose that the sequence{Xj(k)} converges fork → ∞, i.e. {Xj(k)} → X∗j with

(see (2.5))

X∗j =

⎛

⎝

v∗j

x∗j

⎞

⎠ .

Then,x∗j may fail to be a local minimum off(x), i.e. the property

f(x∗j) ≤ f(x), ∀x s.t. ∥x− x∗j∥ ≤ �, � > 0,

may not be satisfied.

Observe that the first issue was addressed and partially investigated in [13, 21, 23]. Here wefocus on the second issue above. On this purpose, we claim that under mild assumptions ifthe functionf(x) is continuously differentiable, it is possible to modify the PSO iteration insuch a way that the sequence{x11, . . . , x

1P , . . . , x

k1 , . . . , x

kP } admitsstationary limit points

for f(x), i.e. either of the following properties holds

lim infk→∞

∥

∥

∥∇f(xkj )∥

∥

∥ = 0

limk→∞

∥

∥

∥∇f(xkj )

∥

∥

∥= 0.

(3.4)

The next sections deal with the latter claim, which will be proved theoretically. We will giveevidence that the satisfaction of condition (3.4) may be metat the expense of a reasonablylarger computational cost, i.e. an increase of the number offunction evaluations.

4. Our Optimization Framework

As described in the Introduction, in the last decades many design optimization and simulation-based optimization problems have claimed for more effective and robust methods, which


do notexplicitly use derivatives. Meanwhile, the advances on parallel and distributed com-puting have considerably helped to reduce the impact of the strong computational burdenof challenging problems. The combination of the latter two trends has yielded a maturefield of research, where efficient algorithms show both a complete convergence analysisand noteworthy performance: namelydirect search methods. In the latter class we include(see [22, 13]) all the optimization methods which do not use derivatives and are simplybased on “the ranks of a countable set of function values”.

In particular, we deal with iterative methods in the subclass of Generating Set Search(GSS), where at each iteration a suitable set of search directions generating a cone is consid-ered, in order to guarantee a decrease of the objective function. Thepattern search methods[22] and thederivative-free methods[4, 7] are both included in GSS. The first provide con-vergence analysis by enforcing at each iteration asimple decreaseof the objective functionf(x), on suitable geometric patterns. On the other hand, the second group imposes asuffi-cient decreaseof the objective function by relying on a local model off(x). We highlightthat the schemes described do not encompass several heuristics, which are often broadlyused in the literature (see [13] and the cited references).

The local convergence analysis of GSS methods may be fruitfully combined with othertechniques, in order to provide globally convergent algorithms. On this purpose, examplesof combined methods where evolutionary strategies and GSS schemes yield globally con-vergent algorithms, can be found in [8, 9, 10, 24, 25]. In particular, in the last two referencesPSO is combined with a pattern search framework, in order to provide methods convergingto stationary points.

Here, we similarly want to combine a PSO-based evolutionaryscheme with a linesearch-based derivative-free algorithm, in order to provide a unified convergence analysis yielding(3.4). The latter approach is motivated by the promising performance of derivative-freemethods when combined with a linesearch technique [14]. We also remark that a PSO-based approach combined with a trust-region framework was already proposed in the liter-ature [24, 25], in order to provide methods converging to stationary points.

In this section we consider the solution of the problem (1.1), by means of a modifiedPSO scheme, combined with a derivative-free globally convergent algorithm, based on alinesearch strategy. We study in particular some convergence properties of the sequences{xkj }, j = 1, . . . , P , under very mild assumptions onf(x). Moreover, we propose fouralgorithms for continuously differentiable functions, whose distinguishing feature is thegeneration of sequences of points, which admit stationary limit points for f(x). To thelatter purpose, here we also impose additional conditions on the sequences of coefficients{�k

j }, {wkj }, {cℎ,j}, {rℎ,j} in (2.3).

We highlight that in accordance with formulae (2.6)-(2.7),here we impose in our anal-ysis someconservativeconditions on the forced responseXjℱ (k) of the particlej. Inparticular, in order to solve (1.1) with PSO, we have to guarantee that suitable PSO param-eters exist such that the sequences{xkj }, for anyj, admit limit points. On this guideline,as remarked above, the Proposition 1 provides onlynecessary conditionsto guarantee theexistence of limit points for the sequences{xkj }, j = 1, . . . , P .

