FEEDBACK STABILIZATION METHODS FOR THE SOLUTION OF NONLINEAR PROGRAMMING PROBLEMS

Transcription

1 FEEDACK STAILIZATION METODS FO TE SOLUTION OF NONLINEA POGAMMING POLEMS Iasso Karafylls Departmet of Evrometal Eeer, Techcal Uversty of Crete,700, Chaa, Greece emal: Abstract I ths wor we show that ve a olear proramm problem, t s possble to costruct a famly of dyamcal systems defed o the feasble set of the ve problem, so that: a the equlbrum pots are the uow crtcal pots of the problem, b each dyamcal system admts the obectve fucto of the problem as a Lyapuov fucto, ad c eplct formulae are avalable wthout volv the uow crtcal pots of the problem. The costructo of the famly of dyamcal systems s based o the Cotrol Lyapuov Fucto methodoloy, whch s used mathematcal cotrol theory for the costructo of stablz feedbac. The owlede of a dyamcal system wth the prevously metoed propertes allows the costructo of alorthms whch uaratee lobal coverece to the set of the crtcal pots. Keywords: Nolear proramm, feedbac stablzato, Lyapuov fuctos, olear systems.. Itroducto Dfferetal equatos have bee used the past for the soluto of Nolear Proramm NLP problems. The reader may cosult [,6,7,,,5,6,7,8] for varous results o the topc. Some methods are teror-pot methods the sese that are defed oly o the feasble set whle other methods are eteror-pot methods the sese that are defed at least a ehborhood of the feasble set. As remared [5], each system of dfferetal equatos that solves a NLP problem whe combed wth a umercal scheme for solv Ordary Dfferetal Equatos ODEs provdes a umercal scheme for solv the NLP problem. I ths wor, we are terested the applcato of feedbac stablzato methods for solv NLP problems. The feedbac stablzato methods ca be appled two ways: frst, for the costructo of the dyamcal system, whch solves the NLP problem ad for the selecto of the step sze of the ue-kutta scheme that s used for the soluto of the result system of ODEs see [,6]. More specfcally, cosder the Nolear Proramm problem: { : S } m. where ad the closed set S s defed by S : : h... h 0, ma m,..., 0. where m < ad all fuctos :, h :,..., m, :,..., are twce cotuously dfferetable. Ispred by the methods employed the boo [5], we would le to costruct a welldefed dyamcal system F o the closed set S, where F : S s a cotuous vector feld wth the follow propertes:

2 Property : For every S, F belos to the cotet coe to S at. Ths property s requred because local estece of solutos of the dyamcal system F ca be uarateed by Naumo s theorem ve o pae 7 of the boo []. Property : F : S s a locally Lpschtz vector feld. Ths property s requred for uqueess of the solutos of the dyamcal system F. Moreover, ths property s requred because we would le to be able to apply st order ue-kutta schemes for the smulato of the solutos of the dyamcal system F. her reularty s also desrable because hh order ue-kutta schemes ca be used for the smulato of the solutos of the dyamcal system F. Property : The equlbrum pots of the dyamcal system F are eactly the pots S for whch there est λ,..., m ad 0,..., such that the ecessary Karush-Kuh-Tucer codtos hold: m 0 λ h 0. * Property : The fucto V s a Lyapuov fucto for the dyamcal system F, where * s oe of the lobal solutos of the NLP descrbed by.,.,.e., { } S m : S. I other words, we would le the equalty F < 0 to hold for all S for whch there are o λ,..., m ad 0,..., such that codtos hold.. Ths property s mportat because t uaratees useful stablty propertes. Furthermore, the fact that the Lyapuov fucto of the dyamcal system F has the * specal form V s mportat for umercal purposes see [,6]: the tme dervatve of the Lyapuov fucto alo the solutos of the system ad the dfferece of the values of the Lyapuov fucto betwee two pots ca be computed wthout owlede of the soluto * S of the NLP problem. Property 5: The vector feld F : S must be eplctly ow. Formulae for the vector feld F : S must be provded: the formulae must ot volve the soluto * S of the NLP descrbed by.,.. Property 6: The vector feld F : S must have free parameters whch ca be selected a approprate way so that the coverece propertes of the correspod umercal schemes to the lobal attractor of the dyamcal system are optmal. I other words, we wat to costruct a famly of vector felds F : S wth all the above propertes. It must be oted that the propertes -6 are rarely satsfed by other dfferetal equato methods for solv NLPs. For eample, [] ad [6], the costructed Lyapuov fucto s V ad ths does ot meet our requremets. Moreover, for [6] the pot s ot a equlbrum pot for the costructed tme-vary system F t,. Atp [] costructs a autoomous system F for whch s a equlbrum pot ad 0 F C ; be locally Lpschtz does ot deped o the locato of the uow pot. owever, the computato of F s volved t requres the soluto of a NLP sce t volves a proecto o the feasble set. The NLP wthout equalty costrats uder addtoal covety hypotheses has bee studed [6]. owever, aa the costructed Lyapuov fucto s of the form V P ad ths does ot meet our requremets. O the other had, the papers [5,7] propose systems of dfferetal equatos that satsfy propertes -6 for systems wthout equalty costrats. Local results are provded the paper [8] ad dfferetal equatos based o barrer methods were cosdered [7]. It s clear that the owlede of the Lyapuov fucto V ca allow us to costruct the vector *

