McMaster Uiversity Advaced Optimizatio Laboratory Title: How good are iterior poit methods? Klee-Mity cubes tighte iteratio-complexity bouds Authors: Atoie Deza, Eissa Nematollahi ad Tamás Terlaky AdvOl-Report No. 2004/20 December 2004, Hamilto, Otario, Caada
How good are iterior poit methods? Klee-Mity cubes tighte iteratio-complexity bouds. Dedicated to Professor Emil Klafszky o the occasio of his 70th birthday Atoie Deza Eissa Nematollahi Tamás Terlaky December 3, 2004 Abstract By refiig a variat of the Klee-Mity example that forces the cetral path to visit all the vertices of the Klee-Mity -cube, we exhibit a early worst-case example for pathfollowig iterior poit methods. Namely, while the theoretical iteratio-complexity upper boud is O(2 5 2), we prove that solvig this -dimesioal liear optimizatio problem requires at least 2 1 iteratios. Key words: Liear programmig, iterior poit method, worst-case iteratio-complexity. MSC2000 Subject Classificatio: Primary: 90C05; Secodary: 90C51, 90C27, 52B12 1 Itroductio While the simplex method, itroduced by Datzig [1], works very well i practice for liear optimizatio problems, i 1972 Klee ad Mity [5] gave a example for which the simplex method takes a expoetial umber of iteratios. More precisely, they cosidered a maximizatio problem over a -dimesioal squashed cube ad proved that a variat of the simplex method visits all its 2 vertices. Thus, the time complexity is ot polyomial for the worst case, as 2 1 iteratios are ecessary for this -dimesioal liear optimizatio problem. The pivot rule used i the Klee-Mity example was the most egative reduced cost but variats of the Klee-Mity -cube allow to prove expoetial ruig time for most pivot rules; see [10] ad the refereces therei. The Klee-Mity worst-case example partially stimulated the search for a polyomial algorithm ad, i 1979, Khachiya s [4] ellipsoid method proved that liear programmig is ideed polyomially solvable. I 1984, Karmarkar [3] proposed a more efficiet polyomial algorithm that sparked the research o polyomial iterior poit methods. I short, while the simplex method goes alog the edges of the polyhedro correspodig to the feasible regio, iterior poit methods pass through the iterior of this polyhedro. Startig at the aalytic ceter, most iterior poit methods follow the so-called cetral path ad coverge to the aalytic ceter of the optimal face; see e.g., [6, 8, 9, 13, 14]. I 2004, Deza, Nematollahi, Peyghami ad Terlaky [2] showed that, by carefully addig a expoetial umber of redudat costraits to the Klee-Mity -cube, the cetral path ca be severely distorted. Specifically, they provided a example for which path-followig iterior poit methods have to take 2 2 sharp turs as the cetral path passes withi a arbitrarily small eighborhood of the correspodig vertices of 2
Atoie Deza, Eissa Nematollahi ad Tamás Terlaky 3 the Klee-Mity cube before covergig to the optimal solutio. This example yields a theoretical lower boud for the umber of iteratios eeded for path-followig iterior poit methods: The umber of iteratios is at least the umber of sharp turs; that is, the iteratio-complexity lower boud is Ω(2 ). O the other had, the theoretical iteratio-complexity upper boud is O( NL) where N ad L respectively deote the umber of costraits ad the bit-legth of the iput-data. The iteratio-complexity upper boud for the highly redudat Klee-Mity -cube of [2] is O(2 3 L) = O(2 9 4 ), as N = O(2 6 2 ) ad L = O(2 6 3 ) for this example. Therefore, these 2 1 sharp turs yield a Ω( 6 N ) iteratio-complexity lower boud. I this paper we show that a refied problem with the same Ω(2 ) iteratio-complexity lower boud exhibits a early worst-case iteratio-complexity as the complexity upper boud is O(2 2). 5 I other words, this ew example, with N = O(2 2 3 ), essetially closes the iteratio-complexity gap with a Ω( l 2 N N l 3 N ) lower boud ad a O( N l N) upper boud. 2 Notatios ad the Mai Results We cosider the followig Klee-Mity variat where ε is a small positive factor by which the uit cube [0,1] is squashed. mi x subject to 0 x 1 1 εx k 1 x k 1 εx k 1 for k = 2,...,. The above miimizatio problem has 2 costraits, variables ad the feasible regio is a -dimesioal cube deoted by C. Some variats of the simplex method take 2 1 iteratios to solve this problem as they visit all the vertices ordered by the decreasig value of the last coordiate x startig from v {} = (0,...,0,1) till the optimal value x = 0 is reached at the origi v. While addig a set h of redudat iequalities does ot chage the feasible regio, the aalytic ceter χ h ad the cetral path are affected by the additio of redudat costraits. We cosider redudat iequalities iduced by hyperplaes parallel to the facets of C cotaiig the origi. The costrait parallel to the facet H 1 : x 1 = 0 is added h 1 times at a distace ad the costrait parallel to the facet H k : x k = εx k 1 is added h k times at a distace d k for k = 2,...,. The set h is deoted by the iteger-vector h = (h 1,...,h ), d = (,...,d ), ad the redudat liear optimizatio problem is defied by mi x subject to 0 x 1 1 εx k 1 x k 1 εx k 1 for k = 2,..., 0 + x 1 repeated h 1 times εx 1 d 2 + x 2 repeated h 2 times. εx 1 d + x. repeated h times.