To sum up, in order to carry on a convergence analysis for PSO,in this section we focuson two main ingredients. First we provide conditions on the PSO iteration, in such a waythat a bounded and closed setℒ0 exists which satisfies{xkj } ⊂ ℒ0, for anyj andk (so that


for any fixedj the sequence{xkj } admits limit points inℒ0). Then, recalling that PSO is aheuristics and therefore the sequence{pkg} may not converge to a stationary point off(x)onℒ0, we modify as slightly as possible the PSO iteration so that either of the stationarityconditions holds

lim infk→∞ ∥∇f(pkg)∥ = 0

limk→∞ ∥∇f(pkg)∥ = 0.

The resulting algorithms are modified PSO schemes, which guarantee that under mild as-sumptions at least a subsequence of the points{xkj } converges to a stationary point (that ispossibly a minimum point) off(x).

5. Preliminary Theoretical Results

We are concerned with analyzing the derivative-free approach in [14], which is based on alocal model of the objective function. The proposals in [14]draw their inspiration from theidea of combining the purepattern searchandderivative-freeapproaches. Indeed, in [14]a suitable pattern of search directions is first identified, as in pattern search methods. Then,a one-dimensional linesearch is possibly performed along these directions, as in derivative-free schemes.We report here some mild (simplified) conditions, which will be considered for generatingsearch directions in modified PSO algorithms (see [14]).

Proposition 2 Let f : IRn → IR, with f ∈ C1(IRn). Suppose that for anyk the points inthe sequence{xk} are bounded. Let for anyk the directions{dkj }, j = 1, . . . , n + 1, beboundedand satisfy one of the following two conditions:

(a) the directionsdk1 , . . . , dkn+1 form a positively spanning set ofIRn, i.e. for anyw ∈

IRn, there existn + 1 coefficients�kj ≥ 0, j = 1, . . . , n + 1, such thatw =∑n+1

j=1 �kj d

kj ;

(b) the directionsdk1 , . . . , dkn areuniformly linearly independent. Moreover, the bounded

directiondkn+1 satisfies

dkn+1 =2n∑

ℓ=1

�kℓ

(

wk1 −wk

ℓ

�kℓ

)

, (5.1)

where

– the sequences{�kℓ }, ℓ = 1, . . . , 2n, arebounded, with�kℓ ≥ 0 and�k2n ≥ � > 0,for all k;

– given the vectorszkj ∈ IRn, j = 1, . . . , n, for anyk the vectors{wk1 , . . . , w

k2n}


in (5.1) are defined by

wkℎ =

⎧

⎨

⎩

zk⌊ℎ/2⌋+1 ℎ = 1, 3, 5, . . . , 2n − 1,

zkℎ/2 + �kℎ/2dkℎ/2 ℎ = 2, 4, 6, . . . , 2n,

(5.2)

�kj > 0 j = 1, . . . , n, (5.3)

limk→∞

�kj = 0 j = 1, . . . , n, (5.4)

and

∗ the points{wkℎ} are reordered and (possibly) relabelled in such a way that

f(wk1) ≤ f(wk

2) ≤ ⋅ ⋅ ⋅ ≤ f(wk2n−1) ≤ f(wk

2n);

∗ there exist constantsc1, c2 > 0 such that the sequences{zkj } and {�kj }satisfy

maxj=1,...,n

{�kj }

minj=1,...,n

{�kj }≤ c1, (5.5)

∥zkj − xk∥ ≤ c2�kj , j = 1, . . . , n; (5.6)

∗ the sequences{�kℓ }, ℓ = 1, . . . , 2n in (5.1), satisfy the condition

minj=1,...,n

{�kj } ≤ �kℓ ≤ maxj=1,...,n

{�kj }. (5.7)

Then, the following stationarity condition holds for the functionf(x)

limk→∞

∥∇f(xk)∥ = 0 if and only if limk→∞

n+1∑

j=1

min{

0,∇f(xk)T dkj

}

= 0. (5.8)

□

Observe that considering the sequence{xk} in (5.8), Proposition 2 suggests that it is pos-sible to provide necessary and sufficient conditions of stationarity. In particular, this can beaccomplished by simply exploiting at any point of the sequence{xk} the objective function(through its directional derivatives), along the directions dk1 , . . . , d

kn+1. Furthermore (see

[14]), in Table 1 we report a derivative-free method for unconstrained minimization, whichuses the results of Proposition 2, to generate sequences with stationary limit points. A fullconvergence analysis was developed for the AlgorithmDF-0a and the following conclusionwas proved (see [14] Proposition 5.1)

Proposition 3 Suppose the directionsdk1 , . . . , dkn+1 satisfy the Proposition 2. Consider the

sequence{xk} generated by the AlgorithmDF-0a and let the level setℒ0 = {x ∈ IRn :f(x) ≤ f(x0)} be compact. Then we have

lim infk→∞

∥∇f(xk)∥ = 0. (5.9)

□


Table 1. The derivative-free Algorithm DF-0a (see [14]).

