A Graph-based Decentralization Proposal for Convex Non Linear Separable Problems with Linear Constraints

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1 Proeedings of the World Congress on Engineering nd Comuter Siene Vol I WCECS, 5 Otober,, Sn Frniso, USA A Grhbsed Deentrlition Proosl for Convex Non Liner Serble Problems with Liner Constrints Jime Cerd, Mrio Grff Abstrt This doument rooses severl deentrlition rohes for the Newton ste grhbsed model for onvex non liner serble roblems with liner onstrints. The Newton ste is well suited for this kind of roblems, but when the roblem sie grows the NLP model will grow in non liner mnner. When this hens, the srse mtrix reresenttion is the th to follow. Furthermore, deentrlition shemes re suitble to kee the roblem from growing exonentilly. In this work grh bsed method to hieve this deentrlition is roosed. To this end, we hve hosen to wek the links, whih re rt of the grh, s n lterntive. These links eventully will guide the solution roess in this roh, whih imlies to wek the links whih re ouling the roblems in order to hieve suh deentrlition. A deeer nlysis of these links is done whih leds to its omlete understnding. It will be seen tht the min effet of the link wekening oertion is to llow the omuttion of the ext grdient. However, the solution will be reinfored by tking into ount the seond order informtion rovided by the linking struture. Finlly, different deentrlistion shemes re resented bsed on the revious nlysis. Index Terms Non Liner Serble Progrmming, Multi Agent Systems, Deentrlised Systems. NOMENCLATURE N Number of deision vribles. L Number of equlity onstrints. M Number of inequlity onstrints. i Deision vrible i. i Uer limit vlue of vrible i. i Lower limit vlue of vrible i. i Slk vrible for i uer bound. i Slk vrible for i lower bound. i Vrible x i inrement. f( Objetive funtion g l ( Equlity onstrint l. h m ( Inequlity onstrint m. li ith oeffiient in equlity onstrint l. b mi ith oeffiient in inequlity onstrint m. r l RHS of equlity onstrint l. s m RHS of inequlity onstrint m. λ l Dul vrible for equlity onstrint l. µ m Dul vrible for inequlity onstrint m. ρ i Dul vrible for i uer bound. ρ i Dul vrible for i lower bound. (S Power set of set S. Γ i Set of vribles onneted to i. Mnusrit reeived July, ; revised August 6,. This work ws suorted by the Sientifi Reserh Coordintion, Universidd Mihon de Sn Niols de Hidlgo J. Cerd (jerd@umih.mx nd M. Grff (mgrffg@de.fie.umih.mx re with the Eletril Engineering Shool, Universidd Mihon de Sn Niols de Hidlgo, Moreli, Mihon, Mexio I. INTRODUCTION Non liner serble roblems (NLSP re NLPs whose objetive funtion n be deomosed s sum of funtions with only one vrible eh. The solution of NLSPs re bsed on very well grounded mthemtil theories []. This doument rooses Newton ste grhbsed model to deentrlie onvex non liner serble roblems (NLSP with liner onstrints. Previous works [], [], [4], [5] hve resented deentrlised roh for NLPs lied to Eletril Power Systems, whih rely on the uxiliry rinile roblem [6]. In this work grh bsed method to hieve this deentrlition is roosed. To this end, we hve hosen to wek the links, whih re rt of the grh, s n lterntive. These links eventully will guide the solution roess in this roh, whih imlies to wek the links whih re ouling the roblems in order to hieve suh deentrlition. A deeer nlysis of these links is done whih leds to its omlete understnding. It will be seen tht the min effet of the link wekening oertion is to llow the omuttion of the ext grdient. However, the solution will be reinfored by tking into ount the seond order informtion rovided by the linking struture. This doument is strutured s follows. First, grh bsed model for onvex NLSPs with liner onstrints is resented. Then, n equivlent grh reresenttion is resented in order to simlify the grh. After this, different deentrlistion shemes re resented bsed on the revious nlysis. Finlly some onlusions re withdrwn from this roosl. II. A GRAPHBASED MODEL FOR CONVEX NLSP In this setion grh toology for the Newton ste method is relited just s roosed in [7] with the urose of omleteness. To this end, let us bse the disussion with the NLSP desribed by model. This NLSP onsists of N vribles, L equlity onstrints, nd M inequlity onstrints. min i N i= f i( i st. g l ( =, l =,,..., L ( h m (, m =,,..., M where f i ( i re non liner funtions. Furthermore, we ssume they hve seond order derivtives. On the other hnd, g l ( re liner equlities while h m ( liner inequlities denoted s follows: ISBN: ISSN: (Print; ISSN: (Online WCECS

