ASYMPTOTIC BEHAVIOR OF CONTINUOUS TRAJECTORIES FOR PRIMAL-DUAL POTENTIAL-REDUCTION METHODS

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1 ASYMPTOTIC EHAVIOR OF CONTINUOUS TRAJECTORIES FOR PRIMAL-DUAL POTENTIAL-REDUCTION METHODS REHA H. TÜTÜNCÜ Abstract. This article cosiders cotiuous trajectories of the vector fields iduced by primaldual potetial-reductio algorithms for solvig liear programmig problems. It is kow that these trajectories coverge to the aalytic ceter of the primal-dual optimal face. We establish that this covergece may be tagetial to the cetral path, tagetial to the optimal face, or i betwee, depedig o the value of the potetial fuctio parameter. Key words. liear programmig, potetial fuctios, potetial-reductio methods, cetral path, cotiuous trajectories for liear programmig. AMS subject classificatios. 90C05 1. Itroductio. Durig the past two decades, iterior-poit methods IPMs emerged as oe of the most efficiet ad reliable techiques for the solutio of liear programmig problems. The developmet of IPMs ad their theoretical covergece aalyses ofte rely o certai cotiuous trajectories associated with the give liear program. The best kow examples of such trajectories are the cetral path ad the weighted ceters the sets of miimizers of the parametrized stadard ad weighted logarithmic barrier fuctios i the iterior of the feasible regio. Primal-dual variats of IPMs, which have bee very successful i practical implemetatios, ot oly solve the give liear program but also its dual. If both the give LP ad its dual have strictly feasible solutios the primal-dual cetral path starts from the aalytic ceter of the primal-dual feasible set ad coverges to the aalytic ceter of the optimal solutio set. Similarly, weighted ceters coverge to weighted aalytic ceters. This property of the cetral trajectories led to the developmet of path followig IPMs: algorithms that try to reach a optimal solutio by geeratig a sequece of poits that are close to a correspodig sequece of poits o the cetral path or the weighted cetral path that coverge to its limit poit. A alterative characterizatio of the cetral path ad weighted ceters ca be obtaied by represetig them as solutios of certai differetial equatios. Usig this perspective, Adler ad Moteiro aalyzed the limitig behavior of cotiuous trajectories associated with primal-oly affie-scalig ad projective-scalig algorithms as well as a primal-oly potetial-reductio method [1, 11, 12]. Kojima et al. studied similar trajectories for primal-dual potetial-reductio methods [7]. Potetial-reductio algorithms use the followig strategy: First, oe defies a potetial fuctio that measures the quality or potetial of ay trial solutio of the give problem combiig measures of proximity to the set of optimal solutios, proximity to the feasible set i the case of ifeasible-iterior-poits, ad a measure of cetrality withi the feasible regio. Potetial fuctios are chose such that oe approaches a optimal solutio of the uderlyig problem by reducig the potetial fuctio. The, the search for a optimal solutio ca be performed by progressive reductio of the potetial fuctio, leadig to a potetial-reductio algorithm. We refer the reader to two excellet surveys for further details o potetial-reductio algorithms [2, 15]. Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA 15213, USA reha@cmu.edu. Research supported i part by NSF through grat CCR

