Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

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1 Apped ad Copuaoa Maeacs 6; 5(5): 3-9 p:// do:.648/j.ac ISS: (Pr); ISS: (Oe) Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras L ao, Juje Bao, S Depare of Maeacs ad Iforao Egeerg, Cogqg Uversy of Educao, Cogqg, PR Ca Ea address: sao@63.co (L ao) Correspodg auor o ce s arce: L ao, Juje Bao, S. Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. Apped ad Copuaoa Maeacs. Vo. 5, o. 5, 6, pp do:.648/j.ac Receved: Augus 7, 6; Acceped: Ocober 5, 6; Pubsed: Ocober 7, 6 Absrac: I s paper, we prese a proved subgrade agor for sovg a geera u-age cove opzao probe a dsrbued way, were e ages are o joy ze a goba objecve fuco subjec o a goba equay cosra, a goba equay cosra ad a goba cosra se. e goba objecve fuco s a cobao of oca age objecve fucos ad e goba cosra se s e erseco of eac age oca cosra se. Our ovao coes fro eworg appcaos were dua ad pra-dua subgrade eods ave araced uc aeo e desg of decerazed ewor proocos. Our a focus s o cosraed probes were e oca cosra ses are deca. us, we propose a dsrbued pra-dua subgrade agor, wc s based o e descrpo of e pra-dua opa souos as e sadde pos of e peay fucos. We sow a, e agor ca be peeed over ewors w cagg opooges bu sasfyg a sadard coecvy propery, ad aow e ages o asypocay coverge o opa souo w opa vaue of e opzao probe uder e Saer s codo. Keywords: Cosesus, Sadde Po, Dsrbued Opzao, Subgrade Agor. Iroduco I rece years, dsrbued opzao ad coro ave deveoped rapdy, ad ave bee wecoed e feds of dusry ad aoa defese, cudg sar grd, sesor ewor, soca ewor ad forao syse (Cyber- Pysca syse). Dsrbued opzao probes of u-age syses appear dffere ds of dsrbued processg ssues suc as dsrbued esao, dsrbued oo pag, dsrbued resource aocao ad dsrbued cogeso coro [-]. e a focus s o sove a dsrbued opzao probe were e goba objecve fuco s coposed of a su of oca objecve fucos, eac of wc s oy ow by oe age. Dsrbued opzao probes were frs suded syseacay [] were e uo of e graps was assued o be srogy coeced aog eac e erva of a cera bouded eg ad e adjacecy arces were douby socasc. A dsrbued subgrade eod was roduced o sove e dsrbued opzao ad error bouds o e perforace de fucos were gve. As a couao of [], a dsrbued subgrade projeco agor was deveoped [] for dsrbued opzao were eac age was cosraed o rea a cosed cove se ad e paper gave correspodg covergece aayss o deca cosed cove ses ad o fuy coeced graps w o-deca cosed cove ses. Ispred by e wors of [, ], e agors proposed [] ad [] were suded e rado evroe [3] ad [4], were e ages ad e sae sae cosra. I [5], e coucao opoogy was udreced ad eac possbe coucao was fucog w a gve probaby. us, e epeced coucao opoogy s esseay fed ad udreced. Dffere fro []-[5], a dua averagg subgrade agor was deveoped ad aayzed for radozed graps uder e assupo a a ages rea e sae cosed cove se [6] ad was sow a e uber of eraos were requred by er agor scaes versey e specra gap of e ewor. Moreover, dsrbued opzao probes w asycroous sep-

2 Apped ad Copuaoa Maeacs 6; 5(5): szes or equay-equay cosras or usg oer agors were suded [7]-[] ad correspodg codos were gve o esure e syse coverge o e opa po or s egborood. However, as []-[5], was assued [6]-[] a e sae ses of ages o be deca or e objecve fuco fay coverge o oy a egborood of e opa se. I s paper our wor s o eed [4] o sudy e peay pra-dua subgrade projeco agor a ore geera eod. I [4], e auors soved a uage cove opzao probe were e ages subjec o a goba equay cosra, a goba equay cosra ad a goba cosra se. I order o soved ese cosras, e auor [4] preseed wo dffere dsrbued projeco agors w ree assupos a e uo of e graps s assued o be srogy coeced aog eac e erva of a cera bouded eg ad e adjacecy arces were douby socasc ad odegeeracy. However, [4] guaraeed e edge weg arces of graps were douby socasc (.e., a ( j j ) for a V ad, ad aj ( ) for a j V ad ). Prevous wor dd o perfor we o e appcao of e dsrbued agors uage ewor. Corbuos: e subgrade agor (we proposed) s dffere w e approac proposed [4] properes ad aayss. I our approac, e coucao opoogy s wou oss of geeray. s paper does o recur o e assupo a e adjacecy arces are douby socasc, ad we oy requre e ewor s weg-baaced, wc aes our agor ore pracca. I s paper, we cosder a geera u-age opzao probe were e a focus s o ze a goba objecve fuco wc s a su of oca objecve fucos, subjec o goba cosras, cudg a equay cosra, a equay cosra ad a (sae) cosra se. Eac oca objecve fuco s cove ad oy ow by oe parcuar age. O e oer ad, e equay (resp. equay) cosra s gve by a cove (resp. affe) fuco ad ow by a ages. Eac ode as s ow cove cosra se, ad e goba cosra se s defed as er erseco. Parcuary, we assue a e oca cosra ses are deca. Our a eres s copug approae sadde pos of e Lagraga fuco of a cove cosraed opzao probe. o se e sage, we frs sudy e copuao of approae sadde pos (as opposed o asypocay eac souos) by usg e subgrade eod w a cosa sep-sze. We cosder cosa sep-sze rue because of s spcy ad pracca reevace, ad because our eres s geerag approae souos fe uber of eraos. e paper s orgazed as foows. I Seco II, we gve soe basc preares ad coceps. e, Seco III, we prese our probe foruao as we as dsrbued cosesus agor preares. We e roduce e dsrbued peay pra-dua subgrade agor w soe supporg eas ad coue w a covergece aayss of e agor Seco IV. Furerore, e properes of e agor are epored by epoyg a uerca eape Seco V. Fay, we cocude e paper w a dscusso Seco VI.. Preares ad oaos I s seco, we frs roduce soe preary resus abou grap eory, e properes of e projeco operao o a cosed cove se ad cove aayss (referrg o [3], [4]). A. Agebrac Grap eory e coucao aog dffere odes a forao erpay ewor ca be odeed as a weged dreced grap G { V, E, A}, were V {,,..., } s e se of odes w represeg e ode, E V V s e edge se, ad A ( a j ) s e weged adjacecy ar of G w oegave adjacecy eees a j ad zero dagoa eees. A dreced edge e ( v, v ) pes a j j ode j ca reac ode or ode ca receve forao fro ode j. If a edge ( j, ) E, e ode j s caed a egbor of ode ad a j >. e egbor ode se of ode s deoed by, we we defe as e uber of egbors of ode. e Lapaca ar L ( j ) assocaed w e adjacecy ar A s defed by j aj, j ; a j, j j, wc esures a j j. e Lapaca ar L as a zero egevaue, ad e correspodg egevecor s. oe a e Lapaca ar L of a dreced grap G s asyerc. e -degree ad ou-degree of ode ca be respecvey defed by e Lapaca ar as : d ( v ) j, j j ad d ou ( v ) j, j j. A dreced pa fro ode j o ode s a sequece of edges ( j, ),(, ),...,(, ) e dreced grap G w dsc odes,,,...,. A dreced grap s srogy coeced f for ay wo dsc odes j ad e se V, ere aways ess a dreced pa fro ode j o ode. A grap s caed a -degrees (or ou-degrees) baaced grap f e -degrees (or ou-degrees) of a odes e dreced grap are equa. A dreced grap w odes s caed a dreced ree f coas edges ad ere ess a roo ode w dreced pas o every oer ode. A dreced spag ree of a dreced grap s a dreced ree a coas a e ewor odes. B. Basc oaos ad Coceps e foowg oo of sadde po pays a crca roe our paper. Defo (Sadde po): Cosder a cove-cocave fuco L : M V R, were, M ad V are cosed cove subses R ad M V R. We are

