A Parametric Kernel Function Yielding the Best Known Iteration Bound of Interior-Point Methods for Semidefinite Optimization

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1 Aerca Joural of Appled Maheacs 6; 4(6): hp://wwwscecepublshggroupco/j/aja do: 648/jaja6468 ISSN: (Pr); ISSN: 33-6X (Ole) A Paraerc Kerel Fuco Yeldg he Bes Kow Ierao Boud of Ieror-Po Mehods for Sedefe Opzao Xyao Luo *, Gag Ma, Xaodog Hu, Yuqg Fu College of Fudaeal Sudes, Shagha Uversy of Egeerg Scece, Shagha, Cha Eal address: vcegod996@63co (Xyao Luo) * Correspodg auhor To ce hs arcle: Xyao Luo, Gag Ma, Xaodog Hu, Yuqg Fu A Paraerc Kerel Fuco Yeldg he Bes Kow Ierao Boud of Ieror-Po Mehods for Sedefe Opzao Aerca Joural of Appled Maheacs Vol 4, No 6, 6, pp do: 648/jaja6468 Receved: Noveber, 6; Acceped: Deceber 6, 6; Publshed: Deceber 6, 6 Absrac: I hs paper, a class of large-updae pral-dual eror-po ehods for sedefe opzao based o a paraerc kerel fuco are preseed The proposed kerel fuco s o oly used for deerg he search drecos bu also for easurg he dsace bewee he gve erae ad he ceer for he algorhs By eas of he Neserov ad Todd scalg schee, he currely bes kow erao bouds for large-updae ehods s esablshed Keywords: Ieror-Po Mehods, Sedefe Opzao, Large-Updae Mehods, Polyoal Coplexy Iroduco I hs paper, we focus o he pral proble of sedefe opzao (SDO) he sadard for { C X A X = b =, X } (P) :,,,,, ad s dual proble T (D) ax b y: ya S = C, S = Here, each A S, b R, C S Throughou he paper, we assue ha he arces A are learly depede Recely, (SDO) has bee oe of he os acve research areas aheacal prograg May eror-po ehods (IPMs) for lear opzao (LO) are successfully exeded o (SDO) due o her polyoal coplexy ad praccal effcecy For a overvew of hese resuls, we refer o [, ] ad he refereces [3, 4, 5, 6, 7, 8, 9,,,, 3] Kerel fucos play a pora role he desg ad aalyss of pral-dual (IPMs) for opzao ad copleeary probles They are o oly used for deerg he search drecos bu also for easurg he dsace bewee he gve erae ad he µ -ceer for he algorhs Currely, (IPM) based o kerel fuco s oe of he os effecve ehods for (LO) ad (SDO) ad s a very acve research areas aheacal prograg Parcularly, Ba e al [4] roduced a varey of o-selfregular kerel fucos, e, he so-called elgble kerel fucos, whch s defed by soe sple codos o he kerel fucos ad her dervaves They provded a sple ad ufed copuaoal schee for he coplexy aalyss of pral-dual kerel fuco based (IPMs) for (LO) Cosequely, a seres of elgble kerel fucos are cosdered for varous opzao probles ad copleeary probles, see, eg, [5, 6, 7, 8] For a survey, we refer o he oograph [9] o he subjec ad he refereces here I hs paper, we cosder he followg paraerc kerel fuco [8] q ϕ( ) =, q >, >, () logq whch s a geeralzao of he fe kerel fuco cosdered [5] for (LO), aely,