2 Proeedings of the World Congress on Engineering nd Comuter Siene Vol I WCECS, 5 Otober,, Sn Frniso, USA L li i = r l l= M b mi i s m m= In generl, NLP solver strts by building the Lgrngin given by Eq.. L( = N f i ( i + i= L λ l g l (+ l= M µ m h m (x ( m= This is the bse to imlement the Newton ste, whose formultion is given by Eq []: H(L( = (L( ( Tble I, shows the involved elements to omute the Newton ste for model. Do notie we hve introdued slk vribles s well dul vribles whih re used to ontrol the bounds of the deision vribles. Here, ρ i nd i re used for the lower bound while ρ i nd i re used for the uer bound of i This model n be reresented, s derived in [7], with the grh shown in Fig.. Fig.. ( % $ ' ' % ' $ ( Proosed toology for the Newton ste method Eh onstrint is reresented by dul vrible nd set of links whih reresent the liner terms within the onstrint. The terms in the onstrints re reresented by links whih join the riml vribles with the dul vribles. The only differene between equlity ontrints nd inequlity onstrints is the kind of links used to build the linking struture. In the se of equlity onstrints, the linking struture will be tive long the whole solution roess. On the other hnd, the linking struture for n inequlity onstrint will be tive only when suh onstrint is binding. This is reresented by the gry olor given to the links belonging to inequlity onstrints s oosed to the blk links whih belong to equlity onstrints. In tble I, the sme riterion hs been imosed in the elements of H(L(. As it n be notied, it is full of emty ses nd gry olor, the first re long term srsity tterns nd the seond ones re temorry srsity tterns witing to be exloited. n III. AN EQUIVALENT GRAPH REPRESENTATION In this setion, one the hrteristis of this grh hve been nlysed, simler grh model reresenttion is derived in order to mke its hndling esier. This simlifition is bsed on two min observtions: first, the bounding strutures re fixed, nd seond the different kind of vribles n be reresented in suh wy tht the ontent of tht node will be inferred by its reresenttion. A. An Equivlent Grh Bounding Struture Reresenttion The subgrhs whih reresent the bound on the vribles re well defined. Therefore seil grh nottion will be derived in order to hndle them, s shown in figure. Here ll the links nd nodes ontined in suh subgrh re embedded within the tringle. The link vlue is defined s follows: if the onstrint is lower binding, link.vlue = if the onstrint is uer binding, if is not binding. Fig.. Bound subgrh reresenttion This leds to the reresenttion shown in figure. l m m N Fig.. l l l Hessin toology modified reresenttion B. An Equivlent Node Tye Reresenttion ln In [7], it hs been lernt tht the grdient is embedded within the grh toology nd the informtion tthed to eh node. Therefore, simlifition for the grh reresenttion will be derived. The lst setion hs resented reresenttion for the bounding strutures whih ontrol the limits on the riml vribles. Therefore, now the only m N m mn N ISBN: ISSN: (Print; ISSN: (Online WCECS

3 Proeedings of the World Congress on Engineering nd Comuter Siene Vol I WCECS, 5 Otober,, Sn Frniso, USA L( = P N i= fi(i P L l= λ lg l ( P M m= µmhm( (L( H(L( i λ l µ m ρ i ρ i i i i i f i( i λ l µ m ρ i + ρ i (L( i il b im λ l g l ( il µ m h m( b im ρ i i i + i / i ρ i i i + i / i i ρ i i i ρ i i ρ i i i ρ i TABLE I THE NEWTON STEP INGREDIENTS nodes in the remining grh re those reresenting the riml vribles nd the dul vribles. As mentioned bove, the riml vribles set n be further divided into two sets. The first one reresents those vribles whih re rt of the objetive funtion. The seond one ontins those riml vribles whih er only within the onstrints. An instne of these would be the vrible reresenting the eletril ngle δ in the eletri ower mrket exmle [], []. These two subsets will be lled objetive nd nonobjetive vribles resetively. Therefore, there re three kinds of vribles whih hve to be reresented within the grh e.g. Priml, Dul, Non