3 5 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. eresed copug a sadde po (,, ) of H (,, ) over e se M V, were a sadde po s defed as a vecor par (,, ) a sasfes H(,, ) H(,, ) H(,, ), for a, M, V I s paper, we do o assue e fuco f a soe pos are o dffereabe, ad e subgrade pays e roe of e grade. Defo : For a gve cove fuco F : R R ad a po R, a subgrade of e fuco F a s a vecor DF( ɶ ) R suc a e foowg subgrade equay ods for ay R : Τ DF( ) ( ) F( ) F( ) Sary, for a gve cocave fuco G : R R ad a po R, a supgrade of e fuco G a s a vecor DG( ) R suc a e foowg supgrade equay ods for ay R : Τ DG( ) ( ) G( ) G( ) We use P [ ] o deoe e projeco of a vecor o a cosed cove se,.e. P arg I e subseque deveope, e properes of e projeco operao o a cosed cove se pay a pora roe. I parcuar, we use e projeco equay,.e., for ay vecor P y P for a y () ( ) ( ) We aso use e sadard o-epasveess propery,.e. P P [ y] y for ay ad y () I addo, we use e properes gve e foowg ea. Lea.: Le be a oepy cosed cove se R. e, we ave for ay R, (a) ( P ) ( y P ) P, for a y. (b) P y y P, for a y. Proof: (a) Le R be arbrary. e, for ay y, we ave ( P ) ( y) ( P ) ( P ) ( P ) ( P y) By e projeco equay [cf. ()], foows a pyg P P y ( ) ( ) ( P ) ( y P ) P, for a y (b) For a arbrary R ad for a y, we ave P y P y P y ( P ) ( y) By usg e equay of par (a), we oba P y y P, for a y Par (b) of e precedg ea esabses a reao bewee e projeco error vecor ad e feasbe drecos of e cove se a e projeco vecor. e foowg oaos besdes ose aforeeoed w be used rougou s paper. R deoes e se of a - desoa rea vecor spaces. Gve a se S, we deoe co( S ) by s cove u. We wre or A o deoe e raspose of a vecor or a ar A. We e e fuco : R deoe e projeco operaor oo e o- R egave ora R. Deoe (,...,) R ad (,...,) R. For a vecor R, we deoe (,..., ), we s e sadard Eucdea or e Eucdea space. I s paper, e quaes (e.g., fucos, scaars ad ses) assocaed w age w be deed by e superscrp [ ]. 3. Probe Saee We cosder a u-age ewor ode. e odes coecvey a e ca be represeed by a dreced weged grap G( ) ( V, E( ), A( )), were E( ) s e se of acvaed edges a e,.e., edge ( j, ) E( ) f ad oy f ode ca receve daa fro ode j, ad A( ) [ a ( )] R s e adjacecy ar, wc j aj ( ) s e weg assged o e edge ( j, ) a e. Pease oe a e se E( ) V V \ dag( V) s e se of edges w o-zero wegs aj ( ). I s paper e ages are o correspodgy sove e foowg opzao probe: (3) f ( ) f ( ), s.. g( ), ( ), R [ were ] f : R R s a cove objecve fuco of age, ad s a oepy, cosed, copac ad cove subse of R. I parcuar, we sudy e cases were e oca cosra ses are deca.e., for eac age, ad s a goba decso vecor. Assue a f s oy ow by age. e fuco g : R R s ow by a e ages w eac copoe g, for {,..., },