2 37 Xyao Luo e al: A Paraerc Kerel Fuco Yeldg he Bes Kow Ierao Boud of Ieror-Po Mehods for Sedefe Opzao ϕ( ) = e, > () The purpose of he paper s o exed he pral-dual large-updae (IPMs) for (LO) based o he paraerc fuco cosdered [5] o (SDO) by usg he NT-scalg schee [, ] Furherore, he coplexy resuls ach he currely bes resul of erao bouds for large-updae ehods s esablshed, aely, O loglog ε, by choosg q = O( ) The oule of he res of he paper s as follows I Seco, we recall soe basc coceps ad resuls o arx heory, he properes of he paraerc kerel (ad barrer) fuco Pral-dual kerel fuco-based (IPMs) for (SDO) are preseed Seco 3 I Seco 4, we gve he coplexy aalyss of he pral-dual (IPMs) for (SDO) Fally, soe cocludg rearks are ade Seco 5 Soe of he oaos used hroughou he paper are as follows R, R ad R deoe he se of vecors wh copoes, he se of oegave vecors ad he se of posve vecors, respecvely R deoes he se of real arces E deoes he dey arx F ad arces, respecvely deoe he Frobeus or ad he specral or for S, S ad S deoe he coe of syerc, syerc posve sedefe ad syerc posve defe arces, respecvely A B = Tr( A T B) deoes he arx er produc of wo arcesa adb, respecvely The Loɺɺwer paral order (or ) o posve sedefe (or posve defe) arces eas A B (ora B ) fa B s posve sedefe (or posve defe) Fally, f g( x) s a real valued fuco of a real oegave varable, he oao g( x) = O( x) eas ha g( x) cx for soe posve cosa c ad g( x) = Θ ( x) ha cx g( x) cx for wo posve cosas c adc Prelares Soe Resuls o Marces ad Marx Fucos I hs seco, soe well kow resuls o arces ad arx fucos fro lear algebra are cosdered Our preseao s aly based o he oograph [] ad he refereces [8, ] LeV S ad T V = Q dag( λ ( V), λ ( V),, λ ( V)) Q, (3) where Q s ay orhooral arx Q = Q ha T ( ) dagoalzes V The arx valued fuco ϕ( V): S S s defed by T ϕ( V) = Q dag ( ϕ( λ ( V)), ϕ( λ ( V)),, ϕ( λ ( V))) Q (4) Le ϕ ( ) be dffereable, e, he dervave ϕ ( ) exss The he arx valued fuco ϕ ( V) s well defed, aely ϕ ( V) = Q T dag( ϕ ( λ ( V)), ϕ ( λ ( V)),, ϕ ( λ ( V))) Q (5) Recall ha a arx A( ) s sad o be a arx of fucos f each ery of A( ) s a fuco of, e, A( ) = [ A ( )] Le A( ) ad B( ) be wo arces of j fucos The d d A ( ) = Aj ( ) A( ), d = d (6) d Tr( A( )) = Tr( A ( )), (7) d d ( A ( ) B ( )) = A ( ) B ( ) A ( ) B ( ) (8) d For ay fuco ϕ ( ), le us deoe by ϕ he dvded dfferece of ϕ ( ) as follows ϕ( ) ( ) (, ) ϕ ϕ =, R (9) If =, we sply wre ϕ(, ) = ϕ ( ) The followg heore provdes o easure he frs-order drecoal dervave of a geeral fuco ϕ ( A( )) ad boud s secod-order dervave wh respec o Theore (Lea 6 []) Suppose ha A( ) s a arx of fucos such ha he arx A( ) s posve defe wh egevalues λ( ) λ( ) λ( ) > If A( ) s wce dffereable wh respec o ( l, u ) ad ϕ ( ) s wce couously dffereable fuco a doa ha coas all he egevalues of A( ), he ad where d Tr( ϕ ( A ( ))) = Tr( ϕ ( A ( )) A ( )), () d d Tr( ϕ ( A ( ))) ω A ( ) Tr( ϕ ( A ( )) A ( )), d () ω = ax{ ϕ ( λ ( ), λ ( )) : ( l, u ), j, k =,,, } () j k s a uber depedg o A( ) ad ϕ ( ) wh

