Avoiding Premature Convergence in a Mixed-Discrete Particle Swarm Optimization (MDPSO) Algorithm

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1 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs and Materals Conference<BR>20th AI Aprl 2012, Honolulu, Hawa AIAA rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs and Materals Conference, Aprl 2012, Honolulu, Hawa SDM 2012 Student Papers Competton Avodng Premature Convergence n a Mxed-Dscrete Partcle Swarm Optmzaton (MDPSO) Algorthm Souma Chowdhury and Je Zhang Rensselaer Polytechnc Insttute, Troy, New York and Achlle Messac Syracuse Unversty, Syracuse, NY, Over the past decade or so, Partcle Swarm Optmzaton (PSO) has emerged to be one of most useful methodologes to address complex hgh dmensonal optmzaton problems - t s popularty can be attrbuted to ts ease of mplementaton, and fast convergence property (compared to other populaton based algorthms). However, a premature stagnaton of canddate solutons has been long standng n the way of ts wder applcaton, partcularly to constraned sngle-objectve problems. Ths ssue becomes all the more pronounced n the case of optmzaton problems that nvolve a mxture of contnuous and dscrete desgn varables. In ths paper, a modfcaton of the standard Partcle Swarm Optmzaton (PSO) algorthm s presented, whch can adequately address system constrants and deal wth mxed-dscrete varables. Contnuous optmzaton, as n conventonal PSO, s mplemented as the prmary search strategy; subsequently, the dscrete varables are updated usng a determnstc nearest vertex approxmaton crteron. Ths approach s expected to avod the undesrable dscrepancy n the rate of evoluton of dscrete and contnuous varables. To address the ssue of premature convergence, a new adaptve dversty-preservaton technque s developed. Ths technque characterzes the populaton dversty at each teraton. The estmated dversty measure s then used to apply () a dynamc repulson towards the globally best soluton n the case of contnuous varables, and () a stochastc update of the dscrete varables. For performance valdaton, the Mxed-Dscrete PSO algorthm s successfully appled to a wde varety of standard test problems: () a set of 9 unconstraned problems, and () a comprehensve set of 98 Mxed-Integer Nonlnear Programmng (MINLP) problems. Keywords: constrant, dscrete varable, mxed-nteger nonlnear programmng (MINLP), Partcle Swarm Optmzaton, populaton dversty I. Introducton A. An Overvew of Partcle Swarm Optmzaton Partcle Swarm Optmzaton (PSO) s a stochastc optmzaton algorthm that mtates the dynamcs of socal behavor observed n nature. Ths algorthm was ntroduced by an Electrcal Engneer, Russel C. Eberhart, and a Socal Psychologst, James Kennedy. 1 The underlyng phlosophy of PSO and swarm ntellgence can be found n the book by Kennedy et al. 2 PSO has emerged over the years to be one of Doctoral Student, Multdscplnary Desgn and Optmzaton Laboratory, Department of Mechancal, Aerospace and Nuclear Engneerng. AIAA Student Member Dstngushed Professor and Department Char. Department of Mechancal and Aerospace Engneerng. AIAA Lfetme Fellow. Correspondng author. Emal: messac@syr.edu Copyrght c 2012 by Achlle Messac. Publshed by the, Inc. wth permsson. 1 of 20 Copyrght 2012 by Achlle Messac. Publshed by the, Inc., wth permsson.

2 the most popular populaton-based heurstc optmzaton approaches. Several varatons of PSO have been reported n the lterature, and appled to dverse optmzaton problems n engneerng, basc scences and fnance. 3 The modfcatons of the PSO algorthm presented n ths paper are nspred by the optmzaton challenges,encounteredntheauthors researchnproductfamlydesgn 4 andwndfarmoptmzaton. 5 Both these optmzaton problems (defned as sngle-objectve) nvolve complex multmodal crteron functons and a hgh dmensonal system of mxed-dscrete desgn varables. These problems generally requre a large number of system-model evaluatons; they also suffer from early premature convergence durng optmzaton when a standard populaton-based algorthm s used. In the case of constraned sngle-objectve optmzaton problems, populaton-based algorthms, e.g., evolutonary and swarm-based optmzaton methods, often suffer from premature stagnaton. 3 Ths undesrable property can be attrbuted to an excessve and mostly unopposed pressure of exploraton or evoluton. The smultaneous presence of contnuous and dscrete desgn varables that may experence dfferng rates of evoluton further complcates the optmzaton scenaro. In ths paper, a new method s developed to both characterze and nfuse dversty, adaptvely, nto the populaton of canddate solutons. Ths method s an evoluton from earler dversty-preservaton methods reported n the PSO lterature, whch are later dscussed n Secton D. The PSO algorthm presented n ths paper can address a mxture of dscrete and contnuous desgn varables. Two dstnct yet coherent approaches are developed to address the dversty-preservaton ssues for dscrete and contnuous varables. A comprehensve revew of the background and the development of Partcle Swarm Optmzaton based algorthms (tll 2007) can be found n the chapter by Banks et al. 6 An extensve follow up revew of the varous attrbutes of PSO, and the applcablty of PSO to dfferent classes of optmzaton problems: unconstraned/constraned, combnatoral, and multcrtera optmzaton, can be found n the book chapter by Banks et al. 3 Bref surveys of reported varatons of PSO that address the followng crtcal optmzaton attrbutes: () mxed-dscrete varables, () populaton dversty preservaton, and () constrant handlng, are provded n Sectons B, D, and C, respectvely. B. Exstng Mxed-Dscrete Optmzaton Approaches A sgnfcant amount of research has been done n developng algorthms for solvng Mxed-Integer Non- Lnear Programmng (MINLP) problems. Most of these algorthms are gradent-based search technques. Three major categores of gradent-based algorthms are () the branch and bound, () the cuttng plane, and () the outer approxmaton algorthms. A lst of these algorthms, related dscusson, and bblography can be found n the webstes of MINLP World 7 and CMU-IBM Cyber-Infrastructure for MINLP. 8 These algorthms possess attractve numercal propertes, namely () fast convergence, () proof of optma, and () an ntrnsc ablty to deal wth constrants. However, gradent-based algorthms do not readly apply to the broad scope of engneerng desgn problems that may nvolve hghly nonlnear, non-smooth and multmodal crteron functons. Amongpopulaton-basedoptmzatonmethods, bnarygenetcalgorthms(gas) 9,10 havebeenreported to be effectve for dscrete optmzaton. Bnary GAs convert the desgn varables nto bnary strngs. Ths process leads to an approxmate dscrete representaton of the contnuous varables. A populaton of canddate solutons, each represented by a bnary strng, evolve over generatons, through the four stages: () ftness assgnment, () selecton, () crossover, and (v) mutaton. One of the most popular bnary GAs s the bn-nsga-ii developed by Deb et al. 11 Genetc algorthms have been successfully mplemented on MINLP problems, such as batch plant desgn. 12,13 Another class of dscrete optmzaton algorthms, whch belong to Ant Colony Optmzaton (ACO), have also been reported n the lterature. 14,15 Applcatons of ACO-based algorthms to dscrete optmzaton problems nclude vehcle routng, sequental orderng, and graph colorng. There exsts n the lterature a handful of varatons of the PSO algorthm that can address dscrete and/or nteger varables. A summary of these varatons of PSO s dscussed n the followng secton. C. Mxed-Dscrete Partcle Swarm Optmzaton: Prncples and Objectves Ths paper presents fundamental modfcatons to the orgnal dynamcs of PSO, wth the am to solve hghly constraned sngle-objectve mxed-dscrete optmzaton problems. The development of ths Mxed-Dscrete PSO (MDPSO) s drven by the followng specfc objectves:. Develop an approxmaton technque that can address mxed-dscrete desgn varables through contnuous optmzaton; 2 of 20