4 Apped ad Copuaoa Maeacs 6; 5(5): beg cove. e equay g( ) s copoe-wse;.e., g ( ), for a {,..., }, ad represes a goba equay cosra. e fuco : R R, represes a goba equay cosra, ad s ow by a e ages. Le f deoe e opa vaue of (3) ad deoe a opa souo of (3). We assue a e opa vaue f o be fe. We aso represe e opa souo se by,.e., { R f ( ) f }. We w assue a geera f s o-dffereabe. o geerae opoa souos o e pra probe of Eq. (3), we cosder opoa souos o s dua probe. Here, e dua probe s e oe arsg fro peay reaao of e equay cosras g( ) ad equay cosras ( ). oe a e pra probe (3) s rvay equvae o e foowg: f ( ), s.. g( ), ( ), R w assocaed dua probe gve by a q (, ), s.., v P R, R v Here qp : R R R s e peay dua fuco Here, e dua fuco, qa (, ) f La (,, ). e defed by qp(, ) f H(,, ) dua opa vaue of probe (7) s deoe by, were a ad e se v of dua opa souos s deoed by Q. Sce s H : R R R R s e peay fuco gve by H(,, ) f ( ) Τ [ g( )] Τ cove, f ad g ( ). We ofe, for {,..., }, are cove, ad f s fe ad e Saer s codo ods, we ca cocude a v refer o vecor R, R w, as wo f a ad Q. We ow proceed o caracerze q uper. We deoe e dua opa vaue by q ad e ad M. Pc ay qa dua opa se by M. We defe e peay fuco (, ) Q. Sce, e Τ Τ a qa (, ) f{ f ( ) ( ) g( ) ( ) ( )} (5) Τ Τ f{ f ( ) ( ) [ g( )] ( ) } q (, ) q v (,, ) : H R R R R for eac age as foows: H (,, ) f ( ) Τ [ g( )] Τ ( ). I s way, we ave a H (,, ) H (,, ). We say a ere s zero duay gap f e opa vaue of e pra ad e dua probes are equa,.e., f q. As prove e foowg ea, e Saer s codo Assupo 3. esures zero duay ad e esece of peay dua opa souos. Assupo 3. (Saer s Codo): ere ess a vecor suc a g( ) < ad ( ). Ad ere ess a eas oe eror of,.e., probe (3) as fe opa souo, ad as oepy eror po. Lea 3.: Le e Saer codo ods, e vaues of f ad q cocde, ad M s o-epy. v Proof: Defe Lagraga fuco La : R R R R as La (,, ) f ( ) Τ g( ) Τ ( ), w e assocaed dua probe defed by v a R, R a q (, ), s.. (4) P O e oer ad, pc ay. e s feasbe,.e., [ g( )] ad ( ). I pes a q(, ) H(,, ) f ( ) f v ad R ods for ay R, ad us q sup q(, ) f a v R R erefore, we ave f q. o prove e o-epy of M, we pc ay (, ) Q. Fro (5) ad a q, we ca see a ad us (, ) M M. rougou s paper, we use e foowg assupo for probe (3). Assupo 3.: Le e foowg codos od: ) e se s cosed ad cove. ) Eac fuco 3) A fucos [ ] f : R R s cove. f ave Lpscz grades w a cosa L : Df ( ) Df ( y) L y for a, y R.. [ 4) e grades Df ] ( ), V are bouded over e se,.e., ad ere ess a cosa G suc a Df ( ) G for a ad a V. We eac f as Lpscz grade w a cosa L, assupo 3.(3) s sasfed w L a L. We s copac, e Assupo 3.(4) ods. We ere ae e foowg assupos o e ewor coucao graps G( ). Assupo 3.3 (o-degeeracy): ere ess a cosa > suc a a ( ), ad a ( ), for j, sasfes a ( ) {} [,], for a. j Assupo 3.4 (Weg-baaced): G( ) s wegbaaced f dou ( v) d ( v), for a v V. Assupo 3.5 (Perodca Srog Coecvy): ere s a posve eger B suc a, for a, e dreced j V B grap ( V, E( )) s srogy coeced.

5 7 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. Lea 3. (Sadde-po eore): e par of (,, ) s a sadde po of e fuco H over v R R f ad oy f s a par of pra ad peay dua opa souos ad e foowg peay a equay ods: sup f H (,, ) f sup H (,, ) v v (, ) R R (, ) R R Based o s caracerzao, we w use e subgrade eod of e foowg seco for fdg e sadde pos of e peay fuco. We deoe w (, ), for eac w R R ad we defe e fuco v [ ] : as Hw R R [ ] [ Hw ( ) H (, w). oe a H ] w ( ) s cove by usg e fac a a oegave weged su of cove fucos s cove. For eac R, we defe e fuco v ( ) : H R R R as cec a H ( w) H (, w). I s easy o [ H ] ( w ) s cocave w. e e peay fuco H (, w ) s e su of cove-cocave oca fucos. Lea 3.3 (Dyac Average Cosesus Agor) [] : e foowg s a vecor verso of e frs-order dyac average cosesus agor w ( ), ξ ( ) R : We se [ j] j j ξ ( ) w ( ) ( ) ( ) ξ ( ) a ξ ( ) ξ ( ) for V V. e sequeces of W( ) [ w ( )] sasfy w ( ) j j ad w ( ). Suppose a perodca j srog coecvy Assupo 3.5 ods. Assue a ξ ( ) for a ad a. e ( ) ( ) for a, j V. Proof: Defe M ( ) a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r ( ) a r ( ) r ( ) r ( ) a V V V D M r r r V j p ( ) () r ( p) [ were r ] ( ) s referred o as e referece sga (or pu) of ode a e. We propose e Frs-Order Dyac Average Cosesus Agor beow o reac e dyac average cosesus: ( ) ( ) w ( )( ( ) ( )) r ( ) j j j Le s ad V be fed. e for every {,..., }, ere ess a rea uber η > suc a for every eger P [ B,( B B )], ad D, ods a for s p p q η (6) ( ) ( s) r ( s q) ( ( s) ( s)) p q a η (7) ( ) M ( s) r ( s q) ( M ( s) ( s)) Wou oss of geeray, we oy cosder e case were s, beg deca w e proof for a geera s. Fg soe, ods a ( ) ( ) w ( )( ( ) ( )) r ( ) j j j Le, we ave a j ( ) () w ()( () ()) r () j j j ( w ()) () w () () r () ( w ()) () w () () r () j () r () Sce (8) ods for a, j j j j j j j (8) ( ) () r () (9) Appyg recursve eod, foows a () p ( ) () r ( p) Sce w ( ) j j a every >, we ave a j j j j j j j p w ( ) ( ) r ( ) w ( ) () w ( ) r ( p) r ( ) j j j p w ( )( ( ) () r ( p)) r ( ) r ( ) p w ( )( ( ) () r ( p) ) p ω( ( ) () r ( p)) were we are usg e propery of () e as wo equaes. Appyg repeaedy (), we ave a, for ay eger P [ B,( B B )], e foowg ods for p ()

