MULTISTART OPTIMIZATION WITH A TRAINABLE DECISION MAKER FOR AVOIDING HIGH-VALUED LOCAL MINIMA

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1 3 rd Internatonal Conference on Experment/Proce/Sytem Modelng/Smulaton & Optmzaton 3 rd IC-EpMO Athen, 8- July, 2009 IC-EpMO MULTISTART OPTIMIZATION WITH A TRAINABLE DECISION MAKER FOR AVOIDING HIGH-VALUED LOCAL MINIMA N. Kyrgo, C. Vogl and I.E. Lagar Unverty of Ioannna, Dept. of Computer Scence, P.O. BOX 86, 450 Ioannna, Greece Keyword: Global optmzaton, Multtart, Neural Network, Molecular Conformaton. Abtract. In th artcle we propoe a tme-avng technque to be ued n conjuncton wth a multtart-baed global optmzaton method, for determnng low-valued local mnma. The man dea to avod the local-earch commencement from non-promng pont. The decon for the tart-pont utablty turn out to be rather nexpenve when compared to the cot of a local-earch. We employ a feedforward neural-network for the decon makng that fed wth functonal and gradent nformaton obtaned from a few elected pont n the neghborhood of the canddate tart-pont. The network traned from data collected durng the optmzaton proce. We report reult for a number of computatonal experment on a multtude of model tet-functon, ung multtart and a pecal local earch that create contguou regon of attracton. Th method can be partcularly ueful for the conformaton problem n molecular mechanc.. INTRODUCTION Global optmzaton (GO) ha receved a lot of attenton n recent year [], due to the ever emergng centfc and ndutral demand. For ntance the collecton of the table conformaton of a molecule, the management of mutual fund, engneerng degn and the degn of drug, to menton a few topc, are n need of effcent global optmzaton technque. There ext everal categore of GO method. We dtnguh two man clae; the determntc and the tochatc cla and refer to [2] for a detaled account on clafcaton. GO method face varou goal; ome am to fnd a ngle global mnmum (Smulated Annealng, Genetc Algorthm, Controlled Random Search), other to fnd all the global mnma (Modfed Partcle Swarm [3] ), whle other (Multtart wth Cluterng [4,5,6,7,8,9] ) am n fndng all the local mnma. Nowaday, wth the avalablty of powerful computer ytem, GO ha become an affordable procedure. GO algorthm that can take advantage of parallel and/or dtrbuted archtecture, are partcularly utable for olvng demandng problem. Among the plethora of uch problem, we dtnguh the determnaton of the table conformaton of a molecule, condered by Molecular Mechanc (MM), due to the far reachng conequence of t oluton. MM employed to tudy molecular properte that are mportant n pharmacology (drug degn), bo cence, materal cence, etc. Gven a realtc nteracton between the conttutng atom, MM am to locate the mnma of the molecular potental energy. When the molecule mall, all the local mnma are rather ealy determned. However, for extended molecule the number of mnma may be notorouly hgh. In uch cae the analy of the molecular properte qute nvolved, and the requrement lowered to the determnaton of the global mnmum and of a lmted number of local mnma wth energy value below an approprate threhold. Mathematcally the problem we are ntereted n may be expreed a: Fnd all x S R n that atfy f ( x ) f g () x arg mn f ( x) S S {x x x } xs where S a bounded doman of fnte meaure, a problem pecfc potve contant and f g the value of the objectve at the global mnmum. Namely the problem to determne all local mnmzer n S wth objectve value not hgher than f g. The artcle organzed n the followng way. In ecton (2), we lay-out the new dea nvolved and we preent the correpondng algorthm, whle n ecton (3), we gve a decrpton of the numercal experment that were performed along wth the correpondng reult. Fnally n ecton (4), our concluon are ummarzed and we gve a recommendaton for future reearch.