3 . Include a constrant handlng technque to deal wth both equalty and nequalty constrants; and. Formulate an explct dversty preservaton technque to avod the stagnaton of partcles. Effcent dversty preservaton provdes a conducve envronment for the frst and the second objectves. Hence, the thrd objectve s consdered to be the prmary contrbuton of ths paper. A method s formulated to characterze the exstng dversty n the populaton and adjust the dversty parameter(s)/coeffcent(s) at every teraton. Ths approach provdes a generalzed adaptve regulaton of the populaton dversty, whch can be mplemented n a majorty of populaton-based optmzaton algorthms and s not restrcted to PSO. For example, the concerned dversty parameter can be () the mutaton probablty n genetc algorthms, 10 or () the tme-varyng acceleraton coeffcents (TVAC) n PSO 16 or () the wndow-sze of the hypercube operator n Predator-Prey algorthms, 17 or (v) the random selecton rate n Ant Colony Optmzaton. 18 A majorty of the exstng Mxed-Dscrete PSO algorthms are hndered by the effects of dfferng rates of evoluton of the contnuous and dscrete desgn varables. To avod ths lmtng scenaro, contnuous optmzaton s appled as the prmary search strategy for all varables, whether they are contnuous or dscrete. After the partcles have moved to ther new locatons, the dscrete component of the desgn vector for each partcle s approxmated to the nearest feasble dscrete doman locaton. In ths case, nearness s determned usng the Eucldan dstance n the dscrete varable space. As a result, although the varables evolve through contnuous search dynamcs, system-functon evaluatons are performed only at the allowed dscrete locatons. Ths approach s partly smlar to the strategy presented by Laskar et al. 19 A schematc of the proposed mxed-dscrete optmzaton approach for each canddate soluton s shown n Fg. 1. Iteraton: t = t + 1 Apply contnuous optmzaton Evaluate system model F (X c, X D-feas ) th canddate soluton X Contnuous varable space locaton X C Dscrete varable space locaton X D Neghborng dscrete-pont selecton crteron Approxmate to nearby feasble dscrete locaton X D-feas Fgure 1. Process dagram of the generalzed approach to MDNLO Constrant handlng n MDPSO s performed usng the prncple of constraned non-domnance that was ntroduced by Deb et al. 11 Ths method has been successfully mplemented n the Non-domnated Sortng Genetc Algorthm-II, 11 Modfed Predator-Prey algorthm, 17 and other standard evolutonary algorthms. The MDPSO algorthm nvolves a set of coeffcents that regulate the nerta, the personal behavor, the socal behavor, and the dversty preservng behavor of the partcles. Parameter selecton n PSO s far from trval, as dscussed n the prevous secton. However, detaled analyss of the selecton of PSO parameters, and the 3 of 20