2 2 REHA H. TÜTÜNCÜ Ofte, implemetatios of potetial-reductio iterior-poit algorithms exhibit behavior that is similar to that of path-followig algorithms. For example, they take about the same umber of iteratios as path-followig algorithms ad they ted to coverge to the aalytic ceter of the optimal face, just like most path-followig variats. Sice potetial-reductio methods do ot geerally make a effort to follow the cetral path, this behavior is surprisig. I a effort to better uderstad the limitig behavior of primal-dual potetial-reductio algorithms for liear programs this paper studies cotiuous trajectories associated with the algorithm proposed by Kojima, Mizuo, ad Yoshise KMY [8], which uses scaled ad projected steepest descet directios for the Taabe-Todd-Ye TTY primal-dual potetial fuctio [14, 16]. Usig earlier results [9, 10, 7], we show that all trajectories of the vector field iduced by the KMY search directios coverge to the aalytic ceter of the primaldual optimal face. Our mai results are o the directio of covergece for these trajectories. We demostrate that their asymptotic behavior depeds o the potetial fuctio parameter. There is a threshold value of this parameter the value that makes the TTY potetial fuctio homogeeous. Whe the parameter is below this threshold, the ceterig is too strog ad the trajectories coverge tagetially to the cetral path. Whe the parameter is above the threshold, trajectories coverge tagetially to the optimal face. However, the directio of covergece of these trajectories depeds o the iitial poit. At the threshold value, the behavior of the trajectories is i betwee these two extremes ad depeds o the iitial poit. Followig this itroductio, Sectio 2 discusses cotiuous trajectories associated with the KMY methods ad proves their covergece. Sectio 3 is devoted to the aalysis of the limitig behavior of these trajectories. Our otatio is fairly stadard: For a -dimesioal vector x, the correspodig capital letter X deotes the diagoal matrix with X ii x i. We will use the letter e to deote a colum vector with all etries equal to 1 ad its dimesio will be apparet from the cotext. We also deote the base of the atural logarithm with e ad sometimes the vector e ad the scalar e appear i the same expressio, but o cofusio should arise. For a give matrix A, we use RA ad N A to deote its ragecolum ad ull space. For a vector-valued differetiable fuctio xt of a scalar variable t, we use the otatio ẋ or ẋt to deote the vector of the derivatives of its compoets with respect to t. For dimesioal vectors x ad s, we write xs to deote their Hadamard compoetwise product. Also, for a dimesioal vector x, we write x p to deote the vector X p e, where p ca be fractioal if x > Primal-Dual Potetial-Reductio Trajectories. We cosider liear programs i the followig stadard form: 2.1 LP mi x c T x Ax = b x 0, where A R m, b R m, c R are give, ad x R. Without loss of geerality we assume that the costraits are liearly idepedet. The, the matrix A has full row rak. Further, we assume that 0 < m < ; m = 0 ad m = correspod to trivial problems.

3 POTENTIAL REDUCTION TRAJECTORIES 3 The liear programmig dual of this primal problem is: 2.2 LD max y,s b T y A T y + s = c s 0, where y R m ad s R. We ca rewrite the dual problem by elimiatig the y variables i 2.2. This is achieved by cosiderig G T, a ull-space basis matrix for A, that is, G is a m matrix with rak m ad it satisfies AG T = 0, GA T = 0. Note also that, A T is a ull-space basis matrix for G. Further, let d R be a vector satisfyig Ad = b. The, 2.2 is equivalet to the followig problem which has a high degree of symmetry with 2.1: 2.3 LD mi s d T s Gs = Gc s 0. Let F ad F 0 deote the primal-dual feasible regio ad its relative iterior: F := {x, s : Ax = b, Gs = Gc, x, s 0}, F 0 := {x, s : Ax = b, Gs = Gc, x, s > 0}. We assume that F 0 is o-empty. This assumptio has the importat cosequece that the primal-dual optimal solutio set Ω defied below is oempty ad bouded: 2.4 Ω := {x, s F : x T s = 0}. We also defie the optimal partitio N = {1,..., } for future referece: := {j : x j > 0 for some x, s Ω}, N := {j : s j > 0 for some x, s Ω}. The fact that ad N is a partitio of {1,..., } is a classical result of Goldma ad Tucker. The aalytic