Step 0. Setk = 0; choosex0 ∈ IRn, set�0 > 0, > 0, � ∈ (0, 1).

Step 1. If there existsyk ∈ IRn such thatf(yk) ≤ f(xk)− �k, then go toStep 4.

Step 2. If there existsj ∈ {1, . . . , n+ 1} and an�k ≥ �k such that

f(xk + �kdkj ) ≤ f(xk)− (�k)2,

then setyk = xk + �kdkj , set�k+1 = �k and go toStep 4.

Step 3. Set�k+1 = ��k andyk = xk.

Step 4. Findxk+1 such thatf(xk+1) ≤ f(yk), setk = k + 1 and go toStep 1.

Observe that the condition (5.9) is met only asymptotically; nevertheless, in the practicalapplication of AlgorithmDF-0a, a stopping condition occurs when�k at Steps 2 and 3becomes too small. We also note that at Step 4 we can possibly choosexk+1 ≡ yk, sincethe convergence analysisdoes notrequiref(xk+1) < f(yk). Furthermore relation (5.9)may be consistently strengthened by adopting a different strategy at Step 2 of the AlgorithmDF-0a. In particular, we highlight that at Step 2 justone directionof sufficient decreasefor the objective function is sought. On the contrary, if we modify (reinforce) Step 2 andconsider an exploitation off(x) alongall the directionsin the set{dk1 , . . . , d

kn+1}, we obtain

the AlgorithmDF-0b in Table 2. The following proposition was also proved in [14]andprovides a stronger result with respect to Proposition 3.

Proposition 4 Suppose the directionsdk1 , . . . , dkn+1 satisfy the Proposition 2. Consider the

sequence{xk} generated by the AlgorithmDF-0b and let the level setℒ0 = {x ∈ IRn :f(x) ≤ f(x0)} be compact. Then we have

limk→∞

∥∇f(xk)∥ = 0. (5.10)

□

Observe that the stronger result is obtained at the cost of a more expensive Step 2, wherethe linesearch procedure and the cyclic use ofall the directionsdk1 , . . . , d

kn+1, may increase

the number of function evaluations required to detect the stationary point. The use of theprocedureLINESEARCH() is aimed to determine the smallest possible steplength�k

j , suchthat a sufficient decrease off(x) is guaranteed.We recall that the Propositions 3 and 4 provide onlylocal convergence properties for theobjective functionf(x), similarly to any gradient method for continuously differentiable


Table 2. The derivative-free Algorithm DF-0b in [14].

Step 0. Setk = 0. Choosex0 ∈ IRn and�0j > 0, j = 1, . . . , n+ 1, > 0,

� ∈ (0, 1), � ∈ (0, 1).

Step 1. Setj = 1 andyk1 = xk.

Step 2. If f(ykj + �kj d

kj ) ≤ f(ykj )− (�k

j )2 then

compute�kj by LINESEARCH(�k

j , ykj , d

kj , , �) and set�k+1

j = �kj ;

else set�kj = 0, and�k+1

j = ��kj .

Setykj+1 = ykj + �kj d

kj .

Step 3. If j < n+ 1 then setj = j + 1 and go toStep 2.

Step 4. Findxk+1 such thatf(xk+1) ≤ f(ykn+1), setk = k + 1 and go toStep 1.

LINESEARCH(�kj , y

kj , d

kj , , �):

Compute the steplength�kj = min

{

�kj /�

ℎ, ℎ = 0, 1, . . .}

such that

f(ykj + �kj d

kj ) ≤ f(xk)− (�k

j )2,

f

(

ykj +�kj

�dkj

)

≥ max

⎡

⎣f(ykj + �kj d

kj ), f(y

kj )−

(

�kj

�

)2⎤

⎦ .

functions. In other words, starting from any initial pointx0 ∈ IRn, as long as the setℒ0

is compact, a stationary point which is possibly only alocal minimumis asymptoticallyapproached.


6. New Algorithms

Now we want to couple the PSO scheme described in Section 2. with the algorithms inTables 1 and 2, in order to possibly obtain new methods endowed with both local conver-gence properties andglobal strategies of exploration. In particular, we use the heuristicexploration of PSO to provide aglobal information onf(x), then a derivative-free schemeis used to enforce thelocal convergence towards stationary points.