3 Aerca Joural of Appled Maheacs 6; 4(6): ϕ ( ) ϕ ( ) ϕ (, ) =,, [ l, u ] The Paraerc Kerel (Barrer) Fuco (3) The frs hree dervaves of ϕ ( ) defed by () wh respec o are gve by ( ) q ϕ =, (4) logq ϕ ( ) = q, (5) 4 log q 6logq 6 ϕ ( ) = q (6) 6 I wha follows, we recall soe useful resuls [5, 6] whou proofs Lea Le :[, ) [, ) be he verse fuco of ϕ( ) for The s ( s) s q Lea Le ϕb( ) = be he barrer er of ϕ ( ), logq ρ ( s) be he verse fuco of ϕ ( ) for (,] ad ρ :[, ) (,] be he verse fuco of he resrco of ϕ b( ) for (,], respecvely The ρ( s) ρ( s); (7) ρ( s) ; log( s) log( q ) ρ( s) log( s) log( q) (8) (9) The followg propery, e, he expoeal covexy, whch plays a pora role he aalyss of kerelfuco based (IPMs) [5,] Lea 3 (Lea [5]) Le > ad > The ϕ( ) ( ϕ( ) ϕ( )) Now, we defe he barrer fuco Ψ( V): S R accordg o he kerel fuco ϕ ( ) as follows Ψ ( X, S, µ ): = Ψ ( V): = Tr( ϕ( V)) () Fro (6), we have Ψ ( V) = ϕ( λ ( V)) () = Oe ca easly verfy ha he dervave of he barrer fuco exacly equal o ϕ ( V), whch s defed by (5) Furherore, we kow ha Ψ ( V) s srcly covex wh respec o V ad vashes a s global al po V = E, e, ϕ( E) = ϕ ( E) =, ad Ψ ( E) = We have he followg heore, by Lea 3 Theore (Proposo 3 (II) []) Le V, V S The Ψ V VV ( Ψ ( V ) Ψ ( V )) The followg heore provdes a esae for he effec of a µ -updae o he value of Ψ ( V), whch s a reforulao of Theore 3 [5] Theore 3 Le S ad β The he V ( V) Ψ( βv) ϕ Ψ Corollary Le θ < ad τ Ψ( V ) ϕ θ V V = θ If Ψ( V) τ, Proof: Wh β = ad Ψ( V) τ, he resul θ follows edaely fro Theore 3 Ths coplees he proof The or-based proxy easure δ ( V): S R s gve by δ ϕ ϕ λ () ( V): = ( V) = ( ( V)) = The lower boud o δ ( V) ers of Ψ ( V) ca be obaed fro he followg heore, whch s a reforulao of Theore 48 [5] Theore 4 Le V S The δ( V) ϕ ( ( Ψ( V))) Corollary Le V S The

4 39 Xyao Luo e al: A Paraerc Kerel Fuco Yeldg he Bes Kow Ierao Boud of Ieror-Po Mehods for Sedefe Opzao Proof: We have ( V )! δ( V) ( Ψ ( V)) Ψ( V) q δ( V) ψ ( ( Ψ( V))) Ψ( V) Ψ( V! Ψ( V) Ψ( ) = ( Ψ( V)) Ψ( V) Ths coplees he proof 3 Pral-Dual Kerel Fuco-Based (IPMs) for (SDO) Whou loss of geeraly, we assue ha boh he pral proble ad s dual proble of (SDO) sasfy he erorpo codo (IPC), e, here exss ( X, y, S ) such ha = =,, = = A X b,,, X y A S C, S (3) The Karush-Kuh-Tucker codos for (P) ad (D) are equvale o he followg syse A X = b, =,,, X, ya S = C, S, XS = (4) = The sadard approach s o replace he hrd equao (4), e, he so-called copleeary codo for (P) ad (D), by he paraeerzed equao XS = µ E wh µ > Ths yelds A X = b, =,,, X, ya S = C, S, XS = µ E (5) = Uder he assupo ha (P) ad (D) sasfy he (IPC), he syse (5) has a uque soluo, deoed by ( X( µ ), y( µ ), S( µ )) Le X ( µ ) be he µ -ceer of (P) ad ( y( µ ), S( µ )) be he µ -ceer of (D) The se of µ -ceers (wh µ rug hrough posve real ubers) gves a hooopy pah, whch s called he ceral pah of (P) ad (D) If µ, he he l of he ceral pah exss, ad sce he l pos sasfy he copleeary codo, e, XS =, aurally yelds a opal soluo for (P) ad (D), see, eg, [] I order o provde he scaled Newo syse has a uque syerc soluo, Zhag [] roduced he followg syerzao operaor T HP( M): = ( PMP ( PMP ) ), M R (6) Oe ca easly verfy ha H ( ), P M = µ E M = µ E (7) for ay osgular arx P, ay arx M wh real specru ad ay µ R For ay gve osgular arx P, he syse (5) s equvale o A X = b, =,,, X, = y A S = C, S, (8) H ( XS) = µ E P By usg Newo ehod o he syse (8), hs yelds A X =, =,,, ya S =, (9) = H ( X S XS) = µ E H ( XS) P The search dreco obaed hrough he syse (9) s called he Moero-Zhag ufed dreco Dffere choces of he arx P resul dffere search drecos (see, eg, [, ]) I hs paper, we cosder he so-called NTsyerzao schee [, ], whch yelds he NT search dreco Le P: = X ( X SX ) X = S ( S XS ) S, (3) ad D: = P The arxd ca be used o rescale X ad S o he sae arx V, defed by Here ad V : = D XD = DSD (3) µ µ Fro (3), afer soe eleeary reducos, we have A D =, =,,, ya DS =, (3) = X = DX DS V V A : = DAD, =,,, µ DX : = D XD, DS : = D SD (33) µ µ Oe ca easly verfy ha V V = Ψ ( V), (34) where Ψ ( ) deoes he grade of Ψ ( ) s gve by c V c P c V

5 Aerca Joural of Appled Maheacs 6; 4(6): ( ) Ψ ( V): = Tr( ϕ ( V)) = ϕ ( λ ( V) c c c = λ = = ( V) log( λ ( V )) Hece, he syse (9) s equvale o A D =, =,,, = X (35) ya DS =, (36) D D = Ψ X S c ( V) Ths eas ha he logarhc barrer fuco esseally deeres he classcal NT search dreco I hs paper, we replace he rgh-had sde Ψ c( V ) he hrd equao (36) by Ψ ( V), e, ϕ ( V) Ths yelds A D =, =,,,, = X ya DS =, (37) D D = Ψ( V) X S The scaled ew search dreco ( DX, y, DS) s copued by solvg he syse (37) so ha X ad S are obaed hrough (33) If ( X, y, S) ( X( µ ), y( µ ), S( µ )), he ( X, y, S) s ozero By akg a defaul sep sze alog he search drecos, we ge he ew erae ( X, y, S ) accordg o X : = X α X, y : = y α y, S : = S α S (38) Oe ca easly verfy ha XS = µ E V = E ϕ ( V) = Ψ ( V) = (39) Hece, he value of Ψ ( V) ca be cosdered as a easure for he dsace bewee he gve erae ( X, y, S ) ad he µ -ceer ( X( µ ), y( µ ), S( µ )) The geerc for of pral-dual kerel fuco-based (IPMs) for (SDO) s show Algorh Algorh Pral-Dual Ieror-Po Algorh for (SDO) Ipu: a hreshold paraeer τ ; a accuracy paraeer ε > ; a fxed barrer updae paraeer θ, < θ < ; a srcly feasble par