6 Apped ad Copuaoa Maeacs 6; 5(5): p p q ω ( ) () r ( q) ( ( ) () r ()) p ω ( () ()) η ( () ()) B were η ω. ow we proceed by duco o. Suppose a (6) ods for soe ; e we soud sow (6) for. By e duco ypoess, we ave a for a eger P [ B,( B B )], ere ess soe η > suc a e foowg ods for p Cosequey, as (), we ave p τ η ( ) () r ( q) ( () ( ) ) j ( ) () ( ) ( )( ( ) () q j q r q w r ( q) ) Foowg aog e sae es as (), we oba ωη ( () ( ) ) p η q for a ( ) () r ( q) ( () ()) ( ) B P [( ) B,( B B )] were η ω η ad p. s esabses (6) for. By duco, we ave sow a (6) ods. e proof for (7) s aaogous. ( ) Le η ω B, e η η for ay {,..., }. By repacg s ad (4) w ad ( LB B ) respecvey. We ave a for every {,..., L} D q {,..., L} ( ) ( ) ( ) r ( q) η ( ( ) ( )) q ( ) r ( q) η( ( ) ( )) Sary, we ca see a a q M ( ) M ( ) r ( q) η( M ( ) ( )) Cobg e above wo equaes gves a q D( ) ( η) D( ) R( q) Deog ( B ) for a eger. Fro (9), we ow a D( ) D( ) R( ). us we ave were D( ) ( η) D() Ω ( ) ( ) ( ) ( )... η R q ( ) q R q q. Ω For ay, e be e arges eger suc a q ( B ), ad Ω ( ) Ω ( ) R( q). us for a foows a q D( ) D( ) R( q) ( η) D() Ω( ) ( B ) ( η) D() Ω( ) () Sce R( ) θ ad D( ) are pu-o-oupu sabe w ( ) B uae boud Ξ 4 θ ( B ) 4 θ ( B ) w ;.e., η ere es Γ > ad suc a D( ) a{ Γ, Ξ}, Coosg as a sae () r ( ) for a {,..., }. Sce ( ) w ( )( ( ) ( )) ( ) r ( ), we ca deduce a j j j

7 9 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. ( ) ( ) r ( ) q () r ( q) () ( r ( ) r ( )) r ( ) (3) I foows fro (3) a ( ) r ( ) ( ) M ad us asup ( ) r ( ) sup D( ) Ξ V Le R( ), for ay >. e peeao of e Dyac Average Cosesus Agors esures a Ξ. So we ca cocude a sup ( ) ( ) sup D( ) j us, sup ( ) ( ) ods. j Cosder e foowg Dsrbued projeced subgrade agor proposed [3]: Suppose Z R s a cosed ad cove se. Le ( ) P [ v ( ) ( ) d ( )]. Deoe Z. e foowg s a e ( ) P [ v ( ) ( ) d ( )] v ( ) Z sg odfcao of Lea 8 ad s proof [3]. Lea 3.4: Le e o-degeeracy Assupo 3.3, e weged-baaced Assupo 3.4, ad e perodc srog coecvy Assupo 3.5 od. e ere es γ > ad β (,) suc a τ ( ) ( ) γ β { ( τ ) d ( τ ) e ( τ ) ( τ ) d ( τ ) } τ γβ () Suppose { d ( )} s ufory bouded for eac V, ad ( ) <, e we ave ( ) a ( ) ( ) V <. 4. Dsrbued Subgrade Meods I s seco, we roduce a dsrbued peay pradua subgrade agor o sove e opzao probe (3), foowed by s covergece properes. Dsrbued Peay Pra-Dua Subgrade Agor We cosder a se V {,..., } of ages. Eac age cooses ay a sae v (), () R, () R, ad y () f ( ()). A ay e, eac age copues e foowg cove cobao: v ( ) ( ) a ( )( ( ) ( )) [ j] j ( ) j v ( ) ( ) a ( )( ( ) ( )) [ j] u j ( ) j v ( ) ( ) a ( )( ( ) ( )) [ j] j ( ) j v ( ) y ( ) a ( )( y ( ) y ( )) [ j] y j ( ) j ad updaes s esaes [ ] [ ] [ ] ( ), ( ), ( ), ad [ y ] ( ) accordg o e foowg ways: ( ) P [ v ( ) ( ) S ( )] ( ) v ( ) ( )[ g( v ( ))] ( ) v ( ) ( ) ( v ( )) y ( ) v ( ) ( f ( ( )) f ( ( ))) y (4) were e scaars a ( ), a ( ),..., a ( ) are oegave wegs ad e posve scaars { ( )} are sep-szes, P s e projecor oo e se. e vecor S ( ) s a subgrade of e age s peay fuco H ( ) [ ] w ( ) a [ v ] ( ), were w ( ) ( v ( ), v ( )) s e cove cobao of dua esaes of age ad s egbors. [ S ] ( ) eeps o e foowg rues: S ( ) Df ( v ( )) v ( ) Dg [( v ( ))] v v ( ) D ( v ( )) Rear 4.: Sce [ j] v ( ) ( ) a ( )( ( ) ( )), foows a j ( ) j [ j] v ( ) ( ) ( ) ( ) j j were L( ) [ ( )] s e Lapaca ar suc a L L., j Proof: Modfyg e secod er o e rg-ad sde e above forua, we e ave [ j] j, j j v ( ) ( ) ( ) ( ) ( ) ( ) [ j] j, j j ( ( )) ( ) ( ) ( ) Le w ( ) ( ( )), w ( ) ( ), oe as j j [ j ] j j v ( ) w ( ) ( ) Sce grap G( ) s baaced, e a ( ) j j ad a ( ) j. We ca cocude a ( ) w j j ad w ( ) j od uder e codo a sasfes ( ) >. Sary, we oba