4 How good are iterior poit methods? By aalogy with the uit cube [0,1], we deote the vertices of the Klee-Mity cube C by usig a subset S of {1,...,}. For S {1,...,}, a vertex v S of C is defied by v S 1 = v S k = { 1, if 1 S 0, otherwise { 1 εv S k 1, if k S εvk 1 S, otherwise k = 2,...,. The δ-eighborhood N δ (v S ) of a vertex v S is defied, with the covetio x 0 = 0, by N δ (v S ) = { { 1 xk εx x C : k 1 ε k 1 δ, if k S x k εx k 1 ε k 1 δ, otherwise } k = 1,...,. v {2} v {1,2} v {1} v Figure 1: The δ-eighborhoods of the 4 vertices of the Klee-Mity 2-cube. I this paper we focus o the followig problem C δ defied by ε = 2(+1), d = (2 +4,...,2 k+5,...,2 5 ), h = ( 22+8 (+1) δ 1 where 0 < δ 1 4(+1). 2+7 (+1) δ,..., 22+8 (+1) +k 1 δ +k 2 2+k+6 (+1) 2k 1 δ 2k 2 ),..., 3 22+6 (+1) 2 1 δ, 2 2 Note that we have: ε+δ < 1 2 ; that is, the δ-eighborhoods of the 2 vertices are o-overlappig, ad that h is, up to a floor operatio, liearly depedet o δ 1. Propositio 2.1 states that, for C δ, the cetral path takes at least 2 2 turs before covergig to the origi as it passes through the δ-eighborhood of all the 2 vertices of the Klee-Mity -cube; See Sectio 3.2 for the proof. Note that the proof give i Sectio 3.2 yields a slightly stroger result tha Propositio 2.1: I additio to pass through the δ-eighborhood of all the vertices, the cetral path is bet alog the edge-path followed by the simplex method. We set δ = 1 4(+1) i Propositios 2.3 ad 2.4 i order to exhibit the sharpest bouds. The correspodig liear optimizatio problem C 1/4(+1) depeds oly o the dimesio.
Atoie Deza, Eissa Nematollahi ad Tamás Terlaky 5 Propositio 2.1. The cetral path P of C δ -cube. itersects the δ-eighborhood of each vertex of the Sice the umber of iteratios required by path-followig iterior poit methods is at least the umber of sharp turs, Propositio 2.1 yields a theoretical lower boud for the iteratiocomplexity for solvig this -dimesioal liear optimizatio problem. Corollary 2.2. For C δ, the iteratio-complexity lower boud of path-followig iterior poit methods is Ω(2 ). Sice the theoretical iteratio-complexity upper boud for path-followig iterior poit methods is O( NL), where N ad L respectively deote the umber of costraits ad the bit-legth of the iput-data, we have: Propositio 2.3. For C 1/4(+1), the iteratio-complexity upper boud of path-followig iterior poit methods is O(2 2L); 3 that is, O(2 3 11 2 ). Proof. We have N = 2 + k=1 h k = 2 + ( k=1 2 2 2+10 ( ) +1 +k 2 +k+8 ( ) ) +1 2k ad, sice ) +k k=1 e 2, we have N = O(2 2 3 ) ad L N l = O(2 2 4 ). ( +1 Noticig that the oly two vertices with last coordiates smaller tha or equal to ε 1 are v ad v {1}, with v = 0 ad v {1} = ε 1, the stoppig criterio ca be replaced by: stoppig duality gap smaller tha ε with the correspodig cetral path parameter at the stoppig poit beig µ = ε N. Additioally, oe ca check that by settig the cetral path parameter to µ0 = 1, we obtai a startig poit which belogs to the iterior of the δ-eighborhood of the highest vertex v {}, see Sectio 3.3 for a detailed proof. I other words, a path-followig algorithm usig a stadard ǫ-precisio as stoppig criterio ca stop whe the duality gap is smaller tha ε as the optimal vertex is idetified, see [8]. The correspodig iteratio-complexity boud O( N log N ǫ ) yields, for our costructio, a precisio-idepedet iteratio-complexity O( N l Nµ0 Nµ ) = O( N) ad Propositio 2.3 ca therefore be stregtheed to: Propositio 2.4. For C 1/4(+1), the iteratio-complexity upper boud of path-followig iterior poit methods is O(2 5 2). Remark 2.5. (i) For C 1/4(+1), by Corollary 2.2 ad Propositio 2.4, the order of the iteratio-complexity of path-followig iterior poit methods is betwee 2 ad 2 5 2 or, equivaletly, betwee N l 3 N ad N l N. (ii) The k-th coordiate of the vector d correspods to the scalar d defied i [2] for dimesio k + 3. (iii) Other settigs for d ad h esurig that the cetral path visits all the vertices of the Klee-Mity -cube are possible. For example, d ca be set to (1.1,22) i dimesio 2. (iv) Our results apply to path-followig iterior poit methods but ot to other iterior poit methods such as Karmarkar s origial projective algorithm [3].