3 feld F : S by the Cotrol Lyapuov Fucto methodoloy of feedbac des see [,9,7,] for the cotrol system u. owever, there are certa obstructos for the drect applcato of the classcal Cotrol Lyapuov Fucto methodoloy: the system s ot defed o but o the closed set S, for every S, F must belo to the cotet coe to S at, ad the posto of the equlbrum pots,.e., the set of pots S for whch there est λ,..., m ad 0,..., such that codtos. hold s uow ths s what we are loo for. The cotrbuto of the paper s twofold: The ma result of the preset wor Theorem. shows that all the prevously metoed obstructos ca be overcome uder approprate assumptos. ased o the deas descrbed [,6], Secto of the preset wor, we preset a alorthm for the soluto of the NLP descrbed by. ad. whch s based o the applcato of the eplct Euler scheme for the umercal soluto of the result system of ODEs wth approprate step selecto Theorem.. The alorthm wll covere for every tal codto lobal coverece. A modfed ad smpler verso of the alorthm ca wor uder slhtly more demad assumptos emar.5. It should be otced that the coverece rates of the proposed alorthms deped o the selecto of certa matrces whch are the free parameters descrbed Property 5 above. owever, sce the proposed alorthms are lobal, t ca be used combato wth ay other local alorthm that uaratees fast coverece based o the follow tutve dea: apply the ewly proposed alorthms whe you are away from a soluto ad apply a fast local alorthm whe you are close to a soluto. It should be emphaszed that o clam s made about the effectveess of the proposed alorthms. The topc of the umercal soluto of NLPs s a mature topc ad t s clear that other alorthms have much better characterstcs tha the alorthms proposed ths paper. owever, the theory used for the costructo of the alorthm s dfferet from other est alorthms. The alorthms cotaed ths wor are derved by us cocepts of dyamcal systems theory ad mathematcal cotrol theory. The structure of the paper s as follows: Secto cotas the statemet ad proof of Theorem., whch provdes the soluto to the problem of the costructo of a vector feld wth propertes -6. Secto provdes umercal alorthms for the eplotato of the costructed vector feld. Secto of the paper provdes some eamples, whch show the performace of the alorthms. Fally, Secto 5 of the paper cotas the coclud remars. The apped provdes the proofs of certa aulary results. Notatos Throuhout ths paper we adopt the follow otatos: 0 Let A be a set. y C A ; Ω, we deote the class of cotuous fuctos o A, whch tae values Ω. y C A ; Ω, where s a teer, we deote the class of dfferetable fuctos o A wth cotuous dervatves up to order, whch tae values Ω. y C A; Ω, we deote the class of dfferetable fuctos o A hav cotuous dervatves of all orders smooth fuctos, whch tae values Ω,.e., C A; Ω C A; Ω. For a vector A we deote by ts usual Eucldea orm ad by ts traspose. For a real matr m m we deote by A ts duced orm,.e., A : ma{ A :, } ad by m A ts traspose. I deotes the detty matr. For every,..., we defe ma 0,,...,ma0,. Notce that the follow property holds for every postve defte ad daoal matr : 0 0. {,..., : 0,..., 0 } : y.. Let, y. We say that y f ad oly f For every scalar cotuously dfferetable fucto V :, V deotes the radet of V at, V V.e., V,..., ad V deotes the essa matr of V at.

4 . Trasform a NLP problem to a feedbac stablzato problem Cosder the NLP problem descrbed by. ad. uder the follow assumptos: The feasble set S defed by. s o-empty ad the level sets of : are compact sets,.e., for every 0 S the level set s compact. { S } : 0 For every S the row vectors h,..., m ad for all,..., for whch 0 actve costrats are learly depedet. Assumpto s a stadard assumpto whch uaratees that the NLP problem descrbed by. ad. s well-posed ad admts at least oe lobal soluto see []. Assumpto s a eteso of the ma assumpto [0]. Assumpto coucto wth the ma result [0] uaratees that for every soluto * S of the NLP problem descrbed by. ad., there est λ,..., m ad 0,..., such that codtos. hold. We defe: h h m h m, A, h m h m,, for all. Assumpto allows us to defe the symmetrc matr: A A A I A, for all a ehborhood of S. The follow facts are drect cosequeces of defto.: Fact :, A 0 ad A 0. Fact : ξ ξ ξ, for all ξ Fact : For every ξ there ests m λ such that ξ ξ A λ. Net, we defe the set of crtcal pots for the NLP problem defed by. ad.. m Defto.: Let Φ S be the set of all pots S for whch there est λ ad such that codtos. hold. I other words, Φ S s the set of crtcal pots or Karush-Kuh-Tucer pots for the problem defed by. ad.. Clearly, assumptos ad uaratee that the set Φ S s o-empty. The follow lemma provdes a useful cosequece of assumpto. Its proof s provded at the Apped. Lemma.: If assumpto holds the the matr s postve defte for all S. : da. We are ow ready to state the ma result of ths secto.