On this guideline, the most obvious way to couple PSO and derivative-free schemesis by performing the PSO iteration in Section 2. up to the finite iterationk ≥ 0. Then,we could apply either AlgorithmDF-0a or Algorithm DF-0b after setting the initial point(see Tables 1 and 2)x0 = pkg . I.e., the local convergence is carried on starting from the

best point detected by PSO up to stepk. Unfortunately, the latter strategy is ablind se-quential application of two different algorithms, which does not join the advantages of thetwo approaches. On the contrary, we want to consider at once both theexploitation(thelocal strategy) and theexploration (the global strategy) of the objective function, at anystep of a new scheme. More explicitly we consider a PSO-type scheme, which provides theinvestigation of a global minimum overIRn, while retaining the asymptotic convergenceproperties of a (local) derivative-free technique.

We propose at first the AlgorithmDF-1a in Table 3. It is a derivative-free methodwhich uses for any iterationk the directionsdk1 , . . . , d

kn+1, described in Proposition 2. We

can prove the following result (see also [14]).

Proposition 5 Consider the AlgorithmDF-1a. Suppose the directionsdk1 , . . . , dkn+1 and

the sequences{zkj }, j = 1, . . . , n, satisfy the hypotheses of Proposition 2. Let the level setℒ0 = {x ∈ IRn : f(x) ≤ f(x0)} be compact. Then, the AlgorithmDF-1a generates thesequence of points{xk} such that

lim infk→∞

∥∇f(xk)∥ = 0. (6.1)

Proof.Observe that the AlgorithmDF-1a and the AlgorithmDF-0a differ only at Step 1 and

Step 4. In particular, observe that the AlgorithmDF-1a is obtained from the AlgorithmDF-0a, where the vectorsyk at Step 1 andxk+1 at Step 4 are computed by means of aPSO method. Thus, the convergence properties of the sequence generated by the AlgorithmDF-1a follow straightforwardly from Proposition 3. □

We remark that the Steps 1 and 4 of the AlgorithmDF-1a include the global explo-ration by using PSO. On the other hand the Steps 2 and 3 are substantially the same of theAlgorithm DF-0a.

In a very similar fashion we can also couple the AlgorithmDF-0b with a PSO method.Consequently, a conclusion as in Proposition 4 trivially holds for the resulting scheme (i.e.the condition (6.1) is reinforced giving condition (5.10)).

Another proposal to join PSO-type schemes and the linesearch-based derivative-freetechnique in Table 1, is the AlgorithmDF-2a reported in Tables 4-5. At Stepk theAlgorithm DF-2a exploitsthe functionf(x) in a neighborhood of the pointxk, along thedirectionsdk1 , . . . , d

kn+1. Then, the new pointxk+1 is generated in such a way that possibly

a sufficient decrease of the objective function is obtained.Observe that at Step0 of the


Table 3. The derivative-free Algorithm DF-1a.

Data. Setk = 0; choosex0 ∈ IRn andz0j ,v0j ∈ IRn, j = 1, . . . , P . Set�0 > 0, > 0,� ∈ (0, 1).

Step 1. Setℎk ≥ 1 integer. Applyℎk PSO iterations considering theP particles withrespective initial velocities and positionsvkj andzkj , j = 1, . . . , P . Set

yk = argmin1≤j≤P, ℓ≤ℎk{f(zℓj)}. If f(yk) ≤ f(xk)− �k, then setvkj = vk+ℎk

j

andzkj = zk+ℎk

j , and go toStep 4.

Step 2. If there existsj ∈ {1, . . . , n + 1} and an�k ≥ �k such that

f(xk + �kdkj ) ≤ f(xk)− (�k)2,

then setyk = xk + �kdkj , �k+1 = �k and go toStep 4.


Step 4. Setqk ≥ 1 integer. Applyqk PSO iterations considering theP particles withrespective initial velocities and positionsvkj andzkj , j = 1, . . . , P . Setxk+1 = argmin1≤j≤P, ℓ≤qk{f(z

ℓj)}; if xk+1 satisfiesf(xk+1) ≤ f(yk), then set

k = k + 1 and go toStep 1.

Algorithm DF-2a, a suitable set ofn+1 search directions is used, which meet the conditionsof Proposition 2. In particular, according with the guidelines of Proposition 2, for anyk we freely set then uniformly linearly independent directionsdk1 , . . . , d

kn, which exploit

local informationon the function. Then, the directiondkn+1 is generated by the resultingapplication of the modified PSO scheme in Table 5, which providesglobal informationonthe objective function. Observe that in particular, we apply here a PSO-based method withexactlyn particles (nevertheless the results can be extended readily to the case ofP > nparticles). Finally, we remark that at Step 4 of AlgorithmDF-2a if f(xk+1) > f(yk), thenthe PSO-based scheme (7.1)-(7.2) is substantially ineffective to improve the current iterateyk.