8 Apped ad Copuaoa Maeacs 6; 5(5): 3-9, j w y. [ j] j j v ( ) w ( ) ( ) v ( ) ( ) ( ) y j j v ( ) w ( ) ( ) ad [ j] j j Assupo 4. (Sep-sze assupo): e sep-szes sasfy ( ), ( ), ( ) < ad ( ) s( ), ( ) ( ) s <, ( ) s( ) <. I e foowg, we sudy e covergece beavor of e subgrade agor roduced s seco were e opa souo ad e opa vaue s asypocay agreed upo. eore 4. (Covergece properes of e DPPDS agor): Cosder e probe (3). Le e odegeeracy Assupo 3.3, e weg-baaced Assupo 3.4 ad e perodc srog coecvy Assupo 3.5 od. Cosder e sequeces of { ( )} ad { y ( )} of e dsrbued peay pra-dua subgrade agor, were e sep-szes { ( )} sasfy e sep-sze Assupo 4.. e ere ess a pra opa souo ɶ suc a ( ) ɶ for a V. Furerore, we ave y ( ) f for a V. Rear 4.: e dsrbued peay pra-dua subgrade agor aes e equay cosra o accou. e presece of e equay cosra ca ae M ubouded. erefore, ue oer subgrade agor, e.g., [5], [6], e dsrbued peay pra-dua subgrade agor does o vove e dua projeco seps oo copac ses. So we do o guaraee e [ subgrade S ] ( ) o o be absouey bouded, we e boudedess of subgrades s a sadard assupo e aayss of subgrade eods, e.g., see [6], [3], [7], [8], [9], []. e sep-sze of Assupo 3. s sroger a e ore sadard dsg sep-sze scee [] ad s w correcy dea w e dffcuy of e [ boudedess of S ] ( ). We gve s codo order o [ prove, e absece of e boudedess of S ] ( ), e esece of a uber of s ad suaby of epaso oward eore 4.. Fay, we adop e peay reaao sead of e Lagraga reaao s paper. Rear 4.3 (Peay subgrade equay): Observe a [ ( ), ( ) ad v ] ( ) (due o e fac a s cove ad v ( ) w ( ) ( ) ). Moreover, ( g[( v ( ))], ( v ( )) ) [ w ] j j s a supgrade of H ( w ( )) ;.e. e foowg peay supgrade ( ) v equay ods for ay R ad R : Τ Τ ( g[( v ( ))] ) ( v ( )) ( v ( )) ( v ( )) H ( v ( ),, ) H ( v ( ), v ( ), v ( )) (5) Proof: Observe a Τ Τ H (,, ) f ( ) [ g( )] ( ) ods for a [ ] ( ), us, [ ] ( ), [ v ] ( ) ad s arbrary. Τ Τ H ( v ( ),, ) f ( v ( )) [ g( v ( ))] ( v ( )) ad Τ ( ( ), ( ), ( )) ( ( )) ( ( )) [ ( ( ))] H v v v f v v g v Τ ( v ( )) ( v ( )) Foowed by e properes of supgrade, we oba H ( v ( ),, ) H ( v ( ), v ( ), v ( )) Τ Τ ( ) ( v ( )) [ g( v ( ))] v ( ) ( v ( )) Τ Τ ( g[( v ( ))] ) ( v ( )) ( v ( )) ( v ( )) Rear 4.4: I s paper, we appy e aroc seres ( ) o our subgrade agor. I s easy Z o cec a ( ) sasfes e sep-sze Z Assupo 4. (for ore deas, oe ay refer o [4]). A. Covergece Aayss I e foowg, we w prove covergece of e dsrbued peay pra-dua subgrade agor. Frs, we rewre our agor o e foowg for: ( ) v ( ) u ( ), ( ) v ( ) u ( ) ( ) v ( ) e ( ), y ( ) v ( ) u ( ) y y [ were e ] ( ) s projeco error descrbed by ad e ( ) P [ v ( ) ( ) S ( )] v ( ) u ( ) ( )[ g( v ( ))] y, u ( ) ( ) ( v ( )), u ( ) ( f ( ( )) f ( ( ))) are soe oca pus. Deoe e au devaos of dua esaes by M ( ) a V ( ) ad M ( ) a V ( ). We furer deoe e averages of pra ad dua esaes as ( ) ( ), ( ) ( ) ad ( ) ( ). Sce s copac, ad f, [ g( )] ad are couous, ere es F, G, H > suc a for a, ods a f ( ) F for a V, [ g ] ( ) G ad ( ) H. Sce s a copac se

9 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. ad f, [ ( )] g, ( ) are cove, e foows fro Proposo 5.4. [9] a ere es DF, D, DH > G suc a for a, we ave a [ ], Df ( ) D ( V ) F D g ( ) D ( ) ad D ( ) DH ( v). Lea 4.: Le K. Cosder e sequece { δ ( )} defed by { δ ( )} ( ) ρ( ) ( ) ( ) > ad ( ). K (a) If ρ( ), e δ ( ) (b) If ρ( ) ρ, e δ ( ) G, were K,. ρ. [4]. Lea 4. (Dsg ad suabe properes): Suppose e weged-baaced Assupo 3.4 ad e sepsze Assupo 4. od. (a) I ods a, ad e sequeces of { ( ) M ( ) }, M { ( ) S ( ) } are suabe. ( ) M ( ), ( ) M ( ), ( ) S ( ) e proof of Lea 4. ca be referred o Lea 5. [ j] M ( ) a ( ), v ( ) w ( ) ( ), w ( ), w ( ) e, we sow a { ( ) ( ) } ad (b) e sequeces of { ( ) ( ) v ( ) }, { ( ) ( ) v ( ) }, { ( ) M ( ) ( ) v ( ) }, { ( ) M ( ) ( ) v ( ) } ad { ( ) ( ) v ( ) } are suabe. Proof: (a) ocg a V j j j j j [ j] [ j] j j j j j j v ( ) w ( ) ( ) w ( ) ( ) w ( ) M ( ) M ( ) Recag a : [ ] v ( ), ( ) ( ) ( )[ ( v g v ( ))]. s pes a e foowg equaes od for a ( ) ( ) ( )[ ( ( ))] v g v v ( ) G ( ) M ( ) G ( ) e we deduce e foowg recursve esae o M ( ) M ( ) G ( ). Repeaedy appyg e above esaes yeds a M ( ) M () G s( ) (6) were s( ) () () ( ). Sar argues ca be epoyed o sow a M ( ) M () Hs( ) (7) Sce ( ) s( ) ad ( ), e we ow a ( ) M ( ) ad ( ) M ( ). ocg a v S ( ) Df ( v ( )) v ( ) Dg [( v ( ))] v ( ) D ( v ( )) v Df ( v ( )) v ( ) Dg [( v ( ))] v ( ) D ( v ( )) e, e foowg esae o [ S ] ( ) ods: [ ] S ( ) D D M ( ) D M ( ) (8) F G H Recag a ( ), ( ) M ( ) ad ( ) M ( ). e we ave (6), we oba ( ) ( ) () () ( ) ( () ( )) M M M G s [ ] ( ) S ( ). By