4 ensung numercal behavor of the partcle dynamcs are not wthn the scope of ths paper. In ths paper, we specfcally ntend to provde. the detaled formulaton of the Mxed-Dscrete PSO algorthm,. the underlyng hypothess supportng the proposed modfcatons, and. the performance of ths modfed algorthm on a wde varety of test cases. It s mportant to note that, consderng the volume of nterestng research n PSO reported n the lterature over the past decade, other exstng characterstc modfcatons mght further advance the performance of the MDPSO algorthm. For valdaton purposes, the MDPSO algorthm s appled to () a set of standard unconstraned nonlnear optmzaton problems, 20,21 and () a comprehensve set of MINLP problems. 22 In the next Secton, the formulaton of the Mxed-Dscrete Partcle Swarm Optmzaton (MDPSO) algorthm s presented. Ths formulaton ncludes redefnng the partcle dynamcs, addressng dscrete varables n optmzaton, ncorporatng the constrant management technque, and developng the new dversty preservaton technque. Results and subsequent dscussons regardng the applcaton of MDPSO to varous standard test problems are provded n Secton III. II. Development of the Mxed-Dscrete Partcle Swarm Optmzaton (MDPSO) A. Dynamcs of Partcle Swarm Optmzaton 1. Lterature Survey: Dynamcs of Partcle Moton n PSO A balance between exploraton, explotaton, and populaton-dversty n PSO requres approprate quantfcaton of the PSO coeffcents, or what s more popularly termed as parameter selecton. One of the earlest strateges to balance exploraton and explotaton was the ntroducton of the nerta weght. 6 Sh and Eberhart 23 nvestgated the nfluences of the nerta weght and the maxmum velocty on the algorthm performance. Usng numercal experments, they proposed partcular values (and/or range of values) for the nerta weght and the maxmum velocty, and also suggested the applcaton of tme varyng nerta weght to further mprove the algorthm performance. Trelea 24 used standard results from dynamc systems theory to provde graphcal parameter selecton gudelnes. The applcatons of control theory by Zhang et al., 25 and chaotc number generaton by Alatas et al. 26 are among the recently proposed methods used to establsh parameter selecton gudelnes (for PSO). 2. Dynamcs of Partcle Moton n MDPSO A general mxed-dscrete sngle-objectve constraned mnmzaton problem nvolvng m dscrete varables and a total of n desgn varables can be expressed as Mn f (X) subject to g j (X) 0, j = 1,2,...,p h k (X) = 0, k = 1,2,...,q where [ ] X = x 1 x 2... x m x m+1... x n (1) where p and q are the number of nequalty and equalty constrants, respectvely. In Eq. 1, X s the desgn varable vector, where the frst m varables are dscrete and the next n m varables are contnuous. To solve ths optmzaton problem, the PSO algorthm s ntalzed wth N random partcles. To ths end, the Sobol s quasrandom sequence generator 27 s appled. Sobol sequences use a base of two to form successvely fner unform parttons of the unt nterval, and then reorder the coordnates n each dmenson. The locaton of each partcle n the swarm s updated usng a velocty vector at each teraton; the velocty vector of a partcle s varable, and s tself updated at every teraton. In the MDPSO algorthm, the velocty vector 4 of 20

5 update formula s redefned to allow for an explct dversty preservaton term. The modfed dynamcs of the partcle moton can be represented as where, X t+1 V t+1 = X t +Vt+1 = αv t and X t s the locaton of the th partcle at the t th teraton; r 1, r 2 and r 3 are random real numbers between 0 and 1; P s the best canddate soluton found for the th partcle; P g s the best canddate soluton for the entre populaton; +β lr 1 (P X t)+β gr 2 (P g X t) (2) t +γ c r 3ˆV α, β l and β g arethe user defned coeffcents that controlthe nertal, the explotve, and the exploratve attrbutes of the partcle moton; and γ c s the dversty preservaton coeffcent for contnuous desgn varables. The determnaton of the dversty preservaton coeffcent (γ c ) s dscussed n Secton D. The global (P g ) and the local best (P ) solutons are updated at every teraton usng the soluton comparson prncple. Ths soluton comparson prncple s based on the values of the correspondng crteron functons: objectve functon and constrant functons. Ths prncple s dscussed n Secton C. The contnuous update process (Eq. 2) s appled to all the desgn varables of a partcle. Followng ths process, the dscrete component of the desgn vector s updated to nearby feasble dscrete locatons. In ths case, feasblty pertans to the constrants mposed by the dscreteness of the varable space, and not to the system constrants. B. Addressng Dscrete Varables n PSO 1. Lterature Survey: Dscrete and Combnatoral PSO Several varatons of the PSO algorthm that can solve combnatoral optmzaton problems have been reported n the lterature. Kennedy and Eberhart 28 presented one of the earlest modfcaton of PSO to address bnary varables. They defned the trajectores of the bnary varables n terms of the change n the probablty that a value of one or zero wll be taken. Tasgetren et al. 29 used constructon/destructon operators to perturb the dscrete component of the varable vector of a partcle n solvng a Travelng Salesman problem. A smlar combnatoral-pso concept was also developed and used by Jarbou et al 30 for resource-constraned project schedulng. These varatons of the PSO algorthm provde effcent and robust performances typcally for combnatoral optmzaton problems that are smlar to the correspondng reported applcatons. A majorty of these methods do not readly apply to the broad scope of mxeddscrete optmzaton that nvolves problems wth: () ntegers and/or real-valued dscrete varables, () non-unformly spaced dscrete varable values (e.g., x [1,3,100,1000,...]) and () wdely dfferent szes of the set of feasble values for the dscrete varables (e.g., x 1 [0,1] and x 2 [1,2,...,1000]). Ktayama et al. 31 developed a more generalzed approach to address dscrete varables usng a penalty functon - dscrete varables are treated as contnuous varables by penalzng at the ntervals. However, the addtonal multmodal constrant n the penalty functon-based approach may undesrably ncrease the complexty of the desgn problem. Sngh et al. 32 presented an nterestng approach to address dscrete varables by manpulatng the random operators n the partcle-velocty update step. Ths approach can be very helpful n mantanng consstency n the rates of evoluton of the contnuous and the dscrete varables. The needed stochastc and mutually ndependent attrbutes of the random operators that regulate the PSO dynamcs are restrcted n ths approach. 2. Updatng Dscrete Desgn Varables n MDPSO In a mxed-dscrete optmzaton scenaro, the desgn space can be dvded nto a contnuous doman and a dscrete doman, whch correspond to the contnuous and the dscrete components of the desgn varable 5 of 20