ceter of Ω is the poit x, s = x, 0, 0, s N where x ad s N are uique maximizers of the followig problems: max j l x j max j N l s j 2.5 A x = b ad G N s N = Gc x > 0, s N > 0. The cetral path C of the primal-dual feasible set F is the set of poits o which the compoet-wise product of the primal ad dual variables is costat: 2.6 C := {xµ, sµ F 0 : xµsµ = µe, for some µ > 0}. The poits o the cetral path are obtaied as uique miimizers of certai barrier problems associated with the primal ad dual LPs ad they coverge to the aalytic ceter of the primal-dual optimal face; see, e.g., [18]. While the cetral path is the mai theoretical tool i the costructio of pathfollowig algorithms, primal-dual potetial-reductio algorithms for liear programmig are derived usig potetial fuctios, i.e., fuctios that measure the quality

4 4 REHA H. TÜTÜNCÜ or potetial of trial solutios for the primal-dual pair of problems. The most frequetly used primal-dual potetial fuctio for liear programmig problems is the Taabe-Todd-Ye TTY potetial fuctio [14, 16]: 2.7 Φ ρ x, s := ρ lx T s lx i s i, for every x, s > 0. i=1 Whe ρ >, the TTY potetial fuctio diverges to alog a feasible sequece {x k, s k } oly if this sequece is covergig to a primal-dual optimal pair of solutios. Therefore, the primal-dual pair of LP problems ca be solved by miimizig the TTY potetial fuctio. Kojima, Mizuo, ad Yoshise KMY developed a primal-dual algorithm that mootoically reduces the TTY potetial fuctio usig a scaled ad projected steepestdescet search directio usig a primal-dual scalig matrix [8]. I the remaider of this article, we will study cotiuous trajectories that are aturally associated with their algorithm. Give a iterate x, s F 0, the search directio used by the KMY method is the solutio of the followig system: 2.8 A x = 0 G s = 0 S x + X s = xt s ρ e xs, where X = diagx, S = diags, ad e is a vector of oes of appropriate dimesio. Whe we discuss the search directio give by 2.8 ad associated trajectories, we will assume that ρ >. For ay give x 0, s 0 F 0, oe ca associate a trajectory {xt, st : t 0} startig from x 0, s 0 with the property that the taget directio to the trajectory at ay of its poits coicides with the KMY directio. I other words, we cosider trajectories that solve the followig system of ordiary differetial equatios: 2.9 A 0 [ ] 0 G ẋ = ṡ S X 0 0 x T s ρ e xs with the iitial coditio x0, s0 = x 0, s 0. I [7, Sectio 4.3], Kojima et al. study these trajectories ad establish that their solutio curves satisfy the followig system of equatios:, 2.10 Axt = b, Gst = Gc, xtst = wt, t 0, where wt = e t w 0 + hte, with w 0 = x 0 s 0, ad, ht = et w 0 exp { 1 β t} e t with β = ρ. Sice w0 = w 0, we will use these two expressios iterchageably. Kojima et al. do ot address the existece ad uiqueess of the solutios to 2.9 rigorously, but these results follow easily from stadard theory of ordiary differetial equatios; see,

5 POTENTIAL REDUCTION TRAJECTORIES 5 e.g., Theorem 1 o p. 162 ad Lemma o p. 171 of the textbook by Hirsch ad Smale [6]. We also ote that Moteiro [12] studies trajectories based o primal-oly potetial-reductio algorithms ad obtais similar but less explicit descriptios of these trajectories. The characterizatio of the potetial-reductio trajectories usig the system 2.10 leads to the followig observatios: Theorem 2.1. Let xt, st for t 0 deote the solutio of the ODE 2.9 with the iitial coditio x0, s0 = x 0, s 0. The, the followig statemets hold: i For ρ >, Φ ρ xt, st is a decreasig fuctio of t. ii Whe w 0 = x 0 s 0 = µe for some µ > 0 i.e., whe x 0, s 0 is o the cetral path the {xt, st : t 0} is a subset of the cetral path C. iii xt, st coverges to the aalytic ceter of the primal-dual optimal face Ω as t. Proof. Lemma 4.14 i [7] proves i. Observig that w 0 = µe implies wt = µe 1 βt e with β = ρ, ii follows