7. How to Generate Search Directions for Global Convergence

In order to prove the global convergence properties of the overall method in Tables 4-5,we remark that a modified PSO scheme must be adopted. In particular we focus on thealgorithm in (7.1)-(7.2), which is supposed to include (at least)n particles. In this scheme


Table 4. The derivative-free Algorithm DF-2a.

Data. Setk = 0, choosez0j ∈ IRn, j = 1, . . . , n. Setx0 = argmax1≤j≤n{f(z0j )}.

Let �0 > 0, �1 > 0, �2 > 0, 1 > 0, c2 > 0, � ∈ (0, 1).

Step 0. Set�k1 = ⋅ ⋅ ⋅ = �kn = �1/(k + 1)�2 . Either setdk1 , . . . , dkn+1 as in (b) of

Proposition 2, or computedk1 , . . . , dkn as in(b) of Proposition 2 and

dkn+1 by using procedurePSO-gen(k; dk1, . . . , dkn; z

k1 , . . . , z

kn; �

k1 , . . . , �

kn).

Step 1. If there existsyk ∈ IRn such thatf(yk) ≤ f(xk)− 1�k, then go to

Step 4.

Step 2. If there existsj ∈ {1, . . . , n + 1} and an�k ≥ �k such that

f(xk + �kdkj ) ≤ f(xk)− 1(�k)2,

then setyk = xk + �kdkj , �k+1 = �k and go toStep 4.


Step 4. Let xk+1 andzk+1j , j = 1, . . . , n, (possibly) satisfy (7.1)-(7.2).

If f(xk+1) ≤ f(yk) then go toStep 0, else setxk+1 = yk and choosezk+1j ∈ IRn, j = 1, . . . , n. Setk = k + 1, and go toStep 0.

we note that the position of the particles is (partially) affected by the choice of the directionsdk1 , . . . , d

kn (see relation (7.2)). Moreover, assuming that for anyk the directionsdk1 , . . . , d

kn

are assigned, in Table 5 we generate the directiondkn+1 by applying the PSO scheme (7.1)-(7.2) and using the sequence{zkj }.

Observe that in formulae (7.1)-(7.2) the projectionPB(c,�)(⋅) onto the convex compactsetB(c, �) is introduced, withc ∈ IRn, � > 0, in order to guarantee that the directiondkn+1 is bounded. For any stepk and any particlej we have (see also (2.1) and (2.3), wherewithout loss of generality we have replaced ‘⊗’ with a simple multiplication)

⎧

⎨

⎩

vk+1j = �k

j

⎡

⎣wkjPB(c,�)(v

kj ) +

n∑

ℎ=1,ℎ ∕=g

ckℎ,jrkℎ,j

[

pkℎ − PB(c,�)(zkj )]

+ckg,jrkg,j

[

xk+1 − PB(c,�)(zkj )]]

,

zk+1j = PB(c,�)(z

kj ) + vk+1

j ,

(7.1)


Table 5. The procedurePSO-gen(k; dk1, . . . , dkn; z

k1 , . . . , z

kn; �

k1 , . . . , �

kn).

Data: k; dk1 , . . . , dkn; zk1 , . . . , z

kn; �k1 , . . . , �

kn; � > 0, �1 > 0, �2 > 0, c ∈ IRn.

Stepk: Compute the vectorswk1 andwk

2n as

wk1 = argmin1≤j≤n

{

f(zkj ), f(zkj + �kj d

kj )}

,

wk2n = argmax1≤j≤n

{

f(zkj ), f(zkj + �kj d

kj )}

.

Compute the directiondkn+1 as

dkn+1 =(k + 1)�2

�1(wk

1 − wk2n).

where

pkℎ = argminℓ≤k

{

f[

PB(c,�)(zℓℎ)]}

, ℎ = 1, . . . , n,

xk+1 = argminℓ≤k, ℎ=1,...,n

{

f[

PB(c,�)(zℓℎ)]

, f[

PB(c,�)(zℓℎ + �ℓℎd

ℓℎ)]}

,(7.2)

andPB(c,�)(y) indicates theorthogonal projectionof vector y ∈ IRn onto thecompactandconvexsetB(c, �). Due to the computational burden which may be involved in theprojection over a convex set, we suggest to chooseB(c, �) among the following possibilities(see Figure 1):

∙ � ∈ IR andB(c, �) = {x ∈ IRn : ∥x− c∥2 ≤ �, � > 0}, which implies that for anyy ∈ IRn,

PB(c,�)(y) =

⎧

⎨

⎩

y if ∥y − c∥2 ≤ �,

c+ �y − c

∥y − c∥2otherwise.