5 Theorem.: Suppose that assumptos ad hold for the NLP problem descrbed by. ad.. Let be the symmetrc postve defte matr defed by.. Let be a arbtrary C, symmetrc ad postve defte matr, be a arbtrary C, symmetrc ad postve semdefte matr, a, b, c,..., be arbtrary C o-eatve fuctos wth b c > 0 for all,...,, S ad at least oe of the matrces a : a,..., a. Defe the follow locally Lpschtz vector feld:, da a be postve defte, where where P : F P da P v : P : da [ P P ] [ ] P P da da a v v p p b c ma0, v,..., b c ma0, v..5 ad p,..., are teers. The the follow propertes hold: a A F 0 for all S, b F < 0, for all S \ Φ, c F 0 Φ d F da w v, for all S, where w : [ P P ] [ da ] da da a v v. Cosder the dyamcal system F.6 o the closed set S. The the follow propertes hold: For every 0 S there ests a uque soluto t of the tal value problem.6 wth 0 0 whch s defed for all t 0 ad satsfes t S for all t 0. Every pot Φ s a equlbrum pot for.6. Every strct local soluto * S of the NLP problem descrbed by. ad. s locally asymptotcally stable for system.6. If we further deote by ω 0 the set of accumulato pots of the set { t: t 0}, where S l such that 0 Φ { S : l} ω 0 s a compact, postvely varat set for whch there ests 5 0 0, the t holds that ω. emar.: Clearly, the matrces, ad the fuctos a, b, c,...,, ca be selected a approprate way so that the coverece propertes of the correspod umercal schemes to the lobal attractor of the dyamcal system are optmal. The stablty propertes of system.6 are aaloous to the stablty propertes of radet systems see []. emar.5: It should be oted that all propertes -6 metoed the Itroducto are satsfed for the dyamcal system.6. Ideed, -- Property s a drect cosequece of a ad d. More specfcally, sce p p b c ma0, v,..., b c ma0, v : da for certa o-eatve fuctos

6 d b, c,..., ad sce F da w v t follows dt that the follow mplcato holds: f 0 for some {,..., } the d dt that for every p b c ma0, v ma0, v 0. The prevous mplcato ad property a uaratee S, F belos to the cotet coe to S at. -- Property s a drect cosequece of deftos.,.,.,.5 ad the fact that all fuctos :, h :,..., m, :,..., are twce cotuously dfferetable. It should be otced that f at least oe of the fuctos b,,..., taes postve values the the vector feld F defed by. s smply locally Lpschtz ad ot C. Whe b 0, for,..., the the vector feld F defed by. s C. her reularty s possble by assum hher reularty for all fuctos ad matrces volved.,.,.,.5, suffcetly lare values for the teers p,..., ad b 0, for,...,. --Property s a drect cosequece of c. Property s a drect cosequece of b. Ideed, otce that the fucto * V, where * S s oe of the lobal solutos of the NLP descrbed by.,. satsfes V F F. --Fally, propertes 5 ad 6 are evdet. emar.6: The sprato for Theorem. s the trasformato of the NLP problem to a feedbac stablzato * problem. Frst, we otce that the Cotrol Lyapuov Fucto see [,9,7,] s selected to be V, where * S s oe of the lobal solutos of the NLP problem descrbed by.,.. The oly problem s that we must defe a approprate way the cotrol system so that S s a postvely varat set for all possble puts. I other words, we must have: d d h A 0 ad da v u dt dt d for all possble puts v, u. Notce that the property da v u for arbtrary dt d v, u uaratees the mplcato: f 0 for some {,..., } the u 0. The dt d property h A 0 mples that w, for arbtrary w. Comb, we et dt w da v u. y redef the put varables w p q ad v p z, we et da z u q p. Cosequetly, the requred cotrol system s I q da z u u wth puts q, z,. The computato of the feedbac law for the above cotrol system wth Cotrol * Lyapuov Fucto V, ves the dyamcal system.6, where F s defed by.,.5. More specfcally, we et: V q da z u The Cotrol Lyapuov Fucto approach requres that each put must be selected so that each term appear the above equato taes eatve values. The feedbac laws da da a q, z, u 6

7 where s a arbtrary C, symmetrc ad postve defte matr, symmetrc ad postve semdefte matr, a, b, c,..., wth b c > 0 for all,...,, S ad at least oe of the matrces be postve defte, where a : a,..., a ad feld F defed by.,.5. 7 are arbtrary s a arbtrary C, C o-eatve fuctos, da a s defed by.5, ve us the vector emar.7: If there are o equalty costrats.e., h 0 the the proof of Theorem. shows that eactly the same results wth that of Theorem. hold wth I. Proof of Theorem.: We frst otce that statemets a ad d are drect cosequeces of deftos.,.,.5 ad Fact. We et prove statemets b ad c. We frst otce that deftos.5 ad the fact that 0 mply that the follow equalty holds for all S : F ξ a v ξ da v ma0, v c ma0, v b ξ [ ] where [ ] da v p ξ. It s clear that equato.7 shows that F 0 for all S. We et vestate the ature of pots S for whch F 0. Equato.7 ad the facts that s a postve defte matr, s a postve semdefte matr, a, b, c,..., are o-eatve fuctos wth b c > 0 for all,...,, S ad at least oe of the matrces, da a s postve defte, where a : a,..., a ad defed by.5, show that F 0 s equvalet to the follow equatos: We defe: 0.7 s v.8 v 0,,...,.9 [ I ] 0 Defto. coucto wth.8 ad.9 mples that Us.0 ad the detty, we obta: Deftos.5,. coucto wth. mply that:.0 v. 0 ad 0. [ I ] 0.. 0