6 vector, respectvely. Followng a contnuous search PSO step (Eq. 2), the locaton of a partcle n the dscrete doman s defned by a local hypercube that s expressed as H d = {( ) ( ) ( )} x L 1,x U 1, x L 2,x U 2,..., x L m,x U m, where x L x x U, = 1,2,...,m (3) In Eq. 3, m s the number of dscrete desgn varables, and x s denote the current locaton of the canddate soluton n the dscrete doman. The parameters x L and x U represent two consecutve feasble values of the th dscrete varable that bound the local hypercube. The total number of vertces n the hypercube s equal to 2 m. Thevalues, x L andx U, canbe obtanedfromthe dscretevectorsthatneed tobe specfedaprorforeach dscrete desgn varable. A relatvely straght-forward crteron, called the Nearest Vertex Approxmaton (NVA), s developed to approxmate the current dscrete-doman locaton of the canddate soluton to one of the vertces of ts local hypercube, H d (Eq. 3). The NVA approxmates the dscrete-doman locaton to the nearest vertex of the local hypercube (H d ), on the bass of the Eucldean dstance. Ths approxmaton s represented by [ ] X = x 1 x 2 x m, x = { x L, f x x L x U, otherwse = 1,2,...,m x x U In Eq. 4, X represents the approxmated dscrete-doman locaton (nearest hypercube vertex). Another crteron, related to the shortest normal dstance between the latest velocty vector of the partcle and the neghborng hypercube vertces, was also tested. However, ths Shortest Normal Approxmaton (SNA) 33 was found to be computatonally expensve, when appled to a wde range of test problems; hence NVA was selected for unversal applcaton n the MDPSO algorthm. An llustraton of the NVA and the SNA n the case of a representatve 2-D dscrete doman are shown n Fg. 2. (4) x 2 U x 1 L NVA vertex X Local hypercube Chld soluton SNA vertex X x 2 L Parent soluton x 1 U Shortest Eucldean Dstance Shortest Normal Dstance Connectng Vector Fgure 2. Illustraton of the Nearest Vertex and Shortest Normal Approxmatons Ths determnstc approxmaton seeks to retan the search characterstcs of the contnuous PSO dynamcs, whle ensurng that the system-model s evaluated only at the allowed dscrete doman locatons. Such an approxmaton strategy can be readly mplemented n other non-gradent based contnuous optmzaton algorthms, as a post process to the usual contnuous search step at every teraton. C. Constrant Handlng n PSO 1. Lterature Survey: Constrant Handlng The basc dynamcs of PSO does not account for system constrants. Several varatons of the PSO algorthm that ncorporate a constrant handlng capablty have been proposed: () a straght-forward method of 6 of 20

7 consderng only feasble partcles for global and local best solutons, 34 () the use of conventonal dynamc penalty functons, 35 () an effectve b-objectve approach where the net constrant serves as the the second objectve, 36 and (v) the use of the effcent constraned non-domnance prncples. 37 In ths paper, we mplement the rules of constraned non-domnance ntroduced by Deb et al. 10 Interestngly, the constraned non-domnance prncple can be perceved as an aspect of natural swarm ntellgence: communcaton of nformaton from partcle to partcle regardng whether they are beyond the feasble doman boundares, and/or how far beyond they are. 2. Soluton Comparson and Constrant Handlng n MDPSO Soluton comparson s essental n PSO at every teraton, to determne and update the global best for the populaton and the local best for each partcle. The prncple of constraned non-domnaton 11 s used to compare solutons. Ths prncple has been successfully mplemented n other major populaton-based optmzaton algorthms. Accordng to ths prncple, soluton- s sad to domnate soluton-j f,. soluton- s feasble and soluton-j s nfeasble or,. both solutons are nfeasble and soluton- has a smaller net constrant volaton than soluton-j or,. both solutons are feasble and soluton- weakly domnates soluton-j. In the case of a mult-objectve problem, t s possble that none of the above scenaros apply, whch mples that the solutons are non-domnated wth respect to each other. The net constrant volaton f c (X) s determned by f c (X) = p max(ḡ j,0)+ j=1 q k=1 max ( hk ǫ,0 ) (5) where ḡ j and h k represent the normalzed values of the j th nequalty constrant and k th equalty constrant, respectvely. In Eq. 5, ǫ represents the tolerance specfed to relax each equalty constrant. The soluton comparson approach n MDPSO favors feasblty over the objectve functon value. Ths approach has a tendency to drve solutons towards and nto the feasble regon durng the ntal teratons of the algorthm. 17,38 Throughout ths ntal phase, domnance scenaros I and II are promnently actve. When a majorty of the partcles have moved nto the feasble space, scenaro III takes over; soluton comparsons are then progressvely determned by the magntude of the objectve functon. In the case of hghly constraned sngle-objectve problems, ths soluton comparson approach, together wth the ntrnsc swarm dynamcs, can lead to an apprecable loss n dversty. Ths undesrable phenomenon occurs prmarly durng the feasblty-seekng process of optmzaton. To counter ths lmtng characterstc t ofthe partcle moton n the MDPSO, an explct dverstypreservatonterm, γ c r 3ˆV, s added to the velocty vector, as shown n Eq. 2. D. Dversty Preservaton n PSO 1. Lterature Survey: Dversty Preservaton Preservaton of the populaton dversty to avod premature convergence has been a long standng challenge for PSO. Rapd swarm convergence, whch s one of the key advantages of PSO over other populaton-based algorthms, can however lead to stagnaton of partcles n a small suboptmal regon. Effcent and tmevarant parameter selecton has been tradtonally used as an mplct method to avod partcle stagnaton, thereby preservng populaton dversty. Over the years, the use of explct dversty preservaton technques have proved to be more effectve. 1 Krnk et al. 39 ntroduced a collson-avodance technque to prevent premature convergence. Partcles comng wthn a defned vcnty of each other were allowed to bounce off; bouncng back along the old velocty vector (U-turn approach) was found to be most effectve. Blackwell and Bentley 40 also developed a dversty preservng swarm based on a smlar collson-avodance concept. The collson avodance schemes however requre an ntutve specfcaton of the threshold radus. A more globally applcable approach was developed by Rget and Vesterstrom, 41 where the usual attracton phase was replaced by a repulson phase, when the entre populaton dversty fell below a predefned threshold. In ths case, the usual PSO locaton update formula s appled wth the drecton reversed. A 7 of 20