10 Apped ad Copuaoa Maeacs 6; 5(5): 3-9 I foows fro e sep-sze Assupo 4. a ( ) M ( ) ( ) M ( ) <. Mupyg bo sdes of M ( ) M ( ) G ( ) foowg recursve esae: F G H ( ) S ( ) () ( D D M () D M ()) <. Sary, oe ca sow a by ( ) ad square, e we deduce e ( ) ( D ( () ( )) ( () ( ))) F DG M G s DH M Hs e e suaby of { ( ) }, { ( ) s( )} ad { ( ) s( ) } esfes a of { ( ) S ( ) }. (b) ocg a ( ) ( ) j v w ( ) ( ) ( ) a j ( ) ( ) (9) j V e foowg fro Lea 3.4 w Z R ad d ( ) [ g( v ( ))], we ave e suaby of { ( )a V ( ) ( ) }. e { ( ) ( ) v ( ) } s suabe. Sary, ods a ( ) ( ) v ( ) <. [ We ow cosder e evouo of [ ( ). Recag a v ( ). By Lea., e Z, z v ( ) ( ) S ( ) ad [ y v ] ( ), we ge Regroupg e above esaes, we oba W e above reao, fro Lea 3.4 w Z < β < : ( ) v ( ) v ( ) ( ) S ( ) v ( ) ( ) ( v ( ) ( ) S ( )) [ ] [ ] [ ] e ( ) ( ) S ( ) ( ) S ( ) ad, e foowg ods for soe γ > ad d ( ) S ( ) τ τ ( ) ( ) γβ () γ β ( τ ) S ( τ ) Mupyg bo sde of () by ( ) M ( ) ad usg (8), for a V, yeds ( ) M ( ) ( ) ( ) γ () ( ) M ( ) β () τ τ F G H γ( ) M ( ) β ( τ )( D D M ( τ ) D M ( τ )) By appyg e reao of ( ) ab a b ad sorg ou, we ge Par (a) gves a τ ( ) M ( )a ( ) ( ) γ ( () ( D ) ) F D D G H β V τ ( ) ( ) M ( ) γ () β β { ( ) M ( ) } s suabe. Meawe, τ F τ G H γ β ( τ ) ( D D M ( τ ) D M ( τ )) (), D,, F D D G H are bouded, ad τ β β τ, e we ca say a e frs er o e rg-ad sde e above esae s suabe.

11 3 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. Recag a γ () β gves a ( ) accordg o e Lea 7 [3] w γ (), s easy o cec a e secod er s aso suabe. Par (a) β ( ) (( DF D M ( ) DHM ( ))) G ad { ( ) (( D ( ) ( )))} F D M D G HM γ γ ( ) ( DF D M ( ) DHM ( )) G s suabe. e, esure a e rd er s suabe. I suary, { ( ) M ( )a ( ) ( ) } s suabe. Foowg e sae es (9), oe ca sow e suaby of V M v. Sary, { ( ) ( ) ( ) ( ) } [ ] { ( ) M ( ) v ( ) ( ) } ad suabe. Lea 4.3 (Basc erao reao): e foowg esaes od for ay ad (, ) R R ν : [ ] { ( ) v ( ) ( ) } are e ( ) ( ) S ( ) ( ) S ( ) ( ( ) ( ) ) ( )( H ( v ( ), ( ), ( )) (, ( ), ( ))) v v H v v () ad ( ( ) ( ) ) ( ( ) ( ) ) ( )( H ( v ( ), v ( ), v ( )) H ( v ( ),, )) ( ) ( [ g( v ( ))] ( ( )) ) v () Proof: By Lea., we ca deduce a Z, z v ( ) ( ) S ( ), y, we ave ad P [ z] z z y P [ z] y. Le Z Z e ( ) ( ) ( ) ( ) ( ) ( ) ( ) S v S Epadg ad regroupg e above forua, we oba Owg o e subgrade equay foows a: Τ e ( ) ( ) S ( ) ( ) S ( ) ( ) S ( ) ( v ( ) ) ( ) ( ) Τ S ( ) ( v ( ) ) H ( v ( ), v ( ), v ( )) H (, v ( ), v ( )), e S S ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) H v v v ( )( ( ( ), ( ), ( )) H (, v ( ), v ( ))) Lea 4.4 (Acevg cosesus): Le assupo ods. Cosder e sequeces of { ( )},{ ( )},{ ( )} { y ( )} of e dsrbued peay pra-dua subgrade agor w e sep-sze sequece { ( )} ad e assocaed { s( )} sasfy ( ), ( ) s( ). e ere ess ɶ suc a ( ) ɶ for a V [ j] [ j] [ ] [ j]. Furerore, ( ) ( ), ( ) ( ) ad y ( ) y ( ) for a, j V. Proof: By Lea 4.3, we see a Owg o e ( ) ( ) S ( ) Τ e ( ) ( ) S ( ) ( ) S ( ) ( ) S ( ) ( v ( ) ) ( ) ( ), oe ca sow a ( ) ( ) S ( ) ( ) S ( ) ( v ( ) ) ( ) [ ]

12 Apped ad Copuaoa Maeacs 6; 5(5): Sce ( ) S ( ), ag e o e above equay, e foows a : ad us sup ( ) f ( ) ( ) ess for ay { : e S S ( ) ( ) ( ) ( ) ( ) { ( ) ( ) } H v v } O e oer ad, ag s o bo sdes of e above equay, we ave ( )( H ( v ( ), v ( ), v ( )) (, ( ), ( ))) e D ad ( ) ( ) ( ) [ j] erefore we deduce a e ( ) for a V. I foows fro Lea 4. a ( ) ( ) for a, j V. Cobg s w e propery, we ca deduce a Sce ( ) ɶ, f s couous, for ( ) ɶ for a V. y u ( ) ( f ( ( )) f ( ( ))), u ( ) ( ) ( v ( )), we ca deduce a u ( ), u ( ), u ( ). Ca : For ay ad y (, ) M, e sequeces of { ( )[ ad { ( )[ H ( v ( ),, ) H ( ( ),, ]} are suabe Proof: Observg a H v v H (, ( ), ( )) (, ( ), ( )) u ( ) ( )[ g( v ( ))] ad H (, v ( ), ( )) (, v H ( ), ( ))]} f v g v f g Τ Τ Τ Τ ( ) ( ( )) [ ( )] ( ( )) ( ) ( ) ( ( )) [ ( )] ( ( )) ( ) Τ Τ Τ Τ (( v ( )) ( ( )) )[ g( )] (( v ( )) ( ( )) ) ( ) Recag a f ( ) F, g( ) G, ( ) H, we e ave H (, v ( ), v ( )) H(, ( ), ( )) Τ Τ Τ Τ (( ( )) ( ( )) )[ ( )] (( ( )) ( ( )) ) ( ) v ( ) ( ) g( ) v ( ) ( ) ( ) v g v G v ( ) ( ) H v ( ) ( ) By usg e suaby of { ( ) ( ) v ( ) } ad { ( ) ( ) v ( ) } par (b) of Lea 4., we ave a are suabe. Sary, e foowg esaes od: { ( ) H (, v ( ), v ( )) H (, ( ), ( )) } H v ( ( ),, ) H( ( ),, ) f v g v v f g Τ Τ Τ Τ ( ( )) ( ) [ ( ( ))] ( ) ( ( )) ( ( ) ( ) [ ( ( )] ( ) ( ( ) Due o Df ( ) D ( V) foowg esaes od:, F D g ( ) D ( ) ad v D ( ) DH ( v) ods for a G, e H ( v ( ),, ) H ( ( ),, ) f ( v ( )) f ( ( )) Τ ( ) ([ g( v ( ))] [ g( ( )] ) Τ ( ) ( ( v ( )) ( ( )) ) ( DF D DH ) v ( ) ( ) G