2 2. DESCRIPTION OF THE METHOD N. Kyrgo, C. Vogl and I.E. Lagar In the followng t wll be aumed that the underlyng GO method to be ued Multtart. Any of the better performng multtart baed cluterng method may be ued wth advantage. Here the empha wll be gven to the new dea of the tmely tart pont rejecton, whle keepng the GO procedure mple. We frt outlne the framework of the new procedure.. Pck at random a pont x S. Apply only a few (ay k ) tep of a local earch procedure, pang ( ) ( ) ( ) ( ) ( ) through pont x k. Let f f ( x ) and g f ( x ). ( ) ( ) 2. From th nformaton,.e. {f } and {g } 0 k predct f, the value of the objectve functon at the mnmum that would be recovered f the local earch wa allowed to converge. 3. If the predcton hgher than a preet threhold: abandon the earch and tart over agan from tep otherwe: contnue wth the local earch untl a mnmum recovered. 4. Repeat from tep. Step 2 need further decrpton. The predcton of the objectve value at the mnmum baed on the followng model f ( x ) f M ( p Y ) f ( x ) N( p Y ) (2) where x L( x ) the mnmum reached by tartng local earch L from pont x. N( p Y ) a feedforward neural network wth one hdden layer and p the et of the network weght and bae whle Y a et of nput data collected durng the run. More pecfcally () () (2) () (2) ( k ) ( k) ( k ) Y f g f f g f f g f f g Each node n the hdden layer requre n 2k 3 parameter (weght). Hyperbolc tangent wa choen for the actvaton n the hdden layer, whle the output actvaton wa taken to be lnear. (Our mplementaton ue k 2 ). 2. Network tranng The weght are determned by tranng the model ung collected data created durng the global optmzaton procedure. Namely, we collect a number (M ) of tartng pont x x 2 xm, and the correpondng local mnma x x 2 xm wth x L( x ). The tranng et for the network gven by ( Y t ) ( Y2 t 2 ) ( YM tm ) where t f ( x ) f ( x ). The tranng performed by mnmzng the error functon M E( p) N( p Y ) t 2 M (3) 2.2 Local earch properte For the predcton model f M ( p Y ) f ( x ) N( p Y ) to be accurate the pont ( ) x x hould be connected va a monotoncally decreang path and even more x hould be the cloet mnmum to x that can be connected wth uch a path. Th enure the local character of the approxmaton. Note, that mot common local earch procedure do not hare th property and hence are not utable n th framework. A method that atfe the above requrement a teepet decent wth an nfntemal tep. However, th only a theoretcal devce and uch a method n practce would be wateful. In fgure we preent a unvarate example of a multmodal functon. Startng pont n a valley hould be aocated wth the urrounded mnmum. In uch a cae the model ha a local character and the approxmaton therefore meanngful. To th end we have mplemented qua-newton (BFGS) local earch wth a modfed lne earch that mantan ntact the Armjo condton. However the lne-earch ue an ncreang tep-ze contrary to the common backtrackng. We gve a bref decrpton of the lne earch n Algorthm.

3 N. Kyrgo, C. Vogl and I.E. Lagar Fgure. Startng pont and aocated mnma Algorthm New lne earch Input: x: Current terate d : Decent drecton from the outer qua-newton local earch : Armjo rule parameter 0 : Method parameter Output: x: Next terate : Lne earch tep fc: Functon call. Intalze: cale fc 0 term fale 2. Man Step: whle term = fale do for =, do max x) d mn cale T f f ( xd ) f ( x) d f ( x) then { Bellow lne} f f ( x d) f ( x d) then {No mprovement} x x d term true, break end f ele { Above lne } x x d term true, break end f fc fc end cale cale mn end max x) d