∙ � ∈ IR andB(c, �) = {x ∈ IRn : ∣xi− ci∣ ≤ �, i = 1, . . . , n, � > 0}, which impliesthat for anyy ∈ IRn and anyi = 1, . . . , n

[

PB(c,�)(y)]

i=

⎧

⎨

⎩

yi if ∣yi − ci∣ ≤ �,

ci + � sgn(yi − ci) otherwise.

∙ � ∈ IRn andB(c, �) = {x ∈ IRn : ∣xi − ci∣ ≤ �i, �i > 0, i = 1, . . . , n}, which


Figure 1. Projections over the compact convex setB(c, �).

implies that for anyy ∈ IRn and anyi = 1, . . . , n

[

PB(c,�)(y)]

i=

⎧

⎨

⎩

yi if ∣yi − ci∣ ≤ �i,

ci + �i sgn(yi − ci) otherwise.

Now, let us consider the following assumption in order to prove the convergence results forthe AlgorithmDF-2a.

Assumption 7.1 Consider the modified PSO scheme (7.1)-(7.2). Suppose the condition(3.3) holds. Let in (7.2) the coefficients�kj , j = 1, . . . , n, k ≥ 0, be chosen as in Proposi-tion 2. Letℒ0 = {x ∈ IRn : f(x) ≤ f(x0)} be compact and let the convex setB(c, �) in(7.1)-(7.2) satisfyB(c, �) ⊇ ℒ0. Assume that in iteration (7.1) the sequences{�k

j }, {wkj },

{ckℎ,j}, {rkℎ,j} satisfy

(1) �kjw

kj = O(�k+1

j ), j = 1, . . . , n;

(2) �kj c

kℎ,jr

kℎ,j = O(�k+1

j ), j = 1, . . . , n, ℎ = 1, . . . , n, ℎ ∕= g;

(3) �kj c

kg,jr

kg,j = 1 +O(�k+1

j ). j = 1, . . . , n.

□

Note that the conditions(1), (2) and (3) in Assumption 7.1 can be readily fulfilled.Thus, the assumption on the coefficients{�kj } is not particularly restrictive. On the other


hand, also the assumptionB(c, �) ⊇ ℒ0 = {x ∈ IRn : f(x) ≤ f(x0)} is not strong, sinceour method is tailored for applications where physical bounds on the unknowns are usuallyeasy to determine.

Now, we are ready to prove the following result, which ensures that under mild assump-tions we can define a globally convergent modification of the PSO scheme in (7.1)-(7.2).

Proposition 6 Supposef : IRn → IR is continuously differentiable and consider the Algo-rithm DF-2a. Let the level setℒ0 = {x ∈ IRn : f(x) ≤ f(x0)} be compact. Assume thatfor any stepk the directionsdk1 , . . . , d

kn are uniformly linearly independent and bounded,

and let the Assumption 7.1 hold. If the directiondkn+1 in AlgorithmDF-2a is generated bythe procedurePSO-gen(⋅), then the AlgorithmDF-2a generates the sequence{xk} suchthat

(a) {xk} ⊂ ℒ0 and lim infk→∞

∥∇f(xk)∥ = 0. (7.3)

(b)

⎧

⎨

⎩

lim infk→∞

∥zkj − xk∥ = 0, j = 1, . . . , n;

lim infk→∞

∥(zkj + �kj dkj )− xk∥ = 0, j = 1, . . . , n.

(7.4)

ProofAs regards(a), by the hypotheses, for anyk the directionsdk1 , . . . , d

kn are bounded and

uniformly linearly independent. Furthermore, consider the Proposition 2, along with theStep 0 and Step 4 of the AlgorithmDF-2a. For anyk, with the settings

c1 = 1

�kj = �kℓ =�1

(k + 1)�2

j = 1, . . . , n, ℓ = 1, . . . , 2n, �1 > 0, �2 > 0,

wk2j = zkj + �kj d

kj j = 1, . . . , n,

wk2j−1 = zkj j = 1, . . . , n,

wk1 , . . . , w

k2n are renamed and relabelled so that f(wk

1) ≤ ⋅ ⋅ ⋅ ≤ f(wk2n),

�kℓ = 0 ℓ = 1, . . . , 2n − 1,�k2n = 1

the sequences{�kj }, {�kl } and{�kℓ } satisfy (5.3), (5.4), (5.5), (5.7) of Proposition 2. More-over, the vectorswk

1 , wk2n computed by the procedurePSO-gen(⋅) are bounded, inasmuch

as by (7.1)-(7.2)zkj is bounded anddkj is bounded by the hypothesis, for anyj. Now weprove that for anyk both (5.6) holds and also the directiondkn+1, generated by the procedurePSO-gen(⋅), is bounded. Thus, by the above choice of the coefficients�kℓ , ℓ = 1, . . . , 2n,relation (5.1) becomes

dkn+1 =1

�k2n(wk

1 − wk2n).