8 modfcaton of the standard devaton of the partcle locatons was used as the measure of dversty. Ths measure, however, does not readly account for the combned effects of the dstrbuton of the partcles and the overall spread of the partcles n the varable space. In other words, wth ther method, 41 nfrequent extreme devatons.e., a hgher kurtoss (e.g., [0,0,0,0,10, 10])may yeld the same measure of dversty as frequent moderate devatons (e.g., [5, 6, 7, 5, 6, 7]), whch s msleadng. Other nterestng methodologes to address populaton dversty nclude: () ntroducton of a predatory partcle, 42 and () ntroducton of the concept of negatve entropy from thermodynamcs. 43 Nevertheless, the consderaton of populaton dversty n a mxed-dscrete/combnatoral optmzaton scenaro (n PSO) has rarely been reported n the lterature. 2. Dversty Preservaton n MDPSO The frst step n dversty preservaton s to characterze and quantfy the exstng populaton dversty wth respect to the desgn varable space. A consstent measure of dversty should smultaneously capture the overall spread and the dstrbuton of solutons n the populaton. A varable space metrc, smlar to the performance metrc (δ parameter) used to measure the spread of solutons along the computed Pareto front n the objectvespace, 11 would be an almostdealchocefor dverstycharacterzaton. However, the requred determnaton of the nearest-neghbor Eucldan dstances for every member of the populaton s lkely to become computatonally prohbtve n the case of hgh dmensonal optmzaton problems. The dversty characterzaton developed n ths paper seeks to effectvely capture the two dversty attrbutes: overall spread and dstrbuton of partcles, and s computatonally nexpensve to mplement, f requred, at every teraton. Separate dversty metrcs and dversty preservaton mechansms are formulated for contnuous and dscrete desgn varables. The dversty metrcs and the correspondng dversty preservaton coeffcents, γ c and γ d, are estmated for the entre populaton at the start of an teraton. These dversty metrcs are then updated usng a common factor that seeks to account for the dstrbuton of solutons. In the case of contnuous desgn varables, the ntal dversty metrc s gven by the normalzed sde length of the smallest hypercube that encloses all the partcles. Ths metrc s expressed as D c = ( n =m+1 x t,max x max x t,mn x mn ) 1 n m (6) where x t,max and x t,mn are, respectvely, the maxmum and the mnmum values of the th desgn varable n the populaton at the t th teraton; and x max and x mn, respectvely, represent the the specfed upper and lower bounds of the th desgn varable. A lkely scenaro s the presence of one or more outler partcles, when the majorty of the partcles are concentrated n a sgnfcantly smaller regon. Occurrence of ths scenaro leads to an apprecable overestmaton of the populaton dversty (D c ). To overcome ths lmtng scenaro, as well as to account for the dstrbuton of canddate solutons, the dversty metrc s further modfed. The number of solutons n a λ fracton of the then occuped varable space s frst determned. Ths λ-fractonal doman s formed around the global best canddate soluton and consders both contnuous and dscrete varables. The boundares of ths doman are gven by x t,max x t,mn [ ] x t,mn = max +λ x t, mn ( P g, +0.5λ x t, ), and xt,max [ x t,max λ x t = mn, ] ( ) max P g, 0.5λ x t, xt,mn = 1,2,...,n where x t = xt,max x t,mn ; the parameters x t,max and x t,mn, respectvely, represent the upper and lower bounds of the fractonal doman for the desgn varable x ; and P g, s the th varable of the global best 8 of 20