6 How good are iterior poit methods? Remark 2.6. (i) Megiddo ad Schub [7] proved, for affie scalig trajectories, a result with a similar flavor as our result for the cetral path, ad oted that their approach does ot exted to projective scalig. They cosidered the o-redudat Klee-Mity cube. (ii) Todd ad Ye [11] gave a Ω( 3 N) iteratio-complexity lower boud betwee two updates of the cetral path parameter µ. (iii) Vavasis ad Ye [12] provided a O(N 2 ) upper boud for the umber of approximately straight segmets of the cetral path. 3 Proofs of Propositio 2.1 ad Propositio 2.4 3.1 Prelimiary Lemmas Lemma 3.1. With b = 4 δ (1,...,1), ε = ( h = 2 2+8 (+1) 2+7 (+1) δ 1 δ,..., 22+8 (+1) +k 1 δ +k 2 we have A h 3b 2. A = 1 +1 1 ε 2(+1), d = (2+4,...,2 k+5,...,2 5 ), 2+k+6 (+1) 2k 1,...,3 22+6 (+1) 2 1 δ 2k 2 d 2 0 0... 0 0 2ε d 2 +1 ε 2 d 3 0... 0 0........... 1 2ε 0 0 k 1 ε k d k +1 d k+1 0 0.......... 0 1 2ε 0 0 0... 2 d 1 +1 ε 1 d 1 2ε 0 0 0... 0 1 d +1 δ 2 2 ) Proof. As ε = 2(+1) ad d = (2+4,...,2 k+5,...,2 5 ), h ca be rewritte as ( ) h = 4 δ ( 4 ε 1 ε ),..., d k ( 4 ε k 1 ε 1 d ),..., 3 ε k ε 1 ε ad A h 3b 2 ca be rewritte as 4 δ ( 4 ε 1 ε ) + 4 δ which is equivalet to ( 1 ε 2 1 ε 3 2 ( 1 ε k+1 2 ε k + 1 ε 3 2 4 δ + 1 ( 4 ε 1 ε ) 4 δ ( 4 ε 1 ε 2) 2d k d k + 1 ( 4 ε 1 ε k ) 4 δ ( 4 ε 1 ε k+1) ( 2 ε + 1 ε 3 2 4 δ ( 4 ε 1 ε ) + 4 δ ) ) 4 ε 1 ε 2 + 3 2 6 δ 6 δ 2d 3 d + 1 ε 6 δ, d k 8 ε 1 ε k+1 1 ε + 3 2 ) d 4 ε 1 ε + 3 2., for k = 2,..., 1 for k = 2,..., 1 ad
Atoie Deza, Eissa Nematollahi ad Tamás Terlaky 7 As 1 ε 1 2 ε 3 2 1 2, 1 ε 3 2 0, 1 ε 3 2 2 0 a + 1 ε k+1 ε 3 2 0, the above system is implied by 1 2 4 ( ε 1 ε k+1 2 ) ε k d k 8 ε for k = 2,..., 1 2 ε d 4 ε, 1 as 2 = 2 a = 2 k ( 1 + 1 ) k ε k+1 ε k ε k ε k 2 k+2, the above system is implied by 2 +5 d k 2 k+4 for k = 2,..., 1 d 2 which is true sice d = (2 +4,...,2 k+5,...,2 5 ). Corollary 3.2. With the same assumptios as i Lemma 3.1 ad h = h, we have Ah b. Proof. Sice 0 h k h k < 1 ad d k = 2 k+5, we have: h 1 h 1 + 2( h k h k )ε k 1 d k +1 h 1 h 1 h 1 h 1 +1 ( h 2 h 2 )ε d 2 2 δ ( h k+1 h k+1 )ε k d k+1 2 δ for k = 2,..., 1 + 2( h h )ε 1 d +1 2 δ thus, A( h h) b 2, which implies, sice A h 3b 2 by Lemma 3.1, that Ah b. Corollary 3.3. With the same assumptios as i Lemma 3.1 ad h = h, we have: h kε k 1 d k +1 h k+1 ε k d k+1 + 4 δ for k = 1,..., 1. Proof. For k = 1,..., 1, oe ca easily check that the first k iequalities of Ah b imply h k ε k 1 d k +1 h k+1ε k d k+1 + 4 δ. The aalytic ceter χ = (ξ1,...,ξ ) of C δ is the uique solutio to the problem cosistig of maximizig the product of the slack variables: s 1 = x 1 s k = x k εx k 1 for k = 2,..., s 1 = 1 x 1 s k = 1 εx k 1 x k for k = 2,..., s 1 = + s 1 repeated h 1 times. s = d + s. repeated h times.