immediately from 2.6 ad For iii, first wt observe that wt e as t. Now, the proof of Theorem 9 i [9] or, Corollary 2 of Theorem 5 i [10] immediately leads to iii. So, these trajectories coverge to a uique poit regardless of their startig poit as they mootoically decrease the potetial fuctio. Further, they iclude the cetral path as a special case givig a theoretical basis for the observatio that cetral path-followig search directios are ofte very good potetial-reductio directios as well. Aother related result is by Nesterov [13] who observes that the eighborhood of the cetral path is the regio of fastest decrease for a homogeeous potetial fuctio. A direct proof of iii i Theorem 2.1 ca be obtaied usig the followig result, which will also be useful i the ext sectio: Lemma 2.2. Let xt, st for t 0 deote the solutio of the ODE 2.9 with the iitial coditio x0, s0 = x 0, s 0. The x t ad s N t solve the followig pair of problems: 2.13 max j w jt l x j A x = b A N x N t x > 0, ad max j N w jt l s j G N s N = Gc G s t s N > 0. Proof. We prove the optimality of x t for the first problem i 2.13 the correspodig result for s N t ca be prove similarly. x t is clearly feasible for the give problem. It is optimal if ad oly if there exists y R m such that w tx 1 t = AT y. From 2.10 we obtai w tx 1 t = s t. Note that, for ay s feasible for LD we have that c s RA T ad therefore, c s RA T. Furthermore, sice s = 0, s N is also feasible for LD we must have that c RA T ad that s t RA T. This is exactly what we eeded. 3. Asymptotic Aalysis of the Trajectories. I the previous sectio, we saw that all primal-dual potetial-reductio trajectories xt, st that solve the differetial equatio 2.9 coverge to the aalytic ceter x, s of the primal-dual optimal face Ω regardless of the iitial poit of the trajectory. I this sectio, we ivestigate the directio of covergece for these trajectories. That is, we wat to aalyze the limitig behavior of the ormalized vectors this aalysis is quite techical. ẋt ẋt, ṡt ṡt. Ievitably,

6 6 REHA H. TÜTÜNCÜ Our strategy for this aalysis is as follows. Usig the optimal partitio ad N we express the basic compoets of the covergece directios of the trajectories i terms of the obasic oes i Lemma 3.1. The, we establish a boud o the covergece speed of the obasic compoets i Lemma 3.3. The depedece of the covergece directio o ρ, the potetial fuctio parameter, becomes apparet at this poit ad a case aalysis is required. Lemma 3.4, Theorems 3.5 ad 3.6 cosider the case ρ 2 while Theorem 3.8 addresses the case ρ > 2. Let β = ρ ad ote that β 0, 1. We ow itroduce some otatio: ŵ t = w te 1 βt, ŵ N t = w N te 1 βt, d t = ŵ 1 2 tx 1 t, d N t = ŵ 1 2 N ts 1 N t, D t = diag d t, D N t = diag d N t, d 1 t = D 1 te, d 1 N t = D 1 N te, Ã t = A D 1 t, GN t = G N D 1 N t, x t = D tẋ t = d tẋ t, s N t = D N tṡ N t = d N tṡ N t, u t = ŵ 1 2 tw 0, u N t = ŵ 1 2 N tw N 0. For our asymptotic aalysis, we express basic compoets of the vectors ẋt ad ṡt i terms of the obasic oes i the ext lemma which forms the backboe of our aalysis. Lemma 3.1. Let xt, st for t 0 deote the solutio of the ODE 2.9 with the iitial coditio x0, s0 = x 0, s 0. The, the followig equalities hold: D tẋ t = Ã+ ta N ẋ N t e βt ρ 1 e βt D N tṡ N t = G + N tg ṡ t e βt ρ 1 e βt I Ã+ tãt u t, I G + N t G N t u N t. Here, Ã+ t ad G + N t deote the pseudo-iverse of Ãt ad G N t, respectively. Proof. We oly prove the first idetity; the secod oe follows similarly. Recall from Lemma 2.2 that x t solves the first problem i Therefore, as i the proof of Lemma 2.2, we must have that 3.3 w tx 1 t RAT. Differetiatig with respect to t we obtai: Observe that w tx 2 w tx 1 tẋ t ẇ tx 1 t RAT, or, tẋ t ẇ t RX ta T. ẇ t = w 0e t + ḣte = w t + et w 0 ρ e 1 βt e. Therefore, from ŵ t = w te 1 βt, we obtai 3.4 ŵ tx 1 tẋ t + ŵ t et w 0 ρ e RX ta T.