9 soluton. Usng the evaluated number of partcles n the fractonal doman, the contnuous dversty metrc (enclosng-hypercube sde length) s adjusted to better account for the dstrbuton of solutons. The adjusted contnuous dversty metrc D c s gven by D c = ( ) 1 N +1 n Dc (7) N λ +1 where N λ s the number of partcles n the λ-fractonal doman. The dversty coeffcent, γ c, for contnuous varables s then defned as a functon of the contnuous dversty metrc, whch s gven by ( ) D2 γ c = γ c0 exp c, where σ c = 2σ 2 c 1 2ln(1/γmn ) (8) and γ c0 and γ mn are specfed constants. The order of magntude of the dversty-scalngconstant γ c0 should be one, or n other words comparable to that of the exploratve coeffcent, β g. In the range 0 to 1 for D c, the dversty coeffcent s a monotoncally decreasng functon. The nature of ths functon for dfferent orders of magntude of γ mn s shown n Fg. 3. In the case of dscrete desgn varables, the dversty s characterzed ndependently for each varable. Ths approach s adopted because of the followng two percepts:. The effectve dversty n the th dscrete varable depends on (1) the avalable number of feasble values for that varable and (2) the dstrbuton of these feasble values.. Dversty preservaton n dscrete varables should seek to avod the stagnaton of partcles nsde a local hypercube H d. The ntal dversty metrc (D d ) for dscrete desgn varables s a vector of the normalzed dscrete varable ranges that span the current populaton. Ths metrc s expressed as D d, = xt,max x max x t,mn x mn, = 1,2,...,m (9) where D d, s the component of the dscrete dversty metrc correspondng to the th dscrete varable. Subsequently, n order to better account for the dstrbuton of solutons, the dscrete dversty metrc s adjusted as ( ) 1 N +1 n D d, = Dd, (10) N λ +1 where D d, s the adjusted dscrete dversty metrc. It s mportant to note how the parameter λ couples the dversty n contnuous and dscrete desgn varables. As a result, the dversty preservaton mechansms for contnuous and dscrete varables are expected to work n coherence wth each other. Dversty preservaton for dscrete varables s accomplshed through modfcaton of the dscrete update process descrbed n Secton B. The otherwse determnstc approxmaton of the partcle to a nearby feasble dscrete locaton s replaced by a stochastc update process. Ths stochastc update gves a partcle the opportunty to jump out of a local hypercube, thereby seekng to prevent the stagnaton of the swarm s dscrete component. A vector of dscrete-varable dversty coeffcents, γ d, s defned to further regulate the updatng of dscrete varables, n order to prevent ther premature stagnaton. A random number (r 4 ) s generated between 0 and 1, and the stochastc update for the generc th dscrete varable (x ) of a partcle s then appled usng the followng rules:. If r 4 s greater than the dversty coeffcent γ d,, then update the dscrete varable usng Eq. 4.. If r 4 s less than equal to γ d,, then randomly approxmate x to ether x L or x U (defned n Eq. 4). 9 of 20

10 The dscrete-varabledversty coeffcent, γ d,, that regulates the stochastc update rules s desgned to adapt to the sze of the set of feasble values for the th dscrete varable. Ths approach avods a false mpresson of consderable dversty, n the case of dscrete varables that take a relatvely small szed set of feasble values. The dscrete dversty coeffcent s defned as ( ) D2 d, γ d, = γ d0 exp, where 1 σ d, = 2lnM = 1,2,...,m 2σ 2 d, and where M represents the sze of the set of feasble values for the th dscrete varable, and γ d0 s a prescrbed constant between 0 and 1. For any estmated value of the populaton dversty, a hgher value of the prescrbed parameter, γ d0, makes the random update of the dscrete doman locaton more lkely. It s mportant to note that whle the contnuous-varable dversty coeffcent (γ c ) drectly regulates the partcle moton (n the locaton update step), the dscrete-varable dversty coeffcents (γ d, ), control the updatng of the dscrete varables as a post-process(durng the NVA applcaton) n every pertnent teraton. In addton, the same value of γ c s used for all desgn varables at a partcular teraton, whereas a dfferent value of γ d, s used for each th dscrete varable. An llustraton of the dscrete dversty coeffcent for dfferent szes of the set of feasble values s shown n Fg. 3. (11) Dversty coeffcents, γ c and γ d, γ mn = 10 1 ; M = 10 γ mn = 10 2 ; M = 10 2 γ mn = 10 3 ; M = 10 3 γ mn = 10 4 ; M = 10 4 γ mn = 10 5 ; M = 10 5 γ mn = 10 6 ; M = 10 6 γ mn = 10 7 ; M = 10 7 γ mn = 10 8 ; M = 10 8 γ mn = 10 9 ; M = 10 9 γ mn = ; M = Dversty metrcs, D and D c d, Fgure 3. Varaton of the dversty coeffcents γ c and γ d, wth the dversty metrcs D c and D d,, respectvely, llustrated at () dfferent values of γ mn for contnuous varables, and () dfferent szes (M) of the feasble set for dscrete varables, wth γ d0 = 1 III. Numercal Experments To valdate the Mxed-Dscrete Partcle Swarm Optmzaton (MDPSO) algorthm, we apply t to two dfferent classes of sngle-objectve optmzaton problems: () standard unconstraned problems, most of whch are multmodal, and () Mxed-Integer Non-Lnear Programmng (MINLP) problems. These two sets of numercal experments are dscussed n the followng sub-sectons. The values of the prescrbed MDPSO parameters for the two sets of numercal experments are gven n Table 1. A. Unconstraned Standard Optmzaton Problems The new MDPSO algorthm s appled to a set of nne standard unconstraned nonlnear optmzaton test problems wth only contnuous varables to compare ts performances wth that of the basc PSO. For a majorty of these test problems, the basc PSO s expected to offer a powerful soluton. The MDPSO s specfcally desgned to address complex constraned and/or mxed-dscrete optmzaton problems. 10 of 20

11 Parameter Table 1. User-defned constants n PSO Unconstraned problems MINLP problems α β g β l γ c 0 0.1,0.5, γ d γ mn 1.0e e-10 Populaton Sze (N) 10 n 10 n Fractonal doman sze (λ N) 0.25 N 0.1 N Allowed number of functon calls 10,000 50,000 Wth ths set of numercal experments, we nvestgate f the addtonal features n MDPSO, partcularly those related to dversty preservaton, ntroduces any unexpected characterstcs. The frst eght test problems have been taken from the lst of sample sngle-objectve optmzaton problems provded n the MATLAB Genetc and Evolutonary Algorthm Toolbox (GEATbx) Documentaton. 20 The GEATbx problems were orgnally developed and reported by dfferent promnent researchers from the desgn and optmzaton communty. The last test problem from Table 2 (Mele-Cantrell functon) has been taken from the paper by Mele and Cantrell. 21 Detals of the standard unconstraned test problems are summarzed n Table 2. Test problem Functon name Table 2. Standard unconstraned optmzaton problems Number of varables Complexty attrbute 1 Rosenbrock s valley 2 long relatvely flat valley 2 Rastrgn s functon 2 hghly multmodal 3 Schwefel s functon 2 hghly multmodal 4 Grewangk s functon 2 hghly multmodal 5 Ackley s Path functon 2 hghly multmodal 6 Mchalewcz s functon 10 flat regons and multmodal 7 Easom s functon 2 mostly flat search space 8 Goldsten-Prce s functon 2 extensve flat regon 9 Mele-Cantrell 4 multmodal The MDPSO algorthm s appled to each test problem, usng three dfferent values of the dversty coeffcent scalng constant: γ c0 = 0.1,0.5,1.0. Each test problem s run 10 tmes, wth a partcular γ c0 value, to compensate for the effects of the random operators on the overall algorthm performance. Results for the conventonal PSO was obtaned smply by specfyng the dversty coeffcent scalng constant, γ c0, to be zero, whle other basc PSO parameters were fxed at the same values as gven n Table 1. The convergence hstores from representatve runs of the MDPSO and a representatve run of the conventonal PSO for the Mele Cantrell test functon are shown n Fg. 4. The actual mnmum of the objectve functon s known for each test problem lsted n Table 3. Usng the actual mnmum objectve functon value, an addtonal algorthm termnaton crteron s specfed, based on the normalzed relatve error (ε f ). Ths error s evaluated as where f comp mn and f act mn ε f = f comp mn fact mn f act mn f comp mn fact mn,, f f act mn 0 f fact mn = 0 are the computed mnmum and the actual mnmum of the objectve functon, (12) 11 of 20

12 respectvely. The algorthms are termnated when the relatve error ε f falls below 1.0e Objectve Functon PSO MDPSO (γ c0 = 0.1) MDPSO (γ c0 = 0.5) MDPSO (γ c0 = 1.0) Number of Functon Evaluatons Fgure 4. Convergence hstores of the MDPSO and the conventonal PSO for the Mele-Cantrell functon It can be observed from Fg. 4 that the algorthms perform very well for the multmodal Mele-Cantrell test functon. Wth the dversty scalng constant equal to 0.1 (green dashed lne), the rate of convergence of MDPSO s approxmately twce that of the conventonalpso - the relatve error ε f reduces to 1.0e-07n half the number of functon calls. Wth the value of the dversty scalng constant equal to 1.0 (black dashed lne), the MDPSO algorthm converges slghtly slower than the conventonal PSO algorthm. Ths phenomenon can be attrbuted to the ncreased reducton n the partcle veloctes towards the global optmum, caused by the ntroducton of a larger amount of populaton dversty among the partcles. The normalzed relatve errors correspondng to the best and the worst optmzed solutons among the 10 runs, obtaned for each test problem by MDPSO and conventonal PSO are shown n Fgs. 5(a) and 5(b). Further detals, regardng the performance of the MDPSO algorthm wth the dversty scalng constant equal to 1.0, are provded n Table Best Soluton: PSO Worst Soluton: PSO 10 2 Best Soluton: MDPSO Worst Soluton: MDPSO Best Soluton: γ c0 = 0.1 Worst Soluton: γ c0 = 0.1 Best Soluton: γ c0 = 0.5 Worst Soluton: γ c0 = 0.5 Best Soluton: γ c0 = 1.0 Worst Soluton: γ c0 = 1.0 Normalzed Error Normalzed Error Test Problem Number Test Problem Number (a) Usng PSO, and MDPSO wth γ c0 = 1.0 (b) Usng MDPSO wth γ c0 = 0.1,0.5,1.0 Fgure 5. Maxmum and mnmum values (among 10 runs) of the normalzed relatve error obtaned for the standard unconstraned test problems Fgure 5(a) shows that the MDPSO algorthm performs as good as or better than the conventonal PSO 12 of 20