Objetive. These reresenttions re shown in figure 4. Bsed on the tye of vrible this node is reresenting, the informtion tthed to it will be known. This informtion is s follows Objetive vribles: Atthed to this node will be the informtion in order to be ble to omute its grdient if there were no other externl informtion, Dul vribles: The informtion tthed to this kind of node will be the right hnd side of the onstrints whih will llow its grdient omuttion. Non objetive vribles: To this node there will be no dditionl informtion sine its oeffiients in the onstrints re given by the vlues of the links whih re tthed to it, i.e. no seond order informtion is vilble, g g $ g Fig. 4. Vrible reresenttion. ( Priml vrible, (b Dul Vrible, ( Non Objetive Vrible This leds to the reresenttion shown in figure 5, where is ssumed s nonobjetive vrible. Bsed on this grh the rorite lsses for eh tye of vrible n be defined. One these definitions hve been imlemented, the oertions to solve the grh n be imlemented strightforwrd. IV. THE GRAPH AND ITS DECENTRALISATION In this setion the grh nd its deentrlistion is ddressed. To this end n oertion over the links of the grh, ( (b ( g Fig. 5. l m l l ln l N m m Hessin toology finl equivlent reresenttion lled link wekenning, is defined. This oertion is bsed on the exression derived in [7] s given in eq. 4 i = i L( i j Γ i (i,j L k mn i j j (4 i where L k denotes the set of links whih re tken into ount for this roess, do notie L k = k. Then, three different rohes to deentrlise the grh re roosed. The first one is omlete deentrlised, the seond roh is bsed on the notion of the kind of vribles within the grh (i.e. riml nd dul vribles; nd the third roh will be bsed on geny definitions [9]. To this end let us refer to the system given in [8] s shown in figure 6(. The grh reresenttion orresonding to this exmle is shown in figure 6(b, where the minus sign reresents vlue for the link. A. Link Wekening Before going into the deentrlistion rohes, let us define the link wekening oertion whih llows the deentrliston roess. This oertion lbels the links with one of the following two lbels. HARD This lbeling will be grnted to those links whih re not rt of the deentrlistion roess. The grh redution roess will tke into ount these links, SOFT These links rovide the mens to deentrlise the grh. If the link osseses this roerty, then the redution roess will not ss through them. Nevertheless, by using its onnetivity, they will rovide ISBN: ISSN: (Print; ISSN: (Online WCECS

4 Proeedings of the World Congress on Engineering nd Comuter Siene Vol I WCECS, 5 Otober,, Sn Frniso, USA 45 MW ( MW 4MW onvergene roess. In rtiulr, ll the nodes relted to riml vribles hve this informtion. Dul vribles do not hve seond order informtion t ll nd therefore they will hve to use Eq. 6 i = κ i L( (6 Gen. [5..6] Gen. [..4] Gen. [5..] ( Two nodes system. The min drwbk of grdient methods known lso s steeest desent methods is the hrdness to estimte κ. Do notie tht for the riml nodes Eq. 7 holds. κ = i (7 C. A Duloriented Aroh The seond nturl roh to deentrlise the grh is the duloriented roh. From the model roosed in setion II it is known the dul vribles re in only one lyer so if line ross both lyers is drwn disseting the grh, the links whih onneted the dul vribles with the riml vribles will be wekened s shown in figure 8. (b Its grh reresenttion. Fig. 6. Two nodes system nd its grh reresenttion mens to retrieve the tul vlue of the vrible t the other end of the link whih will llow the grdient to be omuted, s desribed in [7]. B. A Grdientoriented Aroh The first roh is to deentrlise the grh in n extreme wy by wekening ll the links s shown in figure 7. This method leds to model where the grdient method hs to be lied t eh node in the grh. Fig. 8. An duloriented deentrlistion roh The only dul vribles onsidered in this se re those relted with onstrints involving two or more riml vribles (i.e. bound dul vribles re not slit from the riml vribles set. Let us denote D s the set of those dul vribles. Therefore Eq. 4 beomes Eq. 8, where ll the seond order informtion bout the dul vribles re disregrded by the riml vribles. On the other hnd, s the dul vribles only hve links with riml vribles, they re now isolted just s in the grdient roh. Fig. 7. An