8 8 Equato. coucto wth Fact ad. mples that the codtos. hold. Therefore, Φ. Thus, we have proved the mplcato: Φ F 0. Cosequetly, we have proved statemet b ad oe of the mplcatos of statemet c amely, the mplcato Φ F 0. We wll prove ow the mplcato 0 Φ F. Suppose that Φ. The there est λ m,..., ad 0,..., such that codtos. hold, or vector form: 0 0 A λ.5. It follows from.5, Fact ad deftos.5 that λ A v Us defto. ad the above equalty we obta da v. owever, the facts that 0, 0 ad 0 mply that 0 da. Cosequetly, t follows that v ad that.8 holds. Us.5, defto. ad the facts that v, 0 da, 0 v, we obta: [ ] [ ] P P P P F Us deftos.,.5, Fact,.5, the above equalty ad the fact that 0 da, we et: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 λ da I da I I I I I I A I I F We et tur to the proof of propertes ad. Local estece of the soluto of the tal value problem.6 wth 0 0 s a drect cosequece of propertes a, d ad the Naumo theorem pae 7 []. Global estece of the soluto of the tal value problem.6 wth 0 0 follows from Theorem.. pae 7 [], assumpto ad the fact that t s o-creas a drect cosequece of property b. I fact, assumpto coucto wth the fact that t s o-creas shows that { } 0 : t t s bouded. As the case of dyamcal systems o, t follows that 0 ω s a compact, postvely varat set for system.6 see []. The fact that t s o-creas mples that { } 0 : f lm t t l t t, whch shows that { } l S : 0 ω. We et show that Φ 0 ω. The equalty l ds s t 0 0 γ, for all 0 t.6

9 where γ : F l : f t: t 0. Notce that the mapp s γ s s uformly cotuous sce { t F t: t 0} s bouded a cosequece of the fact that { t: t 0} s bouded ad sce the mapp γ s locally Lpschtz. Us.6 ad apply arbalat s lemma see [8], we coclude that lm γ s 0. The valdty of the mplcato s a drect cosequece of.7 ad the defto { } F γ 0 Φ mples that ω 0 Φ. Fally, the fact that every strct local soluto * S of the NLP problem descrbed by. ad. follows * from property b ad the cosderato of the Lyapuov fucto V. The proof s complete. s. Numercal solutos of NLP problems As remared the Itroducto ad [5], each system of dfferetal equatos that solves a NLP problem whe combed wth a umercal scheme for solv Ordary Dfferetal Equatos ODEs provdes a umercal scheme for solv the NLP problem. owever, whe we try to apply a umercal scheme for the soluto of.6 the we face the problem that the dyamcal system.6 s ot defed o but o the closed set S. I the lterature, proecto schemes have bee proposed see [,]. The proecto schemes preserve the order of the appled umercal scheme see [,] eve f the proecto o the closed set S s ot eact. owever, the applcato of a ue-kutta umercal scheme for.6 ad ts appromate proecto o the closed set S meas that the soluto of a NLP problem s requred at each tme step. The correspod NLP problem may be as dffcult as the tal oe, whch meas that ths approach s ot easly applcable wth the ecepto of cases where the proecto s easy, see []. The ey dea preseted ths wor s that the selecto of the appled tme step ca be used for solv the above problems. Frst we focus o the case wthout equalty costrats. The follow theorem s the ma result of ths secto, whch uaratees lobal coverece of the above alorthm. Theorem.: Suppose that assumptos, hold for the NLP problem descrbed by. ad. wth h 0. Let be a arbtrary C, symmetrc ad postve defte matr, be a arbtrary C, symmetrc ad postve semdefte matr, a, b, c,..., be arbtrary C oeatve fuctos wth b c > 0 for all,...,, S ad at least oe of the matrces da a be postve defte, where a : a,..., a. Let defed by.,.5 wth I. Furthermore, we assume that: Cosder the follow alorthm:, F be the vector feld Alorthm: Gve costats r > 0, ε 0, r, λ 0, ad a tal pot 0 S, we follow the steps for 0,,... Step : Calculate F us.,.5. If F 0 the. If F > 0 the set s r p 0. Moreover, let I {,..., } be the set of all dces {,..., } ma s F >. p p ε wth 0 s ε Step p: Calculate s F. p p Solve m y : ma y 0, for the case I or set y, for the case I I. p If y S ad y λ s F the set y, ad o to Step. 0 ad If p y S or y > λ s F the set p p s s, p p ad o to Step p. The every accumulato pot of the sequece produced by the above alorthm satsfes 9 Φ.

10 emar.: It s clear that the alorthm preseted Theorem. eplots the tme step used for the estmate p p provded by the eplct Euler scheme s F. The costat r > 0 s the mamum allowable tme step. I most teratos, the alorthm wll ot requre the soluto of a NLP problem, provded that ε > 0 s suffcetly small. The fact that most cases the Euler scheme s suffcet s eplaed by statemet d of Theorem.: for all,..., t holds that p b c ma0, v ma0, v F ω. where ω : ω,..., ω s ve by ω : [ ] P P [ ] I da da da a v v. Us the estmate where [ 0, r] that. sf s F s K { } s, K : ma F F F : [0, r], t follows that sf ε s ω s b c ma 0, v provded p ma 0, v s K 0 ε The above equalty s satsfed for all s [ 0, ε ] may cases provded that ε > 0 s suffcetly small. Ths p eplas the addtoal fact that that the NLP problem m y : ma y tal NLP problem: the de set } 0 I I {,..., s epected to be a set wth small cardal umber. p Fally, as remared [], the soluto of the NLP problem m y : ma y eact: t suffces to assume that to fd ay pot C s a costat ad m y p : ma I y y 0. y wth ma y 0 ad I I y s much smpler tha the 0 p p C y eed ot be, where s ay of the lobal solutos of the NLP problem Proof: Defe the sets: Φ : { z Φ, z }, S : { z S, z } 0. Notce that the set Φ s o-empty ad compact by vrtue of property b of Theorem. t follows that the set Φ cocdes wth the set { S : F 0, 0 } for whch assumpto mples that t s bouded; otce that the set Φ cotas the lobal soluto of the NLP problem descrbed by. ad.. Moreover, assumpto uaratees that S s o-empty ad compact. Let d be the dstace of ay pot from the set Φ : { y Φ: y 0 } { y : Φ } 0,.e., d : f y.5 Sce the set Φ s o-empty ad compact, t follows that the fucto d s well-defed, s lobally Lpschtz wth ut Lpschtz costat ad satsfes d > 0 for all Φ. 0

11 Clam: For every η > 0 there ests a costat δ > 0 such that the follow mplcato holds: y If η S, d η ad s δη the y S ad y λ s F, where y sf for the case I ad s ay lobal soluto of m y sf : ma y I The proof of the clam ca be foud at the Apped. 0 for the case I.6 We otce that, by vrtue of mplcato.6 ad the fact that s o-creas, the alorthm s well-defed p.e., for each terato the varable p assumes fte values. Let s 0, r] 0,,... be the appled s for p whch y S ad y λ s F for the case Φ ad s 0 for the case Φ. For every 0,,... t holds that: λ s F.7 Notce that assumpto coucto wth.5 uaratees that the sequece s bouded wth 0 for all 0,,.... Moreover, mplcato.6 mples the follow equalty for every 0,,... wth Φ : s, for all 0, d ] δ η / η.8 where δ η > 0 s the costat volved mplcato.6. If s oe lobal soluto of the NLP problem descrbed by. ad. the by apply.7 ductvely, we coclude that the follow equalty holds for,,... λ s F.9 l 0 l l l 0 The above equalty mples that lm F 0 s. I order to prove that every accumulato pot of the sequece produced by the above alorthm satsfes Φ, we wll use a cotradcto arumet. Let a subsequece of the sequece whch coveres. We wll use the same otato for the subsequece ad let be the uque accumulato pot of the subsequece. We assume that Φ. y cotuty ad us property b of Theorem., we have lm Sce * F F > 0. Sce lm F 0 s, we are a posto to coclude that 0 lm s.0 Φ, t follows that lm d d 0 * >. Therefore, there ests > 0 η such that d η for suffcetly lare. Thus, equalty.8 ves s δ η /, where δ η > 0 s the costat volved mplcato.6. Ths cotradcts.0. The proof s complete. emar.: Whe equalty costrats are preset, the t should be otced that Theorem. s stll useful uder the follow assumpto: There est postve teers, wth ad a smooth fucto φ : such that for every ξ t holds that h 0, where ξ, φ ξ. Ideed, uder assumpto, we may defe for all ξ :

12 ξ wth ξ, φ ξ. ξ : wth ξ, φ ξ for,...,. We ca also defe F ξ to be the vector feld that s made up from the frst compoets of the vector feld F defed by.,.5 evaluated at ξ, φ ξ. The we ca apply Theorem. wth,,...,, F place of,,...,, F. emar.: The alorthm may be modfed a strahtforward way for other hher order eplct ue-kutta umercal schemes. Ths s meaful oly whe the vector feld F has suffcet reularty. More specfcally, p p p p p the term s F may be replaced N s,, for a approprate mapp N s, whch s characterstc of the ue-kutta scheme ad the defto of the set I {,..., } s modfed to be the set of all dces {,..., } ma N s, >. owever, t should be otced that for hher order eplct ε wth 0 s ε ue-kutta schemes, the vector feld F must be computed for varous pots. Sce F s defed oly for a p ehborhood of the set S, t may be eeded to restrct the tme step s so that all pots whch are eeded for the p evaluato of N s, are a ehborhood of the set S. emar.5: Us.,. ad. ad assum that for all,...,, there est postve, cotuous fuctos q : S 0,, : S 0,,...,, such that the follow equaltes hold for all,..., ad S : p b c ma0, v ma0, v K q. where K : ma F F { F : [0, r] },...,, we ca coclude that provded that Defe: sf 0, for all,..., ad S. ω ω q s m r,,,..., q.5 s : sup s [0, r]: ma lf 0 0,...,, for all S.6 l s ω ω q ad otce that.5 mples s m r,,,..., q aaloue of equalty. for,.e., the equalty for all S. Us the s, sf s F s K /.7 { } where [ 0, r] K : ma F F F : [0, r], we ca coclude that the best possble choce for the tme step s [ 0, r] s ve by: s m s F,, for the case K K > 0 ad s s, for the case K 0.8