7 POTENTIAL REDUCTION TRAJECTORIES 7 From 3.3 it also follows that ŵ t RX ta T. Note also that, e T w 0 ρ e = ρ 1 e βt ŵt Combiig these observatios with 3.4 we get 3.5 Next, observe that 3.6 e βt ρ 1 e βt w 0. ŵ tx 1 tẋ t + e βt ρ 1 e βt w 0 RX ta T. A ẋ t = A N ẋ N t. Usig the otatio itroduced before the statemet of the lemma, 3.5 ad 3.6 ca be rewritte as follows: x t + e βt ρ 1 e βt u t RÃT, à t x t = A N ẋ N t. Let Ã+ t deote the pseudo-iverse of à t [3]. For example, if rakãt=m, the Ã+ t = ÃT t à tãt t 1. The, PR à T := Ã+ tãt is the orthogoal projectio matrix oto RÃT ad P N à := I Ã+ tãt is the orthogoal projectio matrix oto N à [3]. From 3.8 we obtai P R à T x t = Ã+ tãt x t = Ã+ ta N ẋ N t, ad from 3.7, usig the fact that RÃT ad N à are orthogoal to each other, we get P N à x t = e βt ρ 1 e βt I Ã+ tãt u t. Combiig, we have x t = P R à T x t + P N à x t = Ã+ ta N ẋ N t e βt ρ 1 e βt I Ã+ tãt u t, which gives 3.1. To determie the covergece directios of the trajectories, we eed to study the relative covergece speeds of ẋ t, ẋ N t, etc. Thus, we compute limits of some of the expressios that appear i equatios 3.1 ad 3.2: lim ŵt = et w 0 t e, lim ŵ N t = et w 0 t e N, lim D et w 0 t = t X 1 et w 0, lim D N t = t S N 1, lim à t = t e T A X w, lim GN t = 0 t e T G N SN, w 0 lim u t = t e T w 0, lim u N t = w 0 t e T w N 0. w 0

8 8 REHA H. TÜTÜNCÜ Lemma lim à + t t = et w 0 A X + lim G + t N t = et w 0 G N SN +. Proof. This result about the limitig properties of the pseudo-iverses is a immediate cosequece of Lemma 2.3 i [4] ad equatios Differetiatig the idetity we obtai 3.15 xtst = wt = e t w 0 + hte, xtṡt + ẋtst = e t w 0 + ḣte = e t w 0 et w 0 1 β e 1 βt e + et w 0 e t e. Next, we will establish that ẋ N t ad ṡ t coverge to zero o slower tha exp{ 1 βt}. For this purpose, we cosider the ormalized directio vectors ˆx, ŝ which are defied as follows: 3.16 ˆxt = exp {1 β t} ẋt, ad ŝt = exp {1 β t} ṡt. From 3.15 it follows that 3.17 xtŝt + ˆxtst = et w 0 1 β e + e βt e T w 0 e w 0. The expressio o the right-had-side of 3.17 is clearly bouded. With some more work, we have the followig coclusio: Lemma 3.3. Let ˆxt, ŝt be as i 3.16 ad assume that ρ >. The, ˆx N t, ŝ t remais bouded as t teds to. Proof. We will prove that xtŝt ad ˆxtst remai bouded as t. The, sice x t ad s N t coverge to x > 0 ad s N > 0 respectively, ad therefore, remai bouded away from zero, we ca coclude that ˆx N t ad ŝ t remai bouded. Sice