4 N. Kyrgo, C. Vogl and I.E. Lagar We menton n pang that n Algorthm the loop over the tep can be performed n parallel. 3. EXPERIMENTS AND COMPARISON We ued Matlab ntegrated envronment to mplement our methodology. Neural network were created and traned ung the Neural Network Toolbox, and the tranng wa performed ung a Levenberg-Marquard algorthm (+ tranlm + opton). 3. Illutratve example In th example we ued the two-dmenonal Shubert functon nde 2 [0 5] gven by: 5 5 f ( x x2 ) " co(( ) x )! #" co(( ) x2 )! # $ %$ % (4) The tranng et wa created by unformly amplng 200 tartng pont, and by performng an equal number of local earche to obtan the aocated mnma, whle mlarly, the tet et ued 600 pont. In Fgure 2 the urface and contour plot of the Shubert functon dplayed. (a) Functon' urface plot Fgure 2. Two dmenonal Shubert functon (b) Functon' contour plot In fgure 3(a) the horzontal axe regter the tartng pont ndce 200 ued for the tranng. The vertcal axe of the top, mddle and bottom row hold the value of the objectve at the aocated mnma x, the predcted value and ther abolute dfference correpondngly. Smlarly n fgure 3(b) the tet et plot are gven, whle n fgure 3(c) the accepted tartng pont are hown. We accepted a tartng pont x when fm ( py ) 2. There are two cae of mclafcaton. One, where a pont erroneouly accepted, and the other when a pont erroneouly ejected. The frt cae cot a local earch, whle the econd cot only a few evaluaton. In fgure 3 the reult refer to a neural network wth 5 hdden node. Fgure 4 llutrate the cae of a 20-hdden node neural network. One may verfy by npecton that the 20-node network obtan a lower MSE over the tranng et, and a hgher MSE over the tet et hnderng that the 5-node network offer a better generalzaton. Namely the 5-node network attan a 84& 33% ucce rate, and the 20-node network a correpondng 82& 83%.

5 N. Kyrgo, C. Vogl and I.E. Lagar (a) Tranng et (b) Tet et (c) Accepted tartng pont Fgure 3. Reult for a neural network wth 5 hdden node

6 N. Kyrgo, C. Vogl and I.E. Lagar (a) Tranng et (b) Tet et (c) Accepted tartng pont Fgure 4. Reult for a neural network wth 20 hdden node

7 N. Kyrgo, C. Vogl and I.E. Lagar Fgure 5. Accepted tartng pont prnted on functon contour We mplemented our approach and teted t on a number of optmzaton problem. Namely we expermented wth well known tet-functon uch a the Ratrgn, Gunta, Boha, Holder and Brd. Our reult were n lne wth thoe of the Schubert tet functon dcued above and wll be reported elewhere. 4. CONCLUSIONS AND FURTHER WORK In th paper we preented an early rejecton crteron utable for multtart baed global optmzaton algorthm. The oberved avng are ubtantal and hence the method may be uggeted for applcaton n tme conumng global optmzaton problem lke thoe appearng n molecular mechanc, where the objectve functon the molecular potental energy whle the atomc coordnate are the adjutable parameter. Molecular mechanc problem are currently under ntenve nvetgaton by our reearch group.

8 N. Kyrgo, C. Vogl and I.E. Lagar REFERENCES [] Pardalo Pano M., Romejn Edwn H., Tuy Hoang (2000), Recent development and trend n global optmzaton, Journal of Computatonal and Appled Mathematc, pp [2] Boender C.G.E. and Romejn Edwn H. (995), Stochatc Method, n Handbook of Global Optmzaton (Hort, R. and Pardalo, P. M. ed.), Kluwer, Dordrecht, pp [3] Paropoulo, K. E., Vrahat M. N. (2004), On the computaton of all Global mnmzer through partcle warm optmzaton, IEEE Tran. Evol. Comp., pp [4] Boender, C.G.E. and Rnnooy Kan, A.H.G and Tmmer, G.T. and Stouge, L., (982), A tochatc method for global optmzaton, Mathematcal Programmng, pp [5] Rnnooy Kan, A.H.G and Tmmer, G.T. (987), Stochatc global optmzaton method. Part I: Cluterng method, Mathematcal Programmng, pp [6] Rnnooy Kan, A.H.G and Tmmer, G.T. (987), Stochatc global optmzaton method. Part II: Mult level method, Mathematcal Programmng, pp [7] Törn, A. A.(978) A earch cluterng approach to global optmzaton, n Dxon, L.C.W and Szegö, G.P. (ed.), Toward Global Optmzaton 2, North-Holland, Amterdam. [8] Törn, A. and Vtanen, S. (994) Topographcal Global Optmzaton Ung Pre Sampled Pont, Journal of Global Optmzaton, pp [9] Al, M.M. and Storey, C. (994), Topographcal Multlevel Sngle Lnkage, Journal of Global Optmzaton, pp