Ψ( X, S, µ ) τ ( X, y, S ) ad beg X : = X ; y: = y ; S : = S ; µ : = µ ; whle µ ε beg µ : = ( θ ) µ ; µ = such ha whle Ψ ( X, S, µ ) > τ do beg copuer he search drecos ( X, y, S); choose a suable sep sze α ; updae ( X, y, S): = ( X, y, S) α( X, y, S) ed ed ed 4 Coplexy Aalyss of Large-Updae Mehods I each er erao he search dreco ( X, y, S) s obaed by solvg he syse (37) ad va (33) Afer a sep wh sze α he ew erae s gve by X = X α X, y = y α y, S = S α S (4) The, we have X = X α X = X α µ DDXD = µ DV ( αdx) D (4) ad S S S S D D D D V D D = α = α µ = S µ ( α S) (4) I follows fro (3) ha V Oe ca easly verfy ha he arx X S X = (43) µ D X S D V s uarly slar o ad hus o X S X ( V αd ) ( V αd )( V αd ) (44) Ths ples ha he egevalues of V are precsely he sae as hose of he arx X S X V : = ( V αd ) ( V αd )( V αd ) (45) Fro he defo of Ψ ( V), oe obas Ψ ( V ) = Ψ ( V ) Hece, by Theore, we have Ψ ( V ) = Ψ( V ) ( Ψ ( V αdx ) Ψ ( V αds) ) (46) Now, we cosder he decrease Ψ ( V) as a fuco of α ad defe Le defe α f ( ): = Ψ( V ) Ψ ( V) = Ψ( V ) Ψ ( V) (47)

6 3 Xyao Luo e al: A Paraerc Kerel Fuco Yeldg he Bes Kow Ierao Boud of Ieror-Po Mehods for Sedefe Opzao ad f( α): = ( Ψ ( V αdx ) Ψ ( V αds) ) Ψ ( V) (48) I follows ha f ( α) f( α) ad f () = f() = By Theore, oe has where f ( α) = Tr ( ψ ( V αdx ) DX ψ ( V αds) DS ) (49) d f ( α) = Tr ϕ( V αd ) ( ) X ϕ V αds dα ( ω DX ω DS ), ( ) { j V DX k V DX j k } (5) ω = ax ϕ ( λ ( α ), λ ( α )) :, =,,,, (5) ad { j V DS k V DS j k } ω = ax φ ( λ ( α ), λ ( α )) :, =,,, (5) Hece, usg he hrd equao of he syse (37), oe has f () = Tr( ϕ ( V)( DX DS )) = Tr( ϕ ( V) ) = δ( V) (53) I order o faclae dscusso, we deoe δ : = δ( V), ad we have he followg resul [8] Theore 3 Oe has f ( ) ( ( ) ) α δ ϕ λ V αδ The defaul sep sze for he algorh should be chose such ha X ad S are feasble ad Ψ( V ) Ψ ( V) decreases suffcely For he deals we leave for he eresed readers (see, eg, [8,5] Followg he sraegy cosdered [8], we brefly recall how o choose he defaul sep sze Suppose ha he sep sze α sasfes ϕ ( λ ( V) αδ) ϕ ( λ ( V)) δ (54) The f ( α) The larges possble value of he sep sze of α sasfyg (53) s gve by α : = ( ρ( δ) ρ( δ)) (55) δ Furherore, we ca coclude ha α ϕ ( ρ( δ)) ϕ ( ρ( δ)) Afer soe eleeary reducos, we have α log( 4 δ) ( 4 δ)( log q) (56) (57) I he sequel, le α : = log( 4 δ) ( 4 δ)( log q) (58) be he defaul sep sze As a cosequece of Lea A ad he fac ha f ( α) f ( α), whch s a wce dffereable covex fuco wh f () =, ad f () = δ <, he followg lea s obaed Lea 4 Le he sep sze α be such ha α ɶ α The f α αδ ( ) The followg heore shows ha he defaul sep sze yelds suffce decrease of he barrer fuco durg each er erao Theore 4 Le Ψ τ 3 ad αɶ be he defaul sep sze as gve by (57) The f ( ɶ α) Ψ( V) log( Ψ ) ( log q) logq Proof: Fro Lea 4 wh (56) ad Corollary, we have f ( ɶ α) ɶ αδ δ log( 4 δ) ( 4 δ)( log q) δ log( 4 δ) δ( log q) δ log( 4 δ) ( log q) Ψ( V) log( Ψ ) ( log q) δ logq Ths coplees he proof A he sar of a ouer erao ad jus before updag he paraeer µ, oe has Ψ( V) τ I follows ha he value of Ψ ( V) exceeds fro he hreshold τ afer updag of µ Therefore, oe eed o cou how ay er eraos are requred o reur o he suao where Ψ( V) τ Le deoe he value of Ψ ( V) afer he µ -updae be Ψ, he subseque values he sae ouer

7 Aerca Joural of Appled Maheacs 6; 4(6): erao are deoed as Ψk, k =,, K, where K deoes he oal uber of er eraos he ouer erao Sce ϕ( ) for, we have τ τ ρ θ θ Ψ Ψ Ψ τ τ τ τ = = O( ) ( θ) θ (59) Accordg o he decrease of f ( αɶ ) Lea 4, we have where γ Ψk Ψk β( Ψ k), k =,,, K, (6) β = log( Ψ ) ( log q) logq, ad γ = (6) The followg lea provdes a esae for he uber of er eraos bewee wo successve barrer paraeer updaes Lea 4 Oe has log( Ψ ) K ( log q) ( Ψ ) logq Proof: Fro Lea A ad (59), he resul of he lea follows Ths coplees he proof I s well kow ha a upper boud of he uber of ouer eraos s bouded above by [3] log θ ε (6) By ulplyg he uber of ouer eraos ad he uber of er eraos, we ge a upper boud for he oal uber of eraos, aely, τ τ log( ) ( log q) θ τ τ log (63) θ logq θ ε Noe ha Ψ O( ) By choosg q = O( ), he bes oal erao boud s obaed Theore 43 For large-updae ehods, we se θ = Θ (), ad τ = O( ) The he erao boud becoes O loglog, ε whch aches he currely bes well-kow coplexy for large-updae ehods 5 Cocluso I hs paper, we have vesgaed a class of large-updae pral-dual (IPMs) for (LO) based o a paraerc kerel fuco preseed [6] ca be exeded o he coex of (SDO) Furherore, he bes resul of erao bouds for large-updae ehods s derved I our fuure sudy, he geeralzaos of he pral-dual (IPMs) for (LO) o syerc coe opzao (SCO) ad syerc coe copleeary probles (SCCP) are eresg Ackowledgees The auhors would lke o hak he edor ad he aoyous referees for her useful coes ad suggesos, whch helped o prove he preseao of hs paper Ths work was suppored by Uversy Sudes' Iovave Trag Progra of Shagha (No CS6) Appedx(Soe Techcal Leas) Lea A (Lea []) Le h( ) be a wce dffereable covex fuco wh h () =, h () < ad le h( ) aa s (global) u a * creasg for [, ], he h () * ( ), h * > If h ( ) s Lea A (Lea 4 []) Suppose,,, K s a sequece of posve ubers such ha, k =,,, K, k k β γ k where β > ad < γ The Refereces γ K βγ [] Ajos M F, Lasserre, J B: Hadbook o Sedefe, Coc ad Polyoal Opzao: Theory, Algorhs, Sofware ad Applcaos Ieraoal Seres Operaoal Research ad Maagee Scece Volue 66, Sprger, New York, USA () [] De Klerk, E: Aspecs of Sedefe Prograg: Ieror Po Algorhs ad Seleced Applcaos Kluwer Acadec Publshers, Dordrech, The Neherlads ()