Now, by the choice of�kℓ and�kℓ , for any indexj the triangular inequality yields

∥dkn+1∥ ≤∥wk

1 − wk2n∥

�k2n=

∥wk1 − wk

2n∥

�kj≤

∥wk1 − xk∥

�kj+

∥wk2n − xk∥

�kj, (7.5)


wherexk is the current iterate in the sequence{xk}, generated in (7.2) by the AlgorithmDF-2a. Then, two cases have to be analyzed.Eitherwk

1 = zkj for somej (similarly if wk2n = zkj ), orwk

1 = zkj +�kj dkj for somej (similarly

if wk2n = zkj + �kj d

kj ). In the first case we have from Assumption 7.1, formulae (7.1)-(7.2)

and the boundedness of the convex setB(c, �)

∥wk1 − xk∥ = ∥zkj − xk∥

=∥

∥

∥PB(c,�)(z

k−1j ) +O(�kj )PB(c,�)(v

k−1j ) +

n∑

ℎ=1,ℎ ∕=g

O(�kj )[

pk−1ℎ − PB(c,�)(z

k−1j )

]

+[

1 +O(�kj )] [

xk − PB(c,�)(zk−1j )

]

− xk∥

∥

∥

≤∥

∥

∥PB(c,�)(z

k−1j ) +O(�kj ) + xk − PB(c,�)(z

k−1j )− xk

∥

∥

∥

≤ c2�kj , c2 > 0, (7.6)

where the first inequality follows from the relation

O(�kj )PB(c,�)(vk−1j ) +

n∑

ℎ=1,ℎ ∕=g

O(�kj )[

pk−1ℎ − PB(c,�)(z

k−1j )

]

+O(�kj )[

xk − PB(c,�)(zk−1j )

]

= O(�kj ).

Otherwise, whenwk1 = zkj + �kj d

kj for somej (similarly if wk

2n = zkj + �kj dkj ), we have

∥wk1 − xk∥ = ∥zkj + �kj d

kj − xk∥ =

∥

∥

∥

(

zkj − xk)

+ �kj dkj

∥

∥

∥≤ c2�

kj , c2 > 0,(7.7)

where the last inequality follows from the boundedness ofdkj and relation (7.6). Therefore,from (7.5)-(7.7) the directiondkn+1 is bounded. Moreover, from (7.6)-(7.7) it is readily seenthat for anyj we have∥zkj − xk∥ ≤ c2�

kj , i.e. (5.6) holds.

Finally, by (7.2) the vectorxk+1 is bounded for anyk. Moreover, from the definitionof x0 we havez0j ∈ ℒ0, j = 1, . . . , n, and consequently from (7.2){xk} ⊂ ℒ0. Indeed,either at Step 4 of AlgorithmDF-2a the vectorxk+1 is computed by (7.1)-(7.2), or it isxk+1 = yk.

Since the directionsdk1 , . . . , dkn+1 satisfy Proposition 2, the results of Proposition 3 hold,

i.e. the AlgorithmDF-2a yields the condition (7.3).As regards(b), the result follows directly by considering the relations (7.3) and (7.6)-

(7.7). □

Remark 7.1 A straightforward set of uniformly linearly independent directionsdk1 , . . . , dkn,

to be used in Proposition 6, is obtained by setting

dkj = ej, j = 1, . . . , n,


whereej is the j-th unit vector. We highlight that for anyk at Step 2 of the AlgorithmDF-2a, the sufficient decrease off(x) is checked along then+ 1 directionsdk1 , . . . , d

kn+1.

Anyway, the first direction which satisfies the test is chosenand the cyclic check stops.Consequently, in order to use as frequently as possible the directiondkn+1 (generated by theprocedurePSO-gen(⋅)), to update the pointyk at Step 2 of AlgorithmDF-2a, the searchoverj ∈ {1, . . . , n+ 1} should preferably be started withj = n+ 1.

Remark 7.2 Observe that by (7.2) the computational cost per iterationk of the procedurePSO-gen(⋅) amounts to2n function evaluations. Furthermore, item(b) of Proposition 6shows that eventually all the particles will cluster aroundthe stationary point detected.

On the guidelines of Proposition 4, we aim at extending the results of Proposition 6 sothat any subsequence of the sequence{xk} possibly converges to a stationary point. Inparticular, the AlgorithmDF-2b in Table 6 meets the latter requirement and the followingproposition holds.