13 5 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. e e propery of ( ) ( ) v ( ) < par (b) Lea 4. pes e suaby of e sequece { ( ) (, ( ), ( )) (, ( ), H v v H ( )) } ad a of { ( )[ ( (, ( ), ( )) (, ( ), H v v H ( )))]}. Ca : Deoe e weged verso of e oca peay fuco H over [, ] as H ( ) ( ) H( v ( ), ( ), ( )) v v. e foowg propery ods: H ( ) f. s( ) Proof: Sug () over [, ] ad repacg by, we ca deduce a H v v v H v v ( ) ( ( ( ), ( ), ( )) (, ( ), ( ))) () ( ) S ( ) (3) e suaby of { ( ) S ( ) } Par (b) pes a e rg-ad sde of () s fe as, ad us sup ( )[ ( H ( v ( ), ( ), ( )) (, ( ), ( )))] v v H v v (4) s( ) v O e oer ad, (,, ) s a sadde po of H over R R. Sce ( ( ), v ( )) R R, e we ave H(, ( ), ( )) H(,, ) f. Ca ad (4) reders a ad us sup ( )[ ( H ( v ( ), ( ), ( )) ] v v f s( ) sup ( )[ ( H ( v ( ), ( ), ( )) (, ( ), ( )))] v v H v v s( ) sup ( )[ H (, v ( ), v ( )) H (, ( ), ( ))] s( ) sup ( H (, ( ), ( )) f ) s( ) H f. O e oer ad, ( ) (due o s cove) pes sup ( ) H( ( ),, ) H(,, ) f. Sary, we ave e foowg esaes H f. ( ) f H ( ) p. us Ca 3: Deoe π ( ) H (( v ( ), v ( ), v ( )) H( ( ), ( ), ( )). Ad we deoe e weged verso of e goba peay fuco H over [, ] as H H ( ) ( ) ( ( ), ( ), ( )) s( ) e foowg propery ods: Proof: ocg a π H ( ) f. ( ) ( f ( v ( )) f ( ( ))) ( v ( )) [ g( v ( ))] v ( ) [ g( ( ))] ) (( v ( )) [ g( v ( ))] ( ) [ ( ( ))] ) g ( v ( ) ( v ( )) ( ) ( ( )) ) v ( v ( ) ( ( )) ( ) ( ( )) ) (5) By usg e boudedess of subgrades ad e pra esaes, we ca see a

14 Apped ad Copuaoa Maeacs 6; 5(5): ocg a π ( ) ( f ( v ( )) f ( ( ))) ( v ( )) [ g( v ( ))] v ( ) [ g( ( ))] ) [ ] (( v ( )) [ g( v ( ))] ( ) [ ( ( ))] ) g ( v ( ) ( v ( )) ( ) ( ( )) ) v ( v ( ) Df ( ) D ( V), e foowg esaes od: F ( ( )) ( ) ( ( )) ), D g ( ) D ( ) ad v D ( ) DH ( v) ods for a F G H π ( ) ( D D M ( ) D M ( )) v ( ) ( ) G G v ( ) ( ) H v ( ) ( ) e foows fro (b) Lea 4. a { ( ) π ( ) } s suabe. oce a ( ) ( ) H( ) H ( ) π s( ). Foowg e Ca, ece, Ca 4: e po ɶ Lea 4.4 s a pra opa souo. H ( ) f., we ge Proof: Le ( ) ( ( ),, ( )) R. By ( ) v ( ) uu ( ) ad w ( ), ( ) j w j j (6) [ j] [ j] ( ) w ( ) ( ) ( ) [ g( v ( ))] j j ( ) ( ) [ g( v ( ))] j (7) s dcaes a e sequece { ( )} s o-decreasg R. Observg a { ( )} s ower bouded by zero. erefore, we gve e foowg wo cases: Case : e sequece { ( )} s upper bouded. e { ( )} s coverge R. e foows fro Lea 4.4 a [ j] ( ) ( ) for a, j V. s pes a ere ess suc a R ( ) for a V. Recag a ( ) ( ) u ( ) u (7). Foowg a recursve sep, we ca ge ( ) () ( ) τ [ g( v ( ))] τ τ. Sce ( ) [ ( ( ))] g v τ < ad ( ), we f[ g ( v ( τ ))] [ j] ( ) ɶ for a V oba. Sce w ( ) ( ) ( ) j v j w j j ɶ for a V, e v ( ) ɶ ad us [ g ( ɶ )]., we ave Case : e sequece { ( )} s o upper bouded. Sce { ( )} s o-decreasg, e ( ) by. Recag a H ( ) ( ) ( ( ), ( ), H ( )) ad H ( ) f, e foows fro (a) Lea 4. a s( ) s possbe a H( ( ), ( ), ( )). Suppose a [ g ( ɶ )] >. e we oba ag s o bo sdes of (8), e we ge H( ( ), ( ), ( )) f ( ( )) ( ) [ g( ( ))] ( ) ( ( )) (8) f ( ( )) ( )[ g ( ( ))] f H ( ( ), ( ), ( )) sup( f ( ( )) ( )[ g ( ( ))] ) Fay, we reac a coradco, pyg a [ g ( ɶ )]. I bo cases, we oba [ g ( ɶ )] for ay. Sary, we ca furer prove ( ɶ ). Sce ɶ, e ɶ s feasbe ad us ɶ f. For aoer, sce f ( ) ( ) ( ) ( ) s a cove cobao of (),, ( ) ad