8 33 Xyao Luo e al: A Paraerc Kerel Fuco Yeldg he Bes Kow Ierao Boud of Ieror-Po Mehods for Sedefe Opzao [3] Ca X Z, Wag G Q, Zhag Z H: Coplexy aalyss ad uercal pleeao of pral-dual eror-po ehods for covex quadrac opzao based o a fe barrer Nuer Algorhs 6(), (3) [4] Cho G M: Large-updae pral-dual eror-po algorh for sedefe opzao Pac J Op (), 9-36 (5) [5] El Gha M, Ba Y Q, Roos C: Kerel-fuco based Algorhs for sedefe opzao RAIRO Oper Res 43(), (9) [6] Lee Y H, Cho Y Y, J J H, Cho G M: Ieror-po algorhs for LO ad SDO based o a ew class of kerel fucos J Nolear Covex Aal 3(3), () [7] Lu, H W, Lu, C H, Yag X M: New coplexy aalyss of a Mehrora-ype predcor-correcor algorh for sedefe prograg Op Mehods Sofw 8(6), (3) [8] Wag G Q, Ba Y Q: A ew pral-dual pah-followg eror-po algorh for sedefe opzao J Mah Aal Appl 353(), (9) [9] Wag G Q, Ba Y Q: Pral-dual eror-po algorhs for covex quadrac sedefe opzao Nolear Aal 7(7-8), (9) [] Wag G Q, Ba Y Q, Gao X Y, Wag D Z: Iproved coplexy aalyss of full Neserov-Todd sep eror-po ehods for sedefe opzao J Op Theory Appl 65(), 4-6 (5) [] Wag G Q, Ba Y Q, Roos C: Pral-dual eror-po algorhs for sedefe opzao based o a sple kerel fuco J Mah Model Algorhs 4(4), (5) [] Wag G Q, Zhu D T: A ufed kerel fuco approach o pral-dual eror-po algorhs for covex quadrac SDO Nuer Algorhs 57(4), () [3] Yag X M, Lu H W, Zhag Y K: A secod-order Mehrora-ype predcor-correcor algorh wh a ew wde eghbourhood for sedefe prograg I J Copu Mah 9(5), 8-96 (4) [4] Zhag M W: A large-updae eror-po algorh for covex quadrac sedefe opzao based o a ew kerel fuco Aca Mah S (Egl Ser) 8(), () [5] Ba, Y Q, El Gha, M, Roos, C: A coparave sudy of kerel fucos for pral-dual eror-po algorhs lear opzao SIAM J Op 5(), -8 (4) [6] Achache M: A ew paraeerzed kerel fuco for LO yeldg he bes kow erao boud for a large-updae eror po algorh Afrka Ma 7(3), 59-6 (5) [7] Ba, Y Q, El Gha, M, Roos, C: A ew effce large-dual eror-po ehod based o a fe barrer SIAM J Op 3(3), (3) [8] Ca X Z, Wag G Q, El Gha M, Yue Y J: Coplexy aalyss of pral-dual eror-po ehods for lear opzao based o a paraerc kerel fuco wh a rgooerc barrer er Absr Appl Aal 4, 758 (4) [9] Wag G Q, Ba Y Q: Ieror-Po Mehods for Syerc Coe Copleeary Probles: Theorecal Aalyss ad Algorh Ipleeao Harb Isue of Techology Press, Harb (4) [] Hor, R A, Johso, C R: Topcs Marx Aalyss Cabrdge Uversy Press, UK (99) [] Peg, J, Roos, C, Terlaky, T: Self-regular fucos ad ew search drecos for lear ad sedefe opzao Mah Progra 93(), 9-7 () [] Zhag, Y: O exedg soe pral-dual eror-po algorhs fro lear prograg o sedefe prograg SIAM J Op 8(), (998) [3] Roos, C, Terlaky, T, Val, J-Ph: Theory ad Algorhs for Lear Opzao Sprger, Chcheser, UK (s Edo, Theory ad Algorhs for Lear Opzao A Ieror- Po Approach Joh Wley & Sos, 997) (5)