13 Table 3. Performance of MDPSO (wth γ c0 = 1.0) on the standard unconstraned test problems Test Problem Actual mnmum Best computed mnmum Worst computed mnmum Standard devaton of the computed mnma E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-08 algorthm for most of the standard unconstraned test problems, except for test problem 2. It s observed from Fg. 5(a) and Table 3 that, nether of the algorthms could provde satsfactory solutons for test problem 6, Mchalewcz s functon; ths test functon has extensve flat regons and s also multmodal. 20 A selectve combnaton of the prescrbed MDPSO parameters may successfully fnd the optmum for such complex nonlnear functons. A detaled parametrc analyss of the senstvty and the varaton of the algorthm performance wth respect to the prescrbed constants s requred for ths purpose. Fgure 5(b) llustrates that the performance of MDPSO s margnally senstve to the specfed value of the dversty scalng constant (γ c0 ) n the case of these unconstraned contnuous problems - the relatve errors gven by MDPSO for the three dfferent values of the dversty scalng constant are close to each other. The standard devaton n the computed mnma obtaned from the 10 runs for each test problem (Table 3) s observed to be relatvely small when compared to the correspondng actual mnma. Ths observaton further llustrates the consstency n the performance of the MDPSO algorthm. B. Mxed-Integer Nonlnear Programmng (MINLP) Problems The MDPSO algorthm s appled to an extensve set of nnety-eght Mxed-Integer Non-Lnear Programmng (MINLP) test problems; these test problems were obtaned from the comprehensve lst of one hundred MINLP problems reported by Schttkowsk. 22 The problems numbered 10 and 100 n the orgnal lst 22 have not been tested n ths paper. A majorty of these MINLP test problems belong to the GAMS Model Lbrary MINLPlb, 44 and have been wdely used to valdate and compare optmzaton algorthms. 22 These MINLP test problems present a wde range of complextes: total number of desgn varables vary from 2 to 50; numbers of bnary desgn varables and nteger desgn varables vary from 0 to 16 and 0 to 50, respectvely; total number of constrants (ncludng equalty and nequalty) vary from 0 to 54; number of equalty constrants vary from 0 to 17. Smlar to the prevous set of numercal experments, each MINLP test problem s run 10 tmes to compensate for the performance effects of the random operators n the algorthm. For each run, the algorthm s termnated when the best global soluton does not mprove by at least 1.0e-06 tmes ts objectve value n 10 consecutve teratons. The normalzed relatve errors correspondng to the best and the worst solutons among the ten runs obtaned for each test problem by MDPSO are llustrated n Fgs. 6(a) and 6(b). Fgs. 6(a) and 6(b) also show () the number of desgn varables and () the number of constrants n each test problem, to provde nsghts nto ther nfluences on the performance of the algorthm. A hstogram of the relatve errors s 13 of 20

14 shown n Fg. 7. It s helpful to note that, n the case of several MINLP test problems that comprse only dscrete varables, a zero relatve error s obtaned through optmzaton; the zero error for each of these test problem runs s replaced by an artfcal error value of 1.0e-12 n the fgures to allow a logarthmc scale representaton of the error. Normalzed Error Best Sol Error Worst Sol Error Total No. of Varables No. of Dscrete Varables Number of Varbales Normalzed Error Best Sol Error Worst Sol Error Total No. of Constrants No. of Eq. Constrants Number of Constrants Test Problem Number (a) Showng the number of desgn varables n each problem Test Problem Number (b) Showng the number of constrants n each problem Fgure 6. Maxmum and mnmum values (among 10 runs) of the normalzed relatve error obtaned by MDPSO for the MINLP test problems Number of Test Problem Runs Logarthm of the Normalzed Error Fgure 7. Hstogram of the order of magntude of the normalzed relatve error obtaned by MDPSO for the MINLP test problems From Fgs. 6(a) and 6(b), t s expectedly observed that, a hgher number of desgn varables and/or a hgher number of constrants generally results n a hgher error n the computed mnmum. The numbers of desgn varables and constrants are two key attrbutes of the complexty of an optmzaton problem. Future advancements n Mxed-Dscrete PSO should focus on reducng the senstvty of the algorthm performance to the problem complexty attrbutes. The hstogram n Fg. 7 llustrates the dstrbuton of the relatve error, on a logarthmc scale, for all the 980 test problem runs, whch ncludes 10 runs for each problem. The same values of the prescrbed algorthm parameters have been used for the entre set of MINLP problems. Ths specfcaton s partly responsble for the hgh relatve error of 10% or more for a sgnfcant number of test problem runs (as seen from Fg. 7). The set of MINLP test problems present a wde varety of non-lnear crtera functons and problem complextes; better performance for ndvdual problems can be obtaned through approprate alteraton of the prescrbed MDPSO parameter values based on the problem complexty. 14 of 20

15 Fgure 8 llustrates the net constrant volaton (Eq. 5) correspondng to the best and the worst solutons obtaned by MDPSO. Fgure 9 llustrates the number of functon evaluatons made by the MDPSO algorthm. Further detals, regardng the performance of the MDPSO algorthm, are provded n Tables 4 for MINLP test problems 1 to 50, and 5 for MINLP test problems 51 to 98. Logarthm of the Constrant Volaton Mn Constrant Volaton Max Constrant Volaton Total No. of Constrants No. of Eq. Constrants Number of Constrants Test Problem Number Fgure 8. Net constrant volaton n the best and the worst solutons obtaned by MDPSO for the MINLP test problems Number of Functon Evaluatons Best Sol Functon Calls Worst Soluton Functon Calls No. of Varables No. of Dscrete Varables Number of Varables Test Problem Number Fgure 9. Number of functon evaluatons made by MDPSO for the MINLP test problems (correspondng to the best and the worst computed solutons) Fgure 8 and the feasblty success % for each problem, lsted n Tables 4 and 5, show that the MDPSO algorthm successfully fnds the feasble space n a majorty of the MINLP test problems. The feasblty success s lower for the test problems that nvolve 20 or more constrant functons, as seen from Fg. 8. Expectedly, the standard devatons n the computed mnma are observed to be hgher n the case of the test problems for whch optmzaton resulted n hgher relatve errors. For complex engneerng desgn applcatons, the computatonal expense of optmzaton s generally domnated by the expense of system-model evaluatons. Therefore, n addton to the ablty to successfully fnd the feasble space, and subsequently the optmum desgn, the number of functon evaluatons nvested n the process s also an mportant per- 15 of 20

16 Table 4. Performance of MDPSO on the MINLP test problems numbered 1 to 50 Test Problem Feasblty success (%) Actual mnmum Best computed mnmum Worst computed mnmum Standard devaton of the computed mnma E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E of 20

17 Table 5. Performance of MDPSO on the MINLP test problems numbered 51 to 98 Test Problem Feasblty success (%) Actual mnmum Best computed mnmum Worst computed mnmum Standard devaton of the computed mnma E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E of 20