Proposition 7 Supposef : IRn → IR is continuously differentiable and consider the Algo-rithm DF-2b, let the level setℒ0 = {x ∈ IRn : f(x) ≤ f(x0)} be compact and Assumption7.1 hold. Let for any stepk the directionsdk1 , . . . , d

kn be uniformly linearly independent

and bounded. If the AlgorithmDF-2b is applied, wheredkn+1 is generated by the procedurePSO-gen, then we have

(a) {xk} ⊂ ℒ0 and limk→∞

∥∇f(xk)∥ = 0. (7.8)

(b)

⎧

⎨

⎩

limk→∞

∥zkj − xk∥ = 0, j = 1, . . . , n

limk→∞

∥(zkj + �kj dkj )− xk∥ = 0, j = 1, . . . , n

(7.9)

ProofThe proof trivially follows by observing the correspondence of the AlgorithmsDF-0a

andDF-0b, with the AlgorithmsDF-2a andDF-2b. Thus, the results of Propositions 4 and6 yield (7.8)-(7.9). □

8. Conclusions

In this paper we have considered four different globally convergent modifications of thePSO iteration, applied for the solution of unconstrained global optimization problems. Wehave proved in Propositions 5 and 6 that under mild assumptions, at least a subsequence ofthe iterates produced by our modified PSO methods converges to a stationary point, whichis possibly a minimum point. This is a relatively strong result, if we consider that by nomeans the standard PSO iteration [12] can guarantee the convergence towards stationarypoints. In addition, the latter result is in our knowledge among the first schemes (see also[24, 25]) where a modified PSO scheme is proved to be globally convergent, i.e. either (7.3)or (7.8) holds. Moreover, this accomplishment contributesto fill the gap between the theoryand the numerical performance of PSO based methods.


Table 6. The derivative-free Algorithm DF-2b.

Data. Setk = 0, choosez0j ∈ IRn, setx0 = argmax1≤j≤n{f(z0j )}, j = 1, . . . , n.

Let �0 > 0, j = 1, . . . , n, �1 > 0, �2 > 0, > 0, c2 > 0, � ∈ (0, 1), � ∈ (0, 1).

Step 0. Set�k1 = ⋅ ⋅ ⋅ = �kn = �1/(k + 1)�2 . Either setdk1 , . . . , dkn+1 as in (b) of

Proposition 2, or computedk1 , . . . , dkn as in(b) of Proposition 2 anddkn+1

by means of the procedurePSO-gen(k; dk1, . . . , dkn; z

k1 , . . . , z

kn; �

k1 , . . . , �

kn).

Step 1. Setj = 1 andyk1 = xk.

Step 2. If f(ykj + �kj d

kj ) ≤ f(ykj )− (�k

j )2 then

compute�kj by LINESEARCH(�k

j , ykj , d

kj , , �) and set�k+1

j = �kj ;

else set�kj = 0, and�k+1

j = ��kj . Setykj+1 = ykj + �k

j dkj .

Step 3. If j < n+ 1 setj = j + 1 and go toStep 2.

Step 4. Let xk+1 andzk+1j , j = 1, . . . , n, (possibly) satisfy (7.1)-(7.2).

If f(xk+1) ≤ f(ykn+1) then go toStep 0, else setxk+1 = ykn+1 and choosezk+1j ∈ IRn, j = 1, . . . , n. Setk = k + 1, and go toStep 0.

Our conclusions have also been reinforced in Proposition 7,whereany subsequencegenerated by the modified PSO iteration was proved to converge to a stationary point. Wehighlight that this stronger result implies an additional computational burden. However, thelatter additional cost may be consistently reduced according with the indication reported inthe Remark 7.1.

Finally, considering the wide range of applications which require the use of efficientderivative-free algorithms, we guess that a new paper will be necessary to describe furthertheoretical results, along with the numerical tests. In particular, the removal of thecontinu-ous differentiabilityassumption for the objective functionf(x), seems the natural extensionof the theory described here. On this purpose, we need to include in our approach severalresults from non-smooth analysis.On the other hand, the extension of our approach to bounded and linearly constrained prob-lems, is another topic of great interest.


Acknowledgements

E.F.Campana and D.Peri would also like to thank the support of the US Office of NavalResearch through Dr. Ki-Han KIM (NICOP project 00014-0810957). G.Fasano wishes tothank the INSEAN research program “VISIR”, and Programma PRIN 20079PLLN7 “Non-linear Optimization, Variational Inequalities, and Equilibrium Problems”.

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globally convergent modifications of particle swarm optimization for unconstrained optimization

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