13 We otce that equaltes. hold automatcally for arbtrary postve, cotuous fuctos q : S 0,, : S 0,,...,, for the case where all fuctos,..., are lear. Therefore, f equalty. holds the we ca smply compute the sequece M, where M sf ad s [ 0, r] s ve by.8. We otce that the mplemetato of a appromato of the umercal scheme M does ot ecessarly requres owlede of the fuctos K, q : S 0,, : S 0,,..., ad K : evaluat sf ad sf for certa s [ 0, r] ca ve us estmates of K ad K satsfy. ad.7. Us these estmates we ca estmate s. Cosequetly, the alorthm s mplemeted as follows: Alorthm: Gve costats r > 0, ε > 0, λ 0,/ ] ad a tal pot 0 S, we follow the steps for 0,,... Step : Calculate F us.,.5. If F 0 the. If F > 0 the K K 0 0, r rf r F ε, r rf r F ma ε for,...,, ma ad set p 0. p F F K p Step p: Compute s m, r for,..., ad p K p p F m, p p s s,...,. Calculate p s F. K If p S ad p p λ s F p the set, ad o to Step. If p S or p p > λ s F p p the set K K ε p p,...,, K K ε, p p ad o to Step p. Us eactly the same procedure wth that of the proof of Theorem., we ca coclude that every accumulato pot of the sequece produced by the above alorthm satsfes Φ, provded that assumptos, ad. hold. owever, umercal epermets show that the alorthm ca covere eve whe assumpto. does ot hold.. Eamples I order to demostrate the performace of the proposed alorthms we have used two eamples from [8]. The frst eample s deal wth the soluto of the problem: m 6 s. t. h 0 0. It ca be show that assumptos, hold for ths problem. Moreover, assumpto holds wth ξ, ad φ ξ. Sce, the equalty costrats are lear, we are a posto to use the alorthm of emar.5. We have used the alorthm of emar.5 wth σ I, where σ > 0,

14 0, a, b, c 0,...,, r, λ 0. ad ε 0 6. It was foud that for all tal pots the feasble set ad for every σ [0.0,00] the alorthm coveres at the pot,, 0,0, o more tha teratos. I ths case, the coverece of the alorthm of emar.5 s very fast. Fure shows the proecto of the phase daram o the plae for the dyamcal system.6, where F s defed by.,.5, for problem. wth σ I, σ, 0, a, b, c 0,...,. Fure was created by solv umercally system.6 wth the eplct Euler method ad tme step 0.0. Fure : The proecto of the phase daram o the plae for the dyamcal system.6, where F s defed by.,.5, for problem. wth σ I, σ, 0, a, b, c 0,...,. The secod eample s deal wth the ose Suzu problem: m s. t. h It ca be show that assumptos, hold for ths problem. Moreover, assumpto holds wth ξ,, ad φ ξ 5. The vector feld F defed by.,.5 s costructed wth σ I, where 0 σ >, 0, a, b, c 0,.