the right-had-side of 3.17 is bouded as t teds to, it is sufficiet to show that [xtŝt] T [ˆxtst] remais bouded below to coclude that both xtŝt ad ˆxtst have bouded orms as t. Let vt = x 1 2 ts 1 2 t = w 1 2 t ad δt = x 1 2 ts 1 2 t. The, xtŝt = vtδtŝt ad ˆxtst = vtδ 1 tˆxt. Note that, [δtŝt] T [ δ 1 tˆxt ] = [δtṡt] T [ δ 1 tẋt ] = 0. Let V t = diagvt, t = diagδt, ad W 0 = diagw 0. The, V 2 t = XtSt = et w 0 e 1 βt e t I+ e t W 0. Now, [xtŝt] T [ˆxtst] = [vtδtŝt] T [ vtδ 1 tˆxt ] = [δtŝt] T V 2 t [ δ 1 tˆxt ]

9 3.18 POTENTIAL REDUCTION TRAJECTORIES 9 = e 21 βt [δtṡt] T V 2 t [ δ 1 tẋt ] = et w 0 e 1 βt e 1 2βt [δtṡt] T [ δ 1 tẋt ] +e 1 2βt [δtṡt] T [ W 0 δ 1 tẋt ] [ ] T [ ] = e βt e 1 βt/2 δtṡt W0 e 1 βt/2 δ 1 tẋt. Recall from 3.9 that lim t ŵ j t = lim t e 1 βt w j t = et w 0, j. Therefore, we have lim t ŵj t = lim t e 1 βt/2 v j t = e T w 0 ad defiig ṽt = e 1 βt/2 vt vtt vt v 1 t, ρ e we have lim t ṽ j t = 1 β T w 0. Now, recallig equatio 2.8 we observe that the vectors e 1 βt/2 δ 1 tẋt ad e 1 βt/2 δtṡt are orthogoal projectios of the vector ṽt ito the ull space of A t ad rage space of [A t] T, respectively. Sice we showed that the vector ṽt is coverget as t teds to, both of these projectios coverge, ad therefore, the expressio i 3.18 coverges to zero. Thus, xtŝt ad ˆxtst have bouded orms as t. It is iterestig that the coclusio of the lemma above holds for ay ρ 0. A alterative proof of Lemma 3.3 ca be obtaied usig the proof techique i [5]. Combiig equatio 3.1, Lemmas 3.2 ad 3.3 we obtai the followig result: Lemma 3.4. Let ˆxt, ŝt be as i 3.16 ad assume that < ρ 2. The, ˆxt, ŝt remais bouded as t teds to. Proof. From 3.1 we have that 3.19 D tˆx t = Ã+ ta N ˆx N t e1 2βt ρ 1 e βt I Ã+ tãt u t. Whe, ρ 2, the factor e1 2βt is coverget as t teds to. Now, usig Lemma ρ1 e βt 3.2 ad the equatios , we coclude that the secod term i the righthad-side of the equatio above remais bouded. Combiig this observatio with the the fact that ˆx N t remais bouded as t teds to, we obtai that D tˆx t remais bouded. Usig 3.10 we coclude that ˆx t is also bouded as t teds to. The fact that ŝt is bouded follows similarly. Now, the followig two results are easy to prove: Theorem 3.5. Let ˆxt, ŝt be as i The, we have that lim t ˆx N t ad lim t ŝ t exist ad satisfy the followig equatios: lim ˆx N t = et w 0 t 1 β s N 1, lim ŝt = et w 0 t 1 β x 1. Proof. From 3.17 we have that x tŝ t + ˆx ts t = et w 0 e 1 β e + e βt T w 0 e w 0.

10 10 REHA H. TÜTÜNCÜ Takig the limit o the right-had-side as t we obtai et w 0 1 βe. Sice s t 0 ad ˆx t is bouded, we must the have that x tŝ t coverges to et w 0 1 βe. Sice x t x, it follows that lim t ŝ t exists ad satisfies The correspodig result for ˆx N t follows idetically. Let π = X ξ = X A X + A N s N 1, I A X + A X w 0, σ N = SN G N SN + G x 1 π N = SN I G N SN + G N SN w N 0. Observe that π = 0 if ad oly if w 0 RX AT which holds, for example, whe x 0, s 0 is o the cetral path ad w0 = µe for some µ > 0 the observatio that e RX AT follows easily from the optimality of x for the first problem i 2.5. Similarly, π N = 0 if ad oly if w N 0 RSN GT N. Theorem 3.6. Let ˆxt, ŝt be as i 3.16 ad assume that ρ 2. The, we have that lim t ˆx t ad lim t ŝ N t exist. Whe ρ < 2 we have the followig idetities: lim ˆx t = et w 0 t 1 β ξ, lim t ŝn t = et w 0 1 β σ N. Whe ρ = 2, the followig equatios hold: lim ˆx t = et w 0 t 2 ξ 2e T w 0 π, lim t ŝn t = et w 0 2 σ N 2e T w 0 π N. Proof. Recall equatio Whe ρ < 2, the secod term o the right-hadside coverges to zero sice e 1 2βt teds to zero ad everythig else is bouded. Thus, usig 3.10 ad 3.11 we have lim t ˆx t = X A X + A N lim t ˆx N t ad 3.22 is obtaied usig Theorem 3.5. Similarly, oe obtais Whe ρ = 2, the factor i frot of the secod term i 3.19 coverges to the positive costat β = 1 2. Therefore, usig Theorem 3.5 ad equatios we get 3.24 ad Limits of the ormalized vectors ẋt ẋt, ṡt ṡt are obtaied immediately from Theorems 3.5 ad 3.6: Corollary 3.7. Let xt, st for t 0 deote the solutio of the ODE 2.9 with the iitial coditio x0, s0 = x 0, s 0 with x 0, s 0 F 0 ad assume that ρ 2. All trajectories of this form satisfy the followig equatios: 3.26 where 3.27 q P = [ [ q P = ξ lim t ξ s N 1 e T w 0 2 π s N 1 ẋt ẋt = q P q P, lim t ṡt ṡt = ] [ x, ad q D = 1 ], ad q D = [ σ N x 1 σ N q D q D ], if ρ < 2, e T w 0 2 πn ], if ρ = 2.

11 POTENTIAL REDUCTION TRAJECTORIES 11 Whe ρ = 2 the TTY potetial-fuctio Φ ρ x, y is a homogeeous fuctio ad exp{φ ρ x, y} is a covex fuctio for all ρ 2 [17]. The value 2 also represets a threshold value for the covergece behavior of the KMY trajectories. Whe ρ = 2 the directio of covergece depeds o the iitial poit x 0, s 0 F 0 as idicated by the appearace of the w 0 = x 0 s 0 terms i the formulas. We ote that, whe ρ < 2 the asymptotic directio of covergece does ot deped o the iitial poit ad is idetical to that of the cetral path. Therefore, whe ρ < 2 all trajectories of the vector field give by the search directio of the Kojima-Mizuo-Yoshise s primaldual potetial-reductio algorithm coverge to the aalytic ceter of the optimal face tagetially to the cetral path. We show below that the asymptotic behavior of the trajectories is sigificatly differet whe ρ > 2: Theorem 3.8. Let xt, st for t 0 deote the solutio of the ODE 2.9 with the iitial coditio x0, s0 = x 0, s 0 ad assume that ρ > 2. Defie 3.28 xt = e βt ẋt, ad st = e βt ṡt. If π 0 ad π N 0, the we have that lim t xt ad lim t st exist ad satisfy the followig equatios: 3.29 where lim t xt xt = q P = q P, ad lim q P t st st = [ ] [ ] π 0, ad q 0 D =. N π N q D q D, Proof. From 3.1 we have that D t x t = Ã+ ta 3.30 N x N t ρ 1 e βt I Ã+ tãt u t. Note that, x N t = e 1 2βtˆx N t. Sice ˆx N t is bouded ad 1 2β < 0, we coclude that x N t 0. Therefore, usig equatios 3.30 ad we observe that x t coverges to a positive multiple of π 0 ad immediately obtai the first equatio i The secod idetity i 3.29 is obtaied similarly. This fial theorem idicates that whe ρ > 2, most trajectories associated with the Kojima-Mizuo-Yoshise algorithm coverge to the aalytic ceter of the optimal face tagetially to the optimal face ad their directio of covergece depeds o the iitial poit. I the exceptioal case of π = 0 or π N = 0 for example, whe x 0, s 0 is o the cetral path, the last term i 3.30 o loger domiates the righthad-side ad i such cases we cojecture that the trajectory coverges tagetially to the cetral path. We coclude by otig the similarity of our asymptotic results to those of Moteiro [12]. I his aalysis of the trajectories based o primal-oly potetial-reductio algorithms, Moteiro also fids that there is a threshold value of the potetial fuctio parameter that leads to differet asymptotic behavior. I his case, this threshold value is 2 N rather tha 2, where N deotes the cardiality of the set N from the optimal partitio of {1,..., }. Just like our case, whe the potetial fuctio

12 12 REHA H. TÜTÜNCÜ parameter is above this value, his trajectories coverge tagetially to the cetral path ad below the threshold the covergece is tagetial to the optimal face. ACKNOWLEDGMENT The coclusio of Lemma 3.3 was stated without a proof i a earlier draft of this paper. The author would like to thak Margaréta Halická whose careful readig led to the correctio of this omissio. She also provided a alterative argumet for the proof. The author would also like to thak a aoymous referee for brigig refereces [7, 9, 10] to his attetio ad makig several costructive commets. Results i these refereces helped make this a more cocise paper. REFERENCES [1] I. Adler ad R. D. C. Moteiro, Limitig behavior of the affie scalig cotiuous trajectories for liear programmig problems, Mathematical Programmig, pp [2] K. M. Astreicher, Potetial reductio algorithms, i: T. Terlaky, ed., Iterior poit methods of mathematical programmig, Kluwer Academic Publishers, Dordrecht, Netherlads, 1996 pp [3] G. H. Golub ad C. F. Va Loa, Matrix Computatios, 2d ed., The Johs Hopkis Uiversity Press, altimore, [4] O. Güler, Limitig behavior of weighted cetral paths i liear programmig, Mathematical Programmig, pp [5] M. Halická, Two simple proofs for aalyticity of the cetral path, Operatios Research Letters, pp [6] M. W. Hirsch ad S. Smale, Differetial equatios, dyamical systems, ad liear algebra Academic Press, New York, [7] M. Kojima, N. Megiddo, T. Noma, ad A. Yoshise, A uified approach to iterior poit algorithms for liear complemetarity problems, Vol. 538 of Lecture Notes i Computer Sciece, Spriger Verlag, erli, Germay, [8] M. Kojima, S. Mizuo, ad A. Yoshise, A O L iteratio potetial reductio algorithm for liear complemetarity problems, Mathematical Programmig, pp [9] L. McLide, A aalogue of Moreau s proximatio theorem, with applicatios to the oliear complemetarity problem, Pacific Joural of Mathematics, pp [10] L. McLide, The complemetarity problem for maximal mootoe multifuctios, i: R. W. Cottle, F. Giaessi, ad J.-L. Lios, eds., Variatioal Iequalities ad Complemetarity Problems, Wiley, New York, 1980 pp [11] R. D. C. Moteiro, Covergece ad boudary behavior of the projective scalig trajectories for liear programmig, Mathematics of Operatios Research, pp [12] R. D. C. Moteiro, O the cotiuous trajectories for potetial reductio algorithms for liear programmig, Mathematics of Operatios Research, pp [13] Yu. Nesterov, Log-step strategies i iterior-poit primal-dual methods, Mathematical Programmig, pp [14] K. Taabe, Cetered Newto method for mathematical programmig, i: M. Iri ad K. Yajima, eds., Lecture Notes i Cotrol ad Iformatio Scieces, Spriger-Verlag, erli, 1988 pp [15] M. J. Todd, Potetial-reductio methods i mathematical programmig, Mathematical Programmig, [16] M. J. Todd ad Y. Ye, A cetered projective algorithm for liear programmig, Mathematics of Operatios Research, [17] R. H. Tütücü, A primal-dual variat of the Iri-Imai algorithm for liear programmig, Mathematics of Operatios Research, , [18] S. Wright, Primal-dual iterior-poit methods SIAM, Philadelphia, 1997.