grdientbsed deentrlistion roh i = i L( i j Γ i j / D i j j (8 i From this figure nd bsed on Eq. 4 we n ssert it will beome Eq. 5, where the seond order informtion of ll of its neighbours is disregrded. In this se L k =. Therefore Eq. 4 beomes Eq. 5. i = i L( i (5 Nevertheless, those nodes whih hve roer seond order informtion will be ble to use it in order to seed u the D. An Agentoriented Aroh In this roh the deentrlistion roess is mde bsed on onets drwn from the multigent ommunity. A lssil definition for n gent is An gent is omuter system situted in n environment, nd ble of flexible utonomous tion in this environment in order to meet its design objetives (dted from [9]. The interrettion in this work for n gent is n entity whih osseses some sttes or vribles nd resents n ISBN: ISSN: (Print; ISSN: (Online WCECS

5 Proeedings of the World Congress on Engineering nd Comuter Siene Vol I WCECS, 5 Otober,, Sn Frniso, USA indeendent nd rotive behviour, reresented by their objetive funtion. Furthermore, it is situted in n environment whih he n sense nd t ordingly. However, he is lso onstrined by its own limittions s well s the onstrins resented by the environment. It is imortnt to remrk tht the gents would be ting on behlf of eh node. To oe with this rdigm, the grh is slit into subsets of riml vribles, links, nd dul vribles. Bsed on the membershi of these omonents, the grh is deentrlised s shown in figure 9. This roh ws the one tken for [], [] [7] J. Cerd nd J. Avlos, A grh model roosl for onvex non liner serble roblems with liner onstrints, in Interntionl Conferene on Comuter Siene nd Alitions. Sn Frniso, CA, USA: Interntionl Assoition for Engineers, 5 Otober. [8] J. Cerd nd D. De Roure, A deentrlised d otiml ower flow, in Proeedings of the Third Interntionl Conferene on Eletri Utility Deregultion nd Restruturing nd Power Tehnologies, I. P. E. Soiety, Ed. Nnjing, Chin: IEEE Power Engineering Soiety, Aril [9] N. Jennings nd M. Wooldridge, Intelligent gents: Theory nd rtie, Knowledge Engineering Review, vol., no.,. 5 5, 995. [] J. Cerd, D. De Roure, nd E. Gerding, An gentbsed eletril ower mrket simultor, in AAMAS 8: Proeedings of the 7th Interntionl Joint Conferene on Autonomous Agents nd Multigent Systems. Rihlnd, SC: Interntionl Foundtion for Autonomous Agents nd Multigent Systems, 8, [] J. Cerd nd D. De Roure, An gentbsed deentrlistion roh for the eletriity ower mrket, in Proeedings of the IEEE Eletronis, Robotis nd Automotive Mehnis Conferene, P. Kellenberger, Ed. Cuernv, Morelos, Mxio: IEEE, Setember, Fig. 9. An gendbsed deentrlistion roh V. CONCLUDING REMARKS This doument hs resented grhbsed roh to deentrlie onvex NLSP with liner onstrints. To this end the min onets on how to deentrlise the grh hve been resented. The underlying deentrlition riniles hve been resented bsed on the nlysis of the eqution reresented by the node nd its links. Finlly, three deentrlistion rohes hve been desribed. The first one is totlly deentrlised roh whih will led us to grdient oriented model reinfored with its roer seond order informtion. The seond one is bsed on the tye of vribles (i.e. riml or dul, nd leds us to horiontl grh slit. The lst roh is bsed on onets drwn from multigents ommunity. Finlly, even when it hs been remrked tht this roh is for onvex roblems, it n be used for non onvex roblems leding to lol otimiers. REFERENCES [] I. Griv, S. G. Nsh, nd A. Soferr, Liner nd nonliner otimition. SIAM Soiety for Industril nd Alied Mthemtis, 9. [] A. G. Bkirtis nd P. N. Bisks, A deentrlied solution of the dof of interonneted ower systems, IEEE Trnstions on Power Systems, vol. 8, no.,. 7, August. [] A. Bkirtis, P. N. Bisks, N. Mhers, nd N. Psilis, A deentrlied imlementtion of d otiml ower flow on network of omuters, IEEE Trnstions on Power Systems, vol., no.,. 5, Februry. [4] P. Bisks nd A. Bkirtis, Deentrlised of of lrge multire ower systems, IEE Power, Genertion, Trnsmission nd Distribution, vol. 5, no., Jnury 6. [5] A. J. Conejo nd J. A. Agudo, Multire oordinted deentrlied d otiml ower flow, IEEE Trnstions on Power Systems, vol., no. 4,. 7 78, November 998. [6] G. Cohen, Otimition by deomosition nd oordintion: A unified roh, IEEE Trnstions on Automti Control, vol., no.,., 978. ISBN: ISSN: (Print; ISSN: (Online WCECS