15 Ths s a problem wth olear equalty costrats. Therefore, we caot assume the valdty of.. Ideed, there are pots the feasble set wth 0, ma 0, v 0 ad for whch ξ sf ξ > 0 for s > 0, where s defed by. ad F ξ s the vector feld that s made up from the frst compoets of the vector feld F evaluated at ξ, φ ξ. Such a pot s,,, ad t s clear that we caot apply the alorthm of emar.5 at ay oe of these pots. owever, the alorthm of emar.5 may be appled wth other tal pots: for eample, f the alorthm of emar.5 s appled wth σ 0., r, λ 0. ad ε 0 6 to the tal pot 0 0.9,,,0.8 whch s close to the problematc pot,,, the the produced sequece reaches the ehborhood N { } 0 5, wth 0,,, teratos. It was also foud that dfferet values of σ > 0 ad r > 0 affect the coverece propertes of the alorthm. For eample, lower values tha for r > 0 ad hher values tha 0.5 for σ > 0 requre more teratos for coverece. The alorthm of emar.5 performs well from almost all pots of the feasble set: for eample, f the alorthm of emar.5 s appled wth σ 0., r, λ 0. ad ε 0 6 to the tal pot,,, the the produced sequece reaches the ehborhood { } 0 5 N, wth 0 0,,, 7 teratos. For the tal pot 0,,, we ca apply the alorthm of Theorem.. If the alorthm of Theorem. s appled wth σ 0., r 0. 5, λ 0. ad ε 0 6 to the tal pot,,, the the produced sequece reaches the ehborhood { } 0 5 N, wth 0 0,,, 9 teratos. I eeral, the coverece of the proposed alorthms s lear. For superlear coverece, we ca ether use dfferet selectos for,, a, b, c, or use a dfferet alorthm oce we are close to the set Φ. The quatty F ca be used order to sal the approach of a ehborhood of Φ. 5. Coclusos I ths wor we have showed that ve a olear proramm problem, t s possble, uder mld assumptos, to costruct a famly of dyamcal systems defed o the feasble set of the ve problem, so that: a the equlbrum pots are the uow crtcal pots of the problem, b each dyamcal system admts the obectve fucto of the problem as a Lyapuov fucto, ad c eplct formulae are avalable wthout volv the uow crtcal pots of the problem. The costructo of the famly of dyamcal systems s based o the Cotrol Lyapuov Fucto methodoloy, whch s used mathematcal cotrol theory for the costructo of stablz feedbac. The owlede of a dyamcal system wth the prevously metoed propertes allows the costructo of alorthms whch uaratee lobal coverece to the set of the crtcal pots. owever, we mae o clam about the effectveess of the proposed alorthms. The topc of the umercal soluto of NLPs s a mature topc ad t s clear that other alorthms have much better characterstcs tha the alorthms proposed ths paper. owever, the theory used for the costructo of the alorthm s dfferet from other est alorthms. The alorthms cotaed ths wor are derved by us cocepts of dyamcal systems theory ad mathematcal cotrol theory. The obtaed results have oth to do wth etremum see see [0,9], but may ope the way of us dfferet etremum see cotrol schemes the future for costraed problems. Fally, t may be beefcal to compare the alorthm wth other lobal alorthms see [] ad refereces there: ths s a future research topc. Acowledemets: The author would le to tha Prof. Lars Grüe for hs valuable commets ad suestos. The cotrbuto of Prof. Lars Grüe to ths wor s maor. efereces [] Atp A. S., Mmzato of cove fuctos o cove sets by meas of dfferetal equatos, Dfferetal Equatos, 09, 99, [] Artste, Z., Stablzato wth relaed cotrols, Nolear Aalyss: Theory, Methods ad Applcatos, 7, 98,

16 [] Aub, J.P., Vablty Theory, rhauser, osto, 99. [] Avrel, M., Nolear Proramm: Aalyss ad Methods, Dover Publcatos, 00. [5] row, A. A. ad M. C. artholomew-s, ODE versus SP methods for costraed optmzato, Joural of Optmzato Theory ad Applcatos, 6, 989, [6] Cabot, A., The Steepest Descet Dyamcal System wth Cotrol. Applcatos to costraed mmzato, ESAIM: Cotrol, Optmsato ad Calculus of Varatos, 0, 00, 58. [7] Evtusheo, Y. G. ad V. G. Zhada, arrer-proectve methods for olear proramm, Computatoal Mathematcs ad Mathematcal Physcs, 5, 99, [8] Facco, A. V., G.P. McCormc, Nolear Proramm: Sequetal Ucostraed Mmzato Techques, Joh Wley, New Yor, 968. [9] Freema,. A. ad P. V. Kootovc, obust Nolear Cotrol Des- State Space ad Lyapuov Techques, rhauser, osto, 996. [0] Ghaffar, A., M. Krstc, ad D. Nesc, Multvarable Newto-based etremum see, Automatca, 8, , 0. [] Goh,.S., Alorthms for Ucostraed Optmzato Problems va Cotrol Theory, Joural of Optmzato Theory ad Applcatos, 9, 997, [] Grüe, L. ad I. Karafylls, Lyapuov fucto based step sze cotrol for umercal ODE solvers wth applcato to optmzato alorthms, Proceeds of Mathematcal Theory of Networs ad Systems MTNS 0, Melboure, Australa, 0. [] arer, E., S. P. Norsett ad G. Waer, Solv Ordary Dfferetal Equatos I Nostff Problems, d Ed., Sprer-Verla, erl-edelber, 99. [] arer, E. ad G. Waer, Solv Ordary Dfferetal Equatos II Stff ad Dfferetal-Alebrac Problems, d Edto, Sprer-Verla, erl-edelber, 00. [5] elme, U. ad J.. Moore, Optmzato ad Dyamcal Systems, d Edto, Sprer-Verla, 996. [6] Karafylls, I. ad L. Grüe, Feedbac Stablzato Methods for the Numercal Soluto of Systems of Ordary Dfferetal Equatos, Dscrete ad Cotuous Dyamcal Systems: Seres, 6, 0, 8-7. [7] Karafylls, I. ad Zho-P Ja, Stablty ad Stablzato of Nolear Systems, Sprer-Verla Lodo Seres: Commucatos ad Cotrol Eeer, 0. [8] Khall,. K., Nolear Systems, d Edto, Pretce-all, 996. [9] Lu, S.-J. ad M. Krstc, Stochastc Avera ad Stochastc Etremum See, Sprer, 0. [0] Maasara, O. L. ad S. Fromovtz, The Frtz Joh Necessary Optmalty Codtos the Presece of Equalty ad Iequalty Costrats, Joural of Mathematcal Aalyss ad Applcatos, 7, 967, 7-7. [] Pa, P.., New ODE methods for equalty costraed optmzato : equatos, Joural of Computatoal Mathematcs, 0, 99, [] Solodov, M. V., Global coverece of a SP method wthout boudedess assumptos o ay of the teratve sequeces, Mathematcal Proramm Seres A, 8, 009,. [] Sota, E.D., A "Uversal" Costructo of Artste's Theorem o Nolear Stablzato, Systems ad Cotrol Letters,, 989, 7-. [] Stuart, A.M. ad A.. umphres, Dyamcal Systems ad Numercal Aalyss, Cambrde Uversty Press, 998. [5] Taabe, K., A alorthm for costraed mamzato olear proramm, Joural of the Operatos esearch Socety of Japa, 7, 97, 8-0. [6] Xa, Y. ad J. Wa, A ecurret Neural Networ for Nolear Cove Optmzato Subect to Nolear Iequalty Costrats, IEEE Trasactos o Crcuts ad Systems, 57, 00, [7] Yamashta,., A Dfferetal Equato Approach to Nolear Proramm, Mathematcal Proramm, 8, 980, [8] Zhou, L., Y. Wu, L. Zha ad G. Zha, Coverece Aalyss of a Dfferetal Equato Approach for Solv Nolear Proramm Problems, Appled Mathematcs ad Computato, 8, 007, Apped Proof of Lemma.: Frst otce that by vrtue of Fact, the follow equalty holds for all ξ ξ,..., ξ : ξ ξ ξ ξ A. Equalty A. mples that s postve semdefte. Suppose that s ot postve defte. 6