15 7 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. ( ) ɶ, e foows fro Ca 3 ad (b) Lea 4. a: Hece, we ave ( ) H ( ( ), ( ), )) ( ) ( ) f H ( ) f ( ) f ( ɶ ) ( ) ( ) ɶ ad us f ( ) f Lea 4.5: I ods a Proof: Sce ɶ. [ ] y ( ) f. j y ( ) v ( ) u ( ), v ( ) w ( ) y ( ), e e foowg ods for ay y y j j [ j] y ( ) w ( ) y ( ) u ( ) e foowg ca be prove by duco o for a fed : j j y ( ) y ( ) ( f ( ( )) f ( ( ))) (9) Le (9) ad reca a a sae y [ ] [ ] [ ] () f ( ()) for a V. e we oba ( ) () ( ( ( )) ( ())) y y f f f ( ( )) (3) Fro (3), we ca oba ( ( ) ( )) ( ( ( )) y y f f ( ( ))) u ( ) (3) Cobg (3) w [ ] [ j] y ( ) y ( ) gves e desred resu. Based o e above fve Leas, we e fs e prove of eore uerca Eape I s seco, we sudy a spe uerca eape o usrae e effecveess of e proposed dsrbued peay pra-dua subgrade agor. Cosder a ewor w fve ages. Suppose eac age as a fuco [ f ] : R R, gve by We sove probe (3) by epoyg e dsrbued peay pra-dua subgrade agor (4) w e sepsze ( ) / ( ). Is suao resus are sow fro [ ] Fgs. o 5. I ca be see fro Fg. a oca pu u eds o we aceves cosesus. Fg. sows e sae evouos of a fve ages, wc deosrae a a 3 ages aes 5 eraes o asypocay aceve cosesus. e sae evouos of dua souo ad are sow Fgs. 3 ad 4, respecvey. We ca observe fro Fg. 5 a a e ages asypocay aceve e opa vaue. f ( ) a b ( c ) d ( e ) [ ] [ ] [ ] [ ] [ ] [ ] 4 were e goba decso vecor 5 [ ] R. e goba equay cosra fuco s gve by g( ) , e goba equay cosra fuco s gve by ( ) ad e goba cosra se s gve as: [ 3 3] [ 3 3] [ 3 3] [ 3 3] [ 3 3]. a, b, c, d, e are paraeers of f, wose vaues are radoy coose fro e ervas (,), (,), (,), (,), (,). Cosder e opzao probe as foows: Fg.. Loca pu [ ] u eds o we aceve cosesus. f g (3) ( ), 5 R V s.. ( ), ( ),

16 Apped ad Copuaoa Maeacs 6; 5(5): Cocuso ad Fuure Wor Fg.. Opa souo of pra probe. I s paper, we foruaed a dsrbued opzao probe w oca objecve fucos, a goba equay, a goba equay ad a goba cosra se defed as e erseco of oca cosra ses. I parcuar, we cosdered e oca cosra ses o be deca. e, we proposed a dsrbued peay pra-dua subgrade agor for e cosraed opzao w a covergece aayss. Moreover, we epoyed a uerca eape o sow a e agor was asypocay coverge o pra souos ad opa vaues. Fuure wor ay a a e aayss a e oca cosra ses of eac age are pares. Aso, we w pay aeo o e covergece raes of e agors s paper. Acowedgees s wor descrbed s paper was suppored par by e aura Scece Foudao Projec of Cogqg CSC uder gra csc4jcyja44, par by e Scefc ad ecoogca Researc Progra of Cogqg Mucpa Educao Cosso uder KJ54. Fg. 3. Opa souo of dua probe. Refereces [] J. C. Duc, A. Agarwa, ad M. J. Wawrg. Dua averagg for dsrbued opzao: covergece aayss ad ewor scag, IEEE rasacos o Auoac coro,, 57(3): [] A. edć, A. Ozdagar, ad P. Parro. Cosraed cosesus ad opzao u-age ewors, IEEE rasacos o Auoac Coro,, 55(4): [3] K. Srvasava ad A. edć. Dsrbued asycroous cosraed socasc opzao, IEEE Joura of Seeced opcs Sga Processg,, 5(4): [4] S. S. Ra, A. edć ad V. V. Veerava, Dsrbued Socasc Subgrade Projeco Agors for Cove Opzao, Joura of opzao eory ad appcaos,, 47(3): Fg. 4. Opa souo of dua probe. [5] S. S. Ra, A. edć, ad V. V. Veerava. Dsrbued socasc subgrade projeco agors for cove opzao, Joura of opzao eory ad appcaos,, 47(3): [6] A. edć ad A. Ozdagar. Dsrbued Subgrade Meods for Muage Opzao, IEEE rasacos o Auoac Coro, 54(): 48-6, 9. [7] S. S. Ra, A. edć, ad V. V. Veerava. Icreea socasc subgrade agors for cove opzao, SIAM Joura o Opzao, 9, (): [8] J. Lu, C. Y. ag, P. R. Reger, ad e a. A gossp agor for cove cosesus opzao over ewors, Aerca Coro Coferece (ACC),. IEEE, : Fg. 5. Opa souo f of objecve fuco f. [9] M. Zu ad S. Maríez. A approae dua subgrade agor for dsrbued o-cove cosraed opzao, e proc. of Coferece o Decso ad Coro,, pp

17 9 L ao e a.: Dsrbued Subgrade Agor for Mu-Age Cove Opzao w Goba Iequay ad Equay Cosras. [] M. Rab ad M. Joasso. A spe peer-o-peer agor for dsrbued opzao sesor ewors, IEEE Coferece o Decso ad Coro. 7, 46: [] E. We, A. Ozdagar, ad A. Jadbabae. A dsrbued ewo eod for ewor uy azao, IEEE Coferece o Decso ad Coro (CDC)., 49: [] A. edć. Asycroous broadcas-based cove opzao over a ewor, IEEE rasacos o Auoac Coro,, 56(6): [3] A. edć, A. Ozdagar, ad P. Parro. Cosraed cosesus ad opzao u-age ewors, IEEE rasacos o Auoac Coro,, 55(4): [4] M. Zu ad S. Maríez. O dsrbued cove opzao uder equay ad equay cosras va pra-dua subgrade eods, arv prepr arv:.6,. [5] B. Joasso,. Kevczy, M. Joasso, ad K. H. Joasso. Subgrade eods ad cosesus agors for sovg cove opzao probes, I IEEE Coferece o Decso ad Coro, pages , Cacu, Meco, Deceber 8. [6] A. edć, A. Osevsy, A. Ozdagar, ad e a. Dsrbued subgrade eods ad quazao effecs, IEEE Coferece o Decso ad Coro. 8: [7] A. edć ad A. Ozdagar. Subgrade eods for saddepo probes, Joura of opzao eory ad appcaos, 9, 4(): 5-8. [8] A. edć ad A. Ozdagar. Approae pra souos ad rae aayss for dua subgrade eods, SIAM Joura o Opzao, 9, 9(4): [9] D. P. Berseas. Cove opzao eory, Aea Scefc, 9. [] D. P. Berseas, A. edć, ad A. Ozdagar. Cove aayss ad opzao, Aea Scefc, 3. [] M. Zu, ad S. Maríez. Dscree-e dyac average cosesus, Auoaca,, 46(): [] A. edć ad A. Ozdagar. Subgrade eods ewor resource aocao: Rae aayss, Iforao Sceces ad Syses, IEEE Coferece o Decso ad Coro, 8: