Continuous Trajectories for Primal-Dual Potential-Reduction Methods

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1 Cotiuous Trajectories for Primal-Dual Potetial-Reductio Methods Reha H. Tütücü September 7, 2001 Abstract This article cosiders cotiuous trajectories of the vector fields iduced by two differet primal-dual potetial-reductio algorithms for solvig liear programmig problems. For both algorithms, it is show that the associated cotiuous trajectories iclude the cetral path ad the duality gap coverges to zero alog all these trajectories. For the algorithm of Kojima, Mizuo, ad Yoshise, there is a a surprisigly simple characterizatio of the associated trajectories. Usig this characterizatio, it is show that all associated trajectories coverge to the aalytic ceter of the primal-dual optimal face. Depedig o the value of the potetial fuctio parameter, this covergece may be tagetial to the cetral path, tagetial to the optimal face, or i betwee. Key words: liear programmig, potetial fuctios, potetial-reductio methods, cetral path, cotiuous trajectories for liear programmig. AMS subject classificatio: 90C05 1 Itroductio Durig the past two decades, iterior-poit methods IPMs emerged as very 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 example of such trajectories is the cetral path the set of miimizers of the parametrized logarithmic barrier fuctio i the iterior of the feasible regio. Primal-dual variats of IPMs ot oly solve the give liear program but also its dual. These variats have bee very successful i practical implemetatios ad form the focus of this article. 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. This property of the cetral path led to the developmet of path followig IPMs: algorithms that try to reach a optimal solutio by Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA 15213, USA reha+@adrew.cmu.edu. Research supported i part by NSF through grat CCR

2 POTENTIAL REDUCTION TRAJECTORIES 2 geeratig a sequece of poits that are close to a correspodig sequece of poits o the cetral path that coverge to its limit poit. Weighted ceters are geeralizatios of the cetral path that are obtaied as miimizers of weighted logarithmic barrier fuctios. They share may properties of the cetral path, icludig covergece to a optimal poit; see, e.g., 11]. Weighted ceters ad the cetral path ca be characterized i alterative ways. For example, these trajectories are obtaied as uique solutios of certai differetial equatios. Usig this perspective, Adler ad Moteiro aalyzed the limitig behavior of cotiuous trajectories associated with primal-oly affie-scalig algorithm 1]. Later, Moteiro aalyzed cotiuous trajectories associated with projective-scalig ad potetial-reductio algorithms 8, 9], oce agai i the primal-oly settig. I this article, we study trajectories of vector fields iduced by primal-dual potetial-reductio algorithms. 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 ofte 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, 13]. 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 two such methods. The first oe is the algorithm proposed by Kojima, Mizuo, ad Yoshise KMY 7], which uses scaled ad projected steepest descet directios for the Taabe-Todd-Ye TTY primal-dual potetial fuctio 12, 14]. The secod algorithm we cosider is a primal-dual variat of the Iri-Imai algorithm that uses Newto search directios for the multiplicative aalogue of the TTY potetial fuctio 15, 16]. For both algorithms, we show that each associated trajectory has the property that the duality gap coverges to zero as oe traces the trajectory to its bouded limit. We also show that the cetral path is a special potetialreductio trajectory associated with both methods. Our aalysis is more extesive for the KMY algorithm, sice we are able to express the poits o the associated trajectories as uique optimal solutios of certai parametrized weighted logarithmic barrier problems. This characterizatio is similar to that obtaied by Moteiro 9] for primal-oly algorithms, but our formulas are explicit ad much simpler. We show that all trajectories of the vector field iduced by the KMY search directios coverge to the aalytic ceter of the primal-dual optimal face. We also aalyze the directio of covergece for these trajectories ad 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

3 POTENTIAL REDUCTION TRAJECTORIES 3 tagetially to the cetral path. Whe the parameter is above the threshold, covergece happes tagetially to the optimal face. 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 the two primal-dual potetial-reductio methods metioed above ad search directios used by these two methods. Sectio 3 establishes the existece of cotiuous trajectories associated with these methods usig stadard results from the theory of differetial equatios. We prove the covergece of the trajectories associated with the Kojima-Mizuo-Yoshise algorithm i Sectio 4 ad aalyze the limitig behavior of these trajectories i Sectio 5. 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 x s 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 > 0. 2 Primal-Dual Potetial-Reductio Directios We cosider liear programs i the followig stadard form: LP mi x c T x Ax = b x 0, 1 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. The dual of this primal problem is: LD max y,s b T y A T y + s = c s 0, 2 where y R m ad s R. We ca rewrite the dual problem by elimiatig the y variables i 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 is equivalet to the followig problem which has a high degree of symmetry with 1: LD mi s d T s Gs = Gc s 0. 3

4 POTENTIAL REDUCTION TRAJECTORIES 4 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: Ω := {x, s F : x T s = 0}. 4 We also defie the optimal partitio B N = {1,..., } for future referece: B := {j : x j > 0 for some x, s Ω}, N := {j : s j > 0 for some x, s Ω}. The fact that B ad N is a partitio of {1,..., } is a classical result of Goldma ad Tucker. Primal-dual feasible iterior-poit algorithms start with a poit x 0, s 0 F 0 ad move through F 0 by geeratig search directios i the ull space of the costrait equatios which ca be represeted implicitly or explicitly, usig ull-space basis matrices: ] A 0 N := { x, s : A x = 0, G s = 0} 0 G = { x, s : x = G T x, s = A T s, x, s R m R m }. Give a search directio x, s, we will call the correspodig x, s a reduced search directio. A feasible iterior-poit algorithm that starts at x 0, s 0 F 0 ca also be expressed completely i terms of the reduced iterates x k, s k rather tha the usual iterates x k, s k it geerates. These two iterate sequeces correspod to each other through the equatios x k = x 0 + G T x k ad s k = s 0 + A T s k. Sice G T ad A T are basis matrices for the ull-spaces of A ad G respectively, x k, s k are uiquely determied give x k, s k ad the iitial feasible iterate x 0, s 0 F 0 : x k = GG T 1 Gx k x 0 ad s k = AA T 1 As k s 0. The reduced feasible set ad its iterior ca be defied as follows: F R x 0, s 0 := {x, s : x 0 + G T x, s 0 + A T s 0}, F 0 Rx 0, s 0 := {x, s : x 0 + G T x, s 0 + A T s > 0}. We will suppress the depedece of these sets o x 0, s 0 i our otatio. Note that F 0 R is a full-dimesioal ope subset of R. This simple alterative view of feasible iterior-poit methods is useful for our purposes sice it allows us use stadard results from the theory of ordiary differetial equatios later i our aalysis. Primal-dual potetial-reductio algorithms for liear programmig are derived usig potetial fuctios, i.e., fuctios that measure the quality 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 12, 14]: Φ x, s := lx T s lx i s i, for every x, s > 0. 5 i=1

5 POTENTIAL REDUCTION TRAJECTORIES 5 As most other potetial fuctios used i optimizatio, the TTY potetial fuctio has the property that it diverges to alog a feasible sequece {x k, s k } if ad oly if this sequece is covergig to a primal-dual optimal pair of solutios, as log as >. Therefore, the primaldual pair of LP problems ca be solved by miimizig the TTY potetial-fuctio. We ow evaluate the gradiet of this fuctio for future referece: Φ x, s = x T s s ] x 1 x T s x. 6 s 1 There are may variats of primal-dual potetial-reductio algorithms that use the TTY potetial fuctio. They differ i the way they geerate the potetial-reductio search directios ad i the lie search strategies they use alog these directios. For example, Todd ad Ye use projective scalig to determie the search directios ad use lie search to keep the iterates approximately cetered 14]. Kojima, Mizuo, ad Yoshise s algorithm uses a steepestdescet search directio after a primal-dual scalig of the LP problem ad does ot require the cetrality of the iterates 7]. Tütücü 15, 16] uses modified Newto directios to reduce the TTY fuctio. All the algorithms metioed have bee prove to have polyomial time worst-case complexities by demostratig that a sufficiet decrease i Φ ca be obtaied i every iteratio of these algorithms. I the remaider of this article, we will study cotiuous trajectories that are aturally associated with primal-dual potetial-reductio algorithms of Kojima-Mizuo-Yoshise 7] ad Tütücü 15]. For this purpose, we first preset the search directios used by these algorithms. Let x, s be a give iterate ad let x, s be the correspodig reduced iterate. The search directio proposed by Kojima, Mizuo, ad Yoshise is the solutio of the followig system: A x = 0 A T y + s = 0 S x + X s = xt s e x s, 7 where X = diagx, S = diags, ad e is a vector of oes of appropriate dimesio. Whe we discuss the search directio give by 7 ad associated trajectories, we will assume that >. The solutio to the system 7 ca be regarded as a scaled ad projected steepest descet directio for the TTY potetial fuctio Φ, usig a primal-dual scalig matrix. The system 7 ca be reduced to the sigle block equatio SG T x + XA T s = xt s e x s ad its solutio ca be preseted explicitly usig orthogoal projectio matrices. First, let D = X 1 2 S 1 2, ad V = X 1 2 S Now defie the primal-dual projectio matrix Ψ as follows: Ψ = Ψx, s = I DA T AD 2 A T 1 AD 9 = D 1 G T GD 2 G T 1 GD 1.

6 POTENTIAL REDUCTION TRAJECTORIES 6 This is the orthogoal projectio matrix ito the ull space of AD which is the same as the rage space of D 1 G T. Fially, let Ψ = I Ψ, v := V e, ad v 1 = V 1 e. The, the solutio of 7 is: x s ] = DΨ D 1 Ψ ] v vt v v 1 =: g 1 x, s. 10 I the reduced space we have the followig solutio: ] ] x GD = 2 G T 1 GD 1 s AD 2 A T 1 v vt v AD v 1 =: f 1 x, s. 11 All the terms o the right-had-side are expressed i terms of x ad s, which are liear fuctios of x ad s. Usig this correspodece, we express the reduced directios as a fuctio of the reduced iterates. The search directio used i 15, 16] is a Newto directio for the multiplicative aalogue of the TTY potetial-fuctio. It ca also be see as a modified Newto directio for the TTY potetial-fuctio itself. I 15], it is observed that the multiplicative aalogue of the TTY potetial-fuctio is a covex fuctio for 2 which motivates the use of Newto directios to reduce this fuctio. Because of this property, we assume that 2 whe we discuss the search directios 19 below ad trajectories associated with them. Its developmet is discussed i detail i 15] ad like the KMY directio, it ca be represeted explicitly usig orthogoal projectio matrices. Let ad Ξ = Ξx := X 1 G T GX 2 G T 1 GX 1 12 = I XA T AX 2 A T 1 AX, Σ = Σs := S 1 A T AS 2 A T 1 AS 1 13 = I SG T GS 2 G T 1 GS. We ote that Ξ ad Σ are orthogoal projectio matrices ito the rage spaces of X 1 G T ad S 1 A T, respectively. These rage spaces are the same as the ull spaces of AX ad GS, respectively. To simplify the otatio, we will suppress the depedece of Ψ, Ξ, ad Σ to x, s i our otatio. We will deote the ormalized complemetarity vector with ν: ν := x s x T s. 14 The followig quatities appear i the descriptio of the search directio used i 15]: β 1 := ν T Ξ + Σ ν, 15 β 2 := ν T Ξ + Σ e, 16 β 3 := e T Ξ + Σ e, 17 := β 1 1 β β

7 POTENTIAL REDUCTION TRAJECTORIES 7 We are ow ready to reproduce the search directios itroduced i 15]: x s ] = 1 XΣ SΞ ] 1 β 2 ν + β 1 1e =: g 2 x, s. 19 Oce agai, there is a correspodig descriptio for the reduced search directios: ] x = 1 ] GX 2 G T 1 GX 1 s AS 2 A T 1 AS 1 1 β 2 ν + β 1 1e =: f 2 x, s Potetial-Reductio Trajectories Let x 0, s 0 F 0 be ay give poit i the iterior of the feasible regio. We will study cotiuous trajectories of the vector fields f i x, s : FR 0 x0, s 0 R for i = 1, 2. That is, we will aalyze the trajectories determied by the maximal solutios of the followig autoomous ordiary differetial equatios: ] ẋ = f ṡ 1 x, s, x0, s0 = 0, 0, 21 ] ẋ = f ṡ 2 x, s, x0, s0 = 0, Note that 0, 0 FR 0 x0, s 0 for ay give x 0, s 0 F 0. Our objective i this sectio is to demostrate that there exist uique solutios to the ODEs defied above ad that their trajectories have limit poits o the primal-dual optimal face. Moteiro 9] performs a similar study of trajectories based o primal-oly potetial-reductio algorithms. We will use some of his results ad develop primal-dual versios of some others. For simplicity, we use z ad z to deote the primal ad dual pair of variables x, s i the origial space ad x, s i the reduced space. Expressios zt ad żt deote ] T ] T x T t s T t ad ẋ T t ṡ T t, respectively. A differetiable path zt : l, u F 0 is a solutio to the ODE 21 or, to the ODE 22, if żt = f 1 z żt = f 2 z for all t l, u for some scalars l ad u such that l < 0 < u. This solutio is called a maximal solutio if there is o solutio ẑt that satisfies the correspodig ODE over a set of t s that properly cotai l, u. The vector fields f 1 z ad f 2 z have the domai FR 0, a ope subset of R ad the image set is R. Note that, both f 1 z ad f 2 z are cotiuous i fact, C fuctios o F 0 R Give these properties, the followig result follows immediately from the stadard theory of ordiary differetial equatios; see, e.g., Theorem 1 o p. 162 ad Lemma o p. 171 of the textbook by Hirsch ad Smale 5]: Theorem 3.1 For each x 0, s 0 F 0 R there is a maximum ope iterval l 1, u 1 l 2, u 2 cotaiig 0 o which there is a uique solutio z 1 t of the ODE 21 z 2 t of the ODE 22.

8 POTENTIAL REDUCTION TRAJECTORIES 8 We defie correspodig trajectories i the origial problem space as follows: z 1 t := x 1 t, s 1 t := x 0, s 0 + G T x 1 t, A T s 1 t, 23 z 2 t := x 2 t, s 2 t := x 0, s 0 + G T x 2 t, A T s 2 t. 24 By differetiatio, we observe that these trajectories satisfy the followig differetial equatios, respectively: ] ẋ = g ṡ 1 x, s, x0, s0 = x 0, s 0, 25 ] ẋ = g ṡ 2 x, s, x0, s0 = x 0, s Now that we established the existece ad uiqueess of potetial-reductio trajectories, we tur our attetio to the aalysis of their covergece. The followig result holds the key to this aalysis: Propositio 3.1 Propositio 2.2, 9] Let U be a ope subset of R ad f be a cotiuously differetiable fuctio from U to R. Assume that there exists a fuctio h : U R such that for ay solutio zt of the ODE ż = fz, the composite fuctio hzt is a strictly decreasig fuctio of t. The, ay maximal solutio z : l, u U of ż = fz satisfies the followig property: Give ay compact subset S of U there exists a umber β l, u such that zt S for all t β, u. The propositio says that, if the solutio of the ODE ca be made mootoic through compositio with aother fuctio, the the trajectory ca ot remai i ay compact subset of the domai ad must ted to the boudary of the domai as t teds to its upper limit. This is a variat of a stadard ODE result see the theorem o p. 171 of 5]. This stadard result requires the fiiteess of the upper ed u of the t domai Moteiro removes this restrictio by placig the coditio about the existece of the fuctio h with the stated mootoicity property 9]. We first show that there is ideed a fuctio h satisfyig the coditio i Propositio 3.1. This fuctio is defied as hx, s = Φ x 0, s 0 + G T x, A T s. I other words, we show that the TTY potetial fuctio is decreasig alog the trajectories of the ODEs 25 ad 26: Lemma 3.1 φ 1 t := Φ x 0, s 0 + G T x 1 t, A T s 1 t = Φ x 1 t, s 1 t ad φ 2 t := Φ x 2 t, s 2 t are both decreasig fuctios of t. Let us start with φ 1 t. Assume that x 1 t, s 1 t satisfies 21 for all t l, u. The, x 1 t, s 1 t satisfies 25 for all t l, u ad for ay such t, φ 1t ] = Φ T ẋ1 t x 1 t, s 1 t = x T s s x 1 x T s x s 1 ṡ 1 t ] T DΨ D 1 Ψ ] v vt v v 1,

9 POTENTIAL REDUCTION TRAJECTORIES 9 where we wrote x ad s i place of x 1 t ad s 1 t to simplify the otatio. Let w = The, φ 1t = xt s wt X 1 DΨV 1 w xt s wt S 1 D 1 ΨV 1 w = xt s V 1 w T Ψ + Ψ = xt s V 1 w 2, V 1 w x T s xs e. where we used X 1 D = S 1 D 1 = V 1. Further, ote that w = 0 would require x T s xs = e which implies =. Sice we assumed that > for the KMY directio, we have that w 0 ad V 1 w 0 ad that φ 1 t < 0. Therefore, φ 1t is a decreasig fuctio of t. We proceed similarly for φ 2 t: ] φ 2t = Φ T ẋ2 t x 2 t, s 2 t. ṡ 2 t By costructio, the directio vector g 2 x, s itroduced i 15] satisfies g 2 x, s = 2 F x, s ] 1 F x, s where F x, s = exp{φ x, s}. Further, F x, s = F x, s Φ x, s. The fuctio F x, s is strictly covex o F 0 if 2 see 15], which we assumed, ad therefore, its Hessia is positive defiite. Thus, φ 2t = 1 1 F x, s F T x, s 2 F x, s] F x, s < 0. Above, we agai wrote x ad s i place of x 2 t ad s 2 t to simplify the otatio. Note also that Φ T x, s 0 because Φ T x, s = 0 would require xs = e which, as above, is ot x T s possible sice we assume 2. Thus, φ 2 t is also a decreasig fuctio of t. We prove two techical results before presetig the mai theorem of this sectio: Lemma 3.2 Give >, β > 0 ad M, the set is compact. S := {x, s F 0 : x T s β, Φ x, s M} 27 We first show that S is a bouded subset of F 0. Give x, s F 0 ad >, the followig iequality is well kow ad follows from the arithmetic mea-geometric mea iequality see, e.g., 17, Lemma 4.2]: Therefore, Φ x, s M implies that Φ x, s lx T s + l. M l x T s exp. 28

10 POTENTIAL REDUCTION TRAJECTORIES 10 Fix ˆx, ŝ F 0. This is possible sice we assumed that F 0 is oempty. Sice x ˆx T s ŝ = 0, we have that x T s = ˆx T s + ŝ T x ˆx T ŝ. Combiig this idetity with 28 we obtai M l ˆx T s + ŝ T x = ˆx j s j + ŝ j x j ˆx T ŝ + exp := M 1 j=1 j=1 for every x, s S. Sice all terms o the left i the iequality above are positive, we have that x j M 1 ŝ j, j, s j M 1 ˆx j, j. Thus, we proved that x ad s are bouded above if they are i S. Let U = max j {max M 1 ŝ j, M 1 ˆx j }. Now, if x, s S. Therefore, l x j = lx T s lx i s i l s j Φ x, s i j l β 2 1 l U M =: M 2, x j e M 2, j, s j e M 2, j, where the iequality for s j follows from idetical argumets. Thus, all compoets of the vectors x, s S are bouded below by a positive costat. This proves that S is a bouded subset of F 0. Sice S is clearly closed with respect to F 0 we must have that S is a compact subset of F 0. Corollary 3.1 Give >, β > 0 ad M, the set is compact. S R := {x, s F 0 R : x 0 + G T x T s 0 + A T s β, Φ x 0 + G T x, s 0 + A T s M} This is immediate sice S R is obtaied by a liear trasformatio of the compact set S. Now, we are ready to establish that the duality gap i our potetial-reductio trajectories coverges to zero: Theorem 3.2 Let z 1 t : l 1, u 1 FR 0 ad z 2t : l 2, u 2 FR 0 be maximal solutios of the ODEs 21 ad 22. Let x 1 t, s 1 t ad x 2 t, s 2 t be as i equatios The, for i = 1, 2. lim x i t T s i t = 0 t u i

11 POTENTIAL REDUCTION TRAJECTORIES 11 We prove the theorem for i = 1; the proof for i = 2 is idetical. Sice z 1 t F 0 R, x 1t, s 1 t remai i F 0 for all t l 1, u 1. Therefore, x 1 t T s 1 t > 0 for all t l 1, u 1. The, if lim t u1 x 1 t T s 1 t 0 we must have a ifiite sequece t k such that lim k t k = u 1 ad that x 1 t k T s 1 t k remais bouded away from zero. That is, there exists a positive β such that x 1 t k T s 1 t k β for all k. Sice we already established that Φ x 1 t, s 1 t is a decreasig fuctio of t, by choosig M = Φ x 1 t 0, s 1 t 0 we have that x 1 t k, s 1 t k S where S is as i 27. Usig Lemma 3.2 ad its corollary we coclude that the assumptio lim t u1 x 1 t T s 1 t 0 implies the existece of a compact subset of R to which the maximal solutio z 1 t of the ODE 21 always returs to as t teds to u 1. But this cotradicts Propositio 3.1. Therefore, the assumptio must be wrog ad we have that This completes the proof. lim x 1 t T s 1 t = 0. t u 1 4 Covergece of the Trajectories The last result of the previous sectio demostrates the covergece of the duality gap to zero alog cotiuous trajectories we associated with two differet primal-dual potetial-reductio algorithms. This theorem, however, does ot guaratee the covergece of these trajectories to a optimal solutio. All accumulatio poits of these trajectories will be optimal solutios ad there will be a uique accumulatio poit if the optimal solutio is uique. However, we eed a more careful aalysis to establish the covergece of these trajectories whe the primal-dual optimal solutio set is ot a sigleto ad this is the purpose of the curret sectio. Not surprisigly, the difficulty metioed i the previous paragraph regardig the proof of covergece of trajectories is similar to the difficulties ecoutered i covergece proofs for potetial-reductio iterior-poit algorithms. To overcome this difficulty, we wat to fid a implicit descriptio of the trajectories, perhaps similar to the descriptio of the cetral path or weighted ceters. We preset such a descriptio for trajectories associated with the Kojima-Mizuo-Yoshise algorithm. The complicated vector field associated with the algorithm i 15, 16] appears difficult to aalyze i a similar fashio ad we leave that as a topic of future research. Before proceedig with the aalysis of the Kojima-Mizuo-Yoshise trajectories, we establish that the cetral path belogs to both families of potetial-reductio trajectories we defied i the previous sectio. 4.1 The Cetral Path The cetral path C of the primal-dual feasible set F is the set of poits o which the compoetwise product of the primal ad dual variables is costat: C := {xµ, sµ F 0 : xµ sµ = µe, for some µ > 0}. 29 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., 17]. The cetral path or its subsets ca be parametrized

12 POTENTIAL REDUCTION TRAJECTORIES 12 i differet ways. For example, let φt : l, u R be a positive valued fuctio. The, we ca defie the followig parameterizatio of the correspodig subset of C: C φ := {xt, st F 0 : xt st = φte, for some t l, u}. 30 As log as the fuctio φ used i the parameterizatio of the cetral path is differetiable, the cetral path ad correspodig subsets of it are differetiable with respect to the uderlyig parameter. Oe ca easily verify that, we have the followig idetities for xt, st C φ : for t l, u. From it follows that ẋt ṡt Aẋt = 0, Gṡt = 0, 31 ẋt st + xt ṡt = φ te, 32 ] = = ] DtΨt Dt 1 φ tvt 1 33 Ψt ] DtΨt φ t Dt 1 e 34 Ψt φt where Dt, vt 1, Ψt, ad Ψt are defied idetically to D, v 1, Ψ, ad Ψ i Sectio 2, except that x ad s are replaced by xt ad st. Next, we make the followig observatio: ] DtΨt g 1 xt, st = Dt 1 vt vtt vt vt 1 Ψt ] DtΨt = Dt 1 φt 1 e. Ψt Therefore, if we choose φt such that φ t = φt 1 φt 35 we will have the subset C φ of the cetral path C satisfy the ODE 25. Let us deote the solutio of the differetial equatio 35 with φ 1 t, which is easily computed: φ 1 t = φ 1 0 exp{ 1 t}. 36 So, for ay give x 0, s 0 o the cetral path C with x 0 s 0 = µe, we ca choose φ 1 0 = µ ad the uique maximal solutio of the ODE 25 is obtaied as the trajectory C φ1 = {xt, st F 0 : xt st = µ exp{ 1 t}e, for t l 1, } for some l 1 < 0. The fact that the upper boud o the rage of t is follows from Theorem 3.2 as t teds to the upper boud, xt T st = µ exp{ 1 t} must approach zero.

13 POTENTIAL REDUCTION TRAJECTORIES 13 We ca use the same approach to show that subsets of the cetral path are solutios to the ODE 26. We first evaluate g 2 xt, st for xt, st C φ. We make the followig observatios usig xt st = φte: Therefore, ν = 1 e, Ξ = Ψt, Σ = Ψt, Ξ + Σ = I, β 1 = 1, β 2 = 1, β 3 =, = 1 1, X = φtdt, S = φtd 1 t. 1 g 2 xt, st = 1 1 φt DtΨt = Dt 1 Ψt This time we eed to choose φt such that ] φt 1 e. DtΨt Dt 1 Ψt ] 1 e φ t φt = φt 1 37 to have the subset C φ of the cetral path C satisfy the ODE 26. The solutio φ 2 t of 37 is: t φ 2 t = φ 2 0 exp{ } As above, for ay give x 0, s 0 o the cetral path C with x 0 s 0 = µe, we ca choose φ 2 0 = µ ad the uique maximal solutio of the ODE 26 is obtaied as the trajectory C φ2 = {xt, st F 0 t : xt st = µ exp{ 1 }e, for t l 2, } for some l 2 < 0. Thus, we proved the followig result: Theorem 4.1 For ay give x 0, s 0 o the cetral path C, the solutio of both the ODEs 25 ad 26 with iitial coditio x0, s0 = x 0, s 0 is a trajectory that is a subset of C. Theorem 4.1 provides a theoretical basis for the practical observatio that cetral pathfollowig search directios are ofte very good potetial-reductio directios as well. Aother related result is by Nesterov 10] who observes that the eighborhood of the cetral path is the regio of fastest decrease for a homogeeous potetial fuctio. Φ is homogeeous whe = A Characterizatio of the Trajectories Now we tur our attetio to the trajectories x 1 t, s 1 t of the ODE 25 that do ot start o the cetral path. From this poit o, we will ot cosider the trajectories x 2 t, s 2 t ad therefore, drop the subscript 1 from expressios like x 1 t, s 1 t for simplicity. Cosider

14 POTENTIAL REDUCTION TRAJECTORIES 14 xt, st for t l, u, the maximal solutio of the ODE 25. fuctio of t: We defie the followig γt := xt st hte 39 where xt st deotes the Hadamard product of the vectors xt ad st, e is a - dimesioal vector of oes, ad ht is a scalar valued fuctio of t that we will determie later. The fuctio γt is defied from l, u to R. Let φt = xt st xtt st e. Differetiatig 39 with respect to t, we obtai: γt = ẋt st + ṡt xt ḣte = V ΨV 1 φt V ΨV 1 φt ḣte = φt ḣte Now, if we were to choose ht such that we would have that This last equatio has a easy solutio: = xt st + xtt st e ḣte. ht = xtt st ḣt 40 γt = xt st + hte = γt. 41 γt = γ0 e t. Therefore, to get a better descriptio of the solutio trajectory xt, st, it suffices to fid a scalar valued fuctio ht that satisfies 40. For this purpose, we use the stadard variatio of parameters techique ad look for a solutio of the form ht = h 1 te t. The, xt T st = ht + ḣt = h 1 te t + h 1 te t h 1 te t = h 1 te t. So, settig the costat that comes from itegratio arbitrarily to 0, we obtai h 1 t = t 0 e τ xτt sτ dτ, ad ht = e t t 0 e τ xτt sτ dτ. 42 Note that, h0 = 0. Give x 0, s 0, let p := γ0 = x 0 s 0. The, γt = p e t. The equatio 42 describig ht is ot very useful to describe xt, st sice it ivolves these terms i

15 POTENTIAL REDUCTION TRAJECTORIES 15 its defiitio. Fortuately, we ca simplify 42. First, observe from 39 that xt T st = e T pe t + ht. Now, cosider t t hte t = e τ xτt sτ dτ = 0 = et p t + t hτe τ dτ. 0 0 e τ et pe τ + hτ dτ This is a itegral equatio ivolvig the fuctio hte t. Applyig the variatio of parameters techique agai ad itegratig by parts we obtai the solutio of this equatio: hte t = et p ht = et p e t 1, ad, { exp 1 } t e t. 43 Oe ca easily verify that whe x 0, s 0 is o the cetral path C with x 0 s 0 = µe, usig 43 oe recovers the followig idetity we derived i the previous subsectio: xt st = µ exp{ 1 t}e, for t l,. Usig 39 ad 43 we obtai the followig idetity: { xt T st = e T pe t + ht = e T p exp } = x0 T s0 exp { 1 t. 1 } t, 44 Propositio 4.1 Cosider xt, st for t l, u, the maximal solutio of the ODE 25 ad ht defied i 43. The, u = +. Furthermore, xt, st remais bouded as t. First part of the propositio follows immediately from 44 usig the coclusio of Theorem 3.2. For the boudedess result, observe that xt x 0 T st s 0 = 0. Therefore, x 0 jst j + s 0 jxt j j=1 { = x 0 T s 0 + x T tst = e T p 1 + exp 1 } t 2e T p. Sice all the terms i the summatio are oegative ad x 0 > 0, s 0 > 0, the result follows. Note also that lim t ht = 0. Let w j t = p j e t + ht ad wt = w j t]. Observe that w j t > 0 j, t. We ow have that Axt = b, Gst = Gc, xt st = wt, t l, u, 45 from which the followig theorem follows:

16 POTENTIAL REDUCTION TRAJECTORIES 16 Theorem 4.2 Cosider xt, st for t l, u, the maximal solutio of the ODE 25. For each t l, u, xt is the uique solutio of the followig primal barrier problem: P BP mi x c T x j=1 w j t l x j Ax = b x > Similarly, for each t l, u, st is the uique solutio of the followig dual barrier problem: LBP mi s d T s j=1 w j t l s j Gs = Gc s > Optimality coditios of both barrier problems i the statemet of the theorem are equivalet to 45; see, eg., 17]. Sice both barrier problems have strictly covex objective fuctios, their optimal solutios are uique ad satisfy 45. Therefore, xt ad st must be the optimal solutios of the respective problems. w Next, we observe that lim j t t ht = 1 for all j. I other words, the weights w j i problems PBP ad DBP are asymptotically uiform. The, problems PBP ad DBP resemble the barrier problems defiig the cetral path, ad oe might expect that the solutios of these problems coverge to the limit poit of the cetral path: the aalytic ceter of the optimal face. We will show below that this is ideed the case. First, we give relevat defiitios: Recall the partitio B ad N of the idex set of the variables that was give i Sectio 2. Let x, s = x B, 0, 0, s N deote the the aalytic ceter of the primal-dual optimal face. That is, x B ad s N are uique maximizers of the followig problems: max j B l x j A B x B = b x B > 0, ad max j N l s j G N s N = Gc s N > 0. The ext lemma will be useful i the proof of the theorem that follows it ad also i the ext sectio: Lemma 4.1 Let xt, st for t l, deote the maximal solutio of the ODE 25. The x B t ad s N t solve the followig pair of problems: max j B w jt l x j A B x B = b A N x N t x B > 0, ad max j N w jt l s j G N s N = Gc G B s B t s N > 0. We prove the optimality of x B t for the first problem i 49 the correspodig result for s N t ca be prove similarly. x B t is clearly feasible for the give problem. It is optimal if ad oly if there exists y R m such that w B t x 1 B t = AT By

17 POTENTIAL REDUCTION TRAJECTORIES 17 From 45 we obtai w B t x 1 B t = s Bt. Note that, for ay s feasible for LD we have that c s RA T ad therefore, c B s B RA T B. Furthermore, sice s = 0, s N is also feasible for LD we must have that c B RA T B ad that s Bt RA T B. This is exactly what we eeded. Now we are ready to preset the mai result of this sectio: Theorem 4.3 Let xt, st for t l, deote the maximal solutio of the ODE 25. The, xt, st coverges to the aalytic ceter of the primal-dual optimal face Ω. Propositio 4.1 idicates that the trajectory xt, st is bouded as t. Therefore, it must have at least oe accumulatio poit, say, ˆx, ŝ. Let t k be a sequece such that xt k, st k ˆx, ŝ as k ad let x k, s k deote xt k, st k. Usig Theorem 3.2, we have that ˆx, ŝ Ω. We wat to show that ˆx, ŝ = x, s. Let x, s = x, s ˆx, ŝ ad x k, s k = x k, s k + x, s. Note that x N, s B = 0 ad therefore, x k N, sk B = xk N, sk B. Note also that, xk, s k x, s, ad therefore, for k large eough x k B, sk N > 0. Thus, for such k, xk B is feasible for the first problem i 49 ad usig Lemma 4.1 we must have w j t ht l xk j w j t ht l xk j. j B j B w This iequality must hold i the limit as well. Recall that lim j t t ht = 1 for all j. Therefore, we must have that ˆx j > 0 for all j B otherwise the right-had term would ted to while the left-had term does ot, ad l x j l ˆx j. j B j B Sice x B is the uique maximizer of the first problem i 48, we coclude that ˆx B = x B. Similarly, ŝ N = s N. 5 Asymptotic Behavior of the Trajectories Our aalysis i the previous sectio revealed that all primal-dual potetial-reductio trajectories xt, st that solve the differetial equatio 25 coverge to the aalytic ceter x, s of the primal-dual optimal face Ω. 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 ẋt ẋt, ṡt ṡt. By defiitio, ẋt, ṡt satisfy equatio 25. However, it turs out to be easier to work with the equatios 45 to aalyze the limitig behavior of ˆx, ŝ. Differetiatig the idetity we obtai xt st = wt = e t p + hte, xt ṡt + ẋt st = e t p + ḣte = e t p et p 1 { exp 1 } t e + et p e t e. 50

18 POTENTIAL REDUCTION TRAJECTORIES 18 For our asymptotic aalysis, we express certai compoets of the vectors ẋt ad ṡt i terms of the others. We first eed to itroduce some otatio: ŵ B t = w B t exp{1 t}, ŵ N t = w N t exp{1 t}, d B t = ŵ 1 2 B t x 1 B t, d N t = ŵ 1 2 N t s 1 N t, D B t = diag d B t, D N t = diag d N t, d 1 B t = D 1 B te, d 1 N t = D 1 N te, à B t = A B DB 1 t, GN t = G N DN 1 t, x B t = D B tẋ B t = d B t ẋ B t, s N t = D N tṡ N t = d N t ṡ N t, p B t = ŵ 1 2 B t p B, p N t = ŵ 1 2 N t p N. Lemma 5.1 Cosider ẋt, ṡt where xt, st for t l, deote the maximal solutio of the ODE 25. The, the followig equalities hold: D B tẋ B t = Ã+ B ta N ẋ N t e t 1 e t D N tṡ N t = G + N tg Bṡ B t e t 1 e t I Ã+ B tãbt p B, 51 I G + N t G N t p N. 52 Here, à + B t ad G + N t deote the pseudo-iverse of à B t ad G N t, respectively. We oly prove the first idetity; the secod oe follows similarly. Recall from Lemma 4.1 that x B t solves the first problem i 49. Therefore, as i the proof of Lemma 4.1, we must have that w B t x 1 B t RAT B. 53 Differetiatig with respect to t we obtai: w B t x 2 B t ẋ Bt ẇ B t x 1 B t RAT B, w B t x 1 B t ẋ Bt ẇ B t RX B A T B. or, Observe that ẇ B t = p B e t + ḣte B = w B t + et p { } Therefore, lettig ŵ B t = w B t exp 1 t, we obtai e 1 t e B. ŵ B t x 1 B t ẋ Bt + ŵ B t et p e B RX B A T B. 54 From 53 it also follows that ŵ B t RX B A T B. Note also that, e T p e B = t 1 e tŵ Bt e 1 e tp B.

19 POTENTIAL REDUCTION TRAJECTORIES 19 Combiig these observatios with 54 we get Note also that ŵ B t x 1 B t ẋ Bt + e t 1 e tp B RX B A T B. 55 A B ẋ B t = A N ẋ N t. 56 Usig the otatio itroduced before the statemet of the lemma, 55 ad 56 ca be rewritte as follows: x B t + 1 e t p B t RÃT B, 57 e t à B t x B t = A N ẋ N t. 58 Let Ã+ B t deote the pseudo-iverse of ÃBt 3]. For example, if rakãbt=m, the Ã+ B t = à T B t ÃB tãt B t 1. The, PR à T B := Ã+ B tãbt is the orthogoal projectio matrix oto RÃT B ad P N ÃB := I Ã+ B tãbt is the orthogoal projectio matrix oto N ÃB 3]. From 58 we obtai P R à T B x Bt = Ã+ B tãbt x B t = Ã+ B ta N ẋ N t, ad from 57, usig the fact that RÃT B ad N ÃB are orthogoal to each other, we get Combiig, we have P N à x B Bt = e t 1 e t x B t = P R à T B x Bt + P N à B x Bt = Ã+ B ta N ẋ N t e t 1 e t I Ã+ B tãbt p B t. I Ã+ B tãbt p B t, which gives 51. Next, we compute limits of some of the expressios that appear i equatios 51 ad 52: lim t ŵ B t = et p e B, lim t D B t = lim t à B t = e T p A BX B, lim t ŵ N t = et p e N, 59 e T p S N 1, 60 e T p X B 1, lim t D N t = lim t p B t = e T p p B, lim t G N t = e T p G N S N, 61 lim t p N t = e T p p N. 62

20 POTENTIAL REDUCTION TRAJECTORIES 20 Lemma 5.2 lim à + t B t = e T p A BXB + 63 lim G + t N t = e T p G N SN This result about the limitig properties of the pseudo-iverses is a immediate cosequece of Lemma 2.3 i 4] ad equatios 61. Cosider the ormalized directio vectors ˆx, ŝ which are defied as follows: ˆxt = exp From 50 it follows that { 1 t xt ŝt + ˆxt st = et p } ẋt, ad ŝt = exp { 1 t } ṡt { e + exp } t e T p e p. 66 This last equatio idicates that xt ŝt ad ˆxt st remai bouded as t. Sice x B t ad s N t coverge to x B > 0 ad s N > 0 respectively, ad therefore, remai bouded away from zero, we must have that ˆx N t ad ŝ B t remai bouded. This observatio leads to the followig coclusio: Lemma 5.3 Let ˆxt, ŝt be as i 65 ad assume that 2. The, ˆxt, ŝt remais bouded as t teds to. From 51 we have that 2 t D B tˆx B t = Ã+ B ta N ˆx N t e 1 1 e t I Ã+ B tãbt p B. 67 Whe, 2, the factor e 1 2 t 1 e t is coverget as t teds to. Now, usig Lemma 5.2 ad the equatios 61-62, we coclude that the secod term i the right-had-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 B tˆx B t remais bouded. Usig 60 we coclude that ˆx B 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 5.1 Let ˆxt, ŝt be as i 65 ad assume that 2. The, we have that lim t ˆx N t ad lim t ŝ B t exist ad satisfy the followig equatios: lim ˆx N t = et p 1 s t N 1, 68 lim ŝbt = et p t 1 x B 1. 69

21 POTENTIAL REDUCTION TRAJECTORIES 21 From 66 we have that x B t ŝ B t + ˆx B t s B t = et p Takig the limit o the right-had-side as t we obtai et p 1 { e B + exp } t e T p e B p B. ad ˆx B t is bouded, we must the have that x B t ŝ B t coverges to et p e B. Sice s B t 0 e B. Sice x B t x B, it follows that lim t ŝ B t exists ad satisfies 69. The correspodig result for ˆx N t follows idetically. Let π B = X B ξ B = XB A BXB + A N s N 1, σ N = SN G N SN + G B x B 1 I A B XB + A B XB p B, π N = SN I G N S N + G N S N p N. Theorem 5.2 Let ˆxt, ŝt be as i 65 ad assume that 2. The, we have that lim t ˆx B t ad lim t ŝ N t exist. Whe < 2 we have the followig idetities: lim ˆx Bt = et p 1 ξ B, 70 t lim t ŝn t = et p Whe = 2, the followig equatios hold: 1 σ N. 71 lim ˆx Bt = et p t 2 ξ B 2e T p π B, 72 lim t ŝn t = et p 2 σ N 2e T p π N. 73 Recall equatio 67. Whe < 2, the secod term o the right-had-side coverges to 1 2 zero sice e t teds to zero ad everythig else is bouded. Thus, usig 59 we have lim t ˆx B t = XB A BXB + A N lim t ˆx N t ad 70 is obtaied usig Theorem 5.1. Similarly, oe obtais 71. Whe = 2, the factor i frot of the secod term i 67 coverges to the positive costat = 1 2. Therefore, usig Theorem 5.1 ad equatios we get 72 ad 73. Limits of the ormalized vectors 5.1 ad 5.2: ẋt ẋt, ṡt ṡt are obtaied immediately from Theorems Corollary 5.1 Let xt, st for t l, deote the maximal solutio of the ODE 25 for a give x 0, s 0 F 0 ad assume that 2. All trajectories of this form satisfy the followig equatios: lim t ẋt ẋt = q P q P, lim t ṡt ṡt = q D q D 74

22 POTENTIAL REDUCTION TRAJECTORIES 22 where q P = ξ q P = B s N 1 2 ξ B e p T πb s N 1 ], ad q D = x B 1 σ N, ad q D = x B 1 ], if < 2, e T p 2 πn σ N, if = 2. Whe = 2 the TTY potetial-fuctio Φ x, y is a homogeeous fuctio ad exp{φ x, y} is a covex fuctio for all 2. The value 2 also represets a threshold value for the covergece behavior of the KMY trajectories. Whe = 2 the directio of covergece depeds o the iitial poit x 0, s 0 F 0 as idicated by the appearace of the p = x 0 s 0 terms i the formulas. We ote that, whe < 2 the asymptotic directio of covergece does ot deped o the iitial poit ad is idetical to that of the cetral path. Therefore, whe < 2 all trajectories of the vector field give by the search directio of the Kojima- Mizuo-Yoshise s primal-dual 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 > 2: Theorem 5.3 Cosider ẋt, ṡt where xt, st for t l, deote the maximal solutio of the ODE 25 ad assume that > 2. Defie xt = exp{ t}ẋt, ad st = exp{ t}ṡt. 75 The, we have that lim t xt ad lim t st exist ad satisfy the followig equatios: lim t xt xt = q P, ad lim q P t st st = q D q D, 76 where πb ] 0B ] q P = 0 N, ad q D = π N. From 51 we have that D B t x B t = Ã+ B ta N x N t 1 e t I Ã+ B tãbt p B. 77 Note that, x N t = e 1 2 tˆx N t. Sice ˆx N t is bouded ad 1 2 < 0, we coclude that x N t 0. Now, the first equatio i 76 follows from equatios 77 ad The secod idetity i 76 is obtaied similarly. This fial theorem idicates that whe > 2, the trajectories associated with the Kojima- Mizuo-Yoshise algorithm will coverge to the aalytic ceter of the optimal face tagetially to the optimal face.

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24 POTENTIAL REDUCTION TRAJECTORIES 24 16] R. H. Tütücü, Quadratic covergece of potetial-reductio methods for degeerate problems, Mathematical Programmig, pp ] S. Wright, Primal-dual iterior-poit methods SIAM, Philadelphia, 1997.