17 The there ests a o-zero ξ ξ,..., ξ wth ξ ξ 0. Cosequetly, equalty A. shows that we m must have ξ 0 ad ξ 0 for all,..., wth < 0. Fact mples that there ests λ such that ξ A λ. The prevous equalty mples that m ξ λ h 0 A. Sce ξ 0 for all,..., wth < 0 ad sce ξ ξ,..., ξ s o-zero, we coclude from A. that assumpto s volated. The proof s complete. Proof of the Clam: Let η > 0 be arbtrary. We dstush two cases. s empty, where S S case, mplcato.6 holds trvally wth arbtrary δ > 0. Case : The set { S : d η } Case : The set { S : d η } s o-empty. Cotuty of the dstace fucto d ad compactess of S S compact. Statemets b ad c of Theorem. uaratee that the quatty η s defed by. ad d s defed by.5. I ths mples that the set { S : d η } s ρ : m η F F : S, d F A. s well-defed ad s postve. Let S wth d η be a arbtrary pot. We deote by z t the uque soluto of the tal value problem z Fz wth z 0. We also otce that the vector feld F as defed by.,.5 s locally Lpschtz o a ehborhood of S. y vrtue of compactess of S S, we are a posto to assume the estece of a costat L 0 that satsfes: F y F L y, for all, y S A. Iequalty A., the fact that z s belos to the compact set of all z S wth z ad stadard arumets show that the follow equalty holds for all s 0 : Net we otce that the problem m y sf : ma y soluto sce the mapp m y sf : ma I Ls s z s sf Le F A.5 I 0 wth y y sf s radally ubouded. Ay soluto y 0 wth I satsfes for all s 0 : I admts at least oe y of the problem y sf z s sf y sf s F A.6 7

18 Notce that the above equaltes hold trvally for the case I ad y sf. Defe: q : ma z : z, z β, where β : ma{ r F : S, d η } A.7 Lr { z : z, β } Le : ma z A.8 We wll show et that mplcato.6 holds wth δ > 0 defed by: Frst, we show the mplcato: / ε λ ρ : m ε,, qlγe Lr η δ η, where γ : ma{ : S, d η } F A.9 If S, d η ad s δη the y S A.0 It suffces to show that y 0 for all I. Notce that by vrtue of the defto of the set I {,..., } t follows that s F ε, for all s [ 0, ε ] ad I. Us A.6 we obta for all s [ 0, r] : Sce s F ε y s F y s F ma z { : z r F }., for all s [ 0, ε ] ad I, we obta from A.5, A.6, A. ad A.7 for all s [ 0, ε ] ad I : y Lr s ε ql F e A. Iequalty A. coucto wth defto A.9 shows that y 0 for all I, provded that s δη. y vrtue of mplcato A.0, we are left wth the tas of prov the equalty y λ s F for all s δη ad S wth d η. Us A.6, we obta for all s [ 0, r] : where K ma{ z : z β } y sf y s F K y A.. The dervato of A. follows from maorz the secod dervatve of the mapp w p w w y ad us the equalty y s F whch s a drect cosequece of A.6. It follows from A.6, A.5, A. ad A.8 that the follow equalty holds for all s [ 0, r] : F F y s F s A. Deftos A., A.9 ad equalty A. allow us to coclude that y λ s F for all s δη. The proof s complete. 8