8 How good are iterior poit methods? Equivaletly, χ is the solutio of the followig maximizatio problem: max x i.e., with the covetio x 0 = 0, max x k=1 (l s k + l s k + h k l s k ), k=1 ( ) l(x k εx k 1 ) + l(1 εx k 1 x k ) + h k l(d k + x k εx k 1 ). The optimality coditios (the gradiet is equal to zero at optimality) for this cocave maximizatio problem give: where 1 σ k ε σk+1 1 σ k ε σ k+1 + h k σ k h k+1ε σ k+1 1 σ 1 σ + h = 0 for k = 1,..., 1 σ = 0 (1) σk > 0, σ k > 0, σ k > 0 for k = 1,...,, σ 1 = ξ 1 σ k = ξ k εξ k 1 for k = 2,..., σ 1 = 1 ξ 1 σ k = 1 εξ k 1 ξ k for k = 2,..., σ k = d k + σ k for k = 1,...,. The followig lemma states that, for C δ, the aalytic ceter χ belogs to the eighborhood of the vertex v {} = (0,...,0,1). Lemma 3.4. For C δ, we have: χ N δ (v {} ). Proof. Addig the -th equatio of (1) multiplied by ε 1 to the j-th equatio of (1) multiplied by ε j 1 for j = k..., 1, we have, for k = 1,..., 1, ε k 1 σ k εk 1 σ k 2ε 1 σ 2 2 i=k ε i σ i+1 + h kε k 1 σ k 2h ε 1 σ = 0, implyig: 2h ε 1 σ h kε k 1 σ k = εk 1 σ k ( ε k 1 σ k + 2ε 1 σ 2 + 2 i=k ε i σ i+1 ) εk 1 σk, which implies, sice σ d + 1, σ k d k ad h 1 h kε k 1 d k by Corollary 3.3, 2h ε 1 d + 1 h 1 εk 1, σ k
Atoie Deza, Eissa Nematollahi ad Tamás Terlaky 9 implyig, sice 2hε 1 d +1 h 1 -th equatio of (1) implies: h ε 1 σ 1 δ by Corollary 3.2, σ k εk 1 δ for k = 1,..., 1. The ; that is, sice σ < d + 1 ad hε 1 d +1 1 δ by ε 1 σ Corollary 3.2, we have: 1 δ hε 1 d +1 ε 1 σ, implyig: σ ε 1 δ. The cetral path P of C δ ca be defied as the set of aalytic ceters χ (α) = (x 1,...,x 1,α) of the itersectio of the hyperplae H α : x = α with the feasible regio of C δ where 0 < α ξ, see [8]. These itersectios Ω(α) are called the level sets ad χ (α) is the solutio of the followig system: where { 1 s k ε s k+1 1 s k ε s + h k k+1 s h k+1ε k s = 0 for k = 1,..., 1 k+1 s k > 0, s k > 0, s k > 0 for k = 1,..., 1, (2) s 1 = x 1 s k = x k εx k 1 for k = 2,..., 1 s = α εx 1 s 1 = 1 x 1 s k = 1 εx k 1 x k for k = 2,..., 1 s = 1 α εx 1 s k = d k + s k for k = 1,...,. Lemma 3.5. For C δ, Ck δ = {x C : s k ε k 1 δ,s k ε k 1 δ} ad Ĉk δ = {x C : s k 1 ε k 2 δ,s k 2 ε k 3 δ,...,s 1 δ}, we have: Cδ k P Ĉk δ for k = 2,...,. Proof. Let x Cδ k P. Addig the (k 1)-st equatio of (2) multiplied by εk 2 to the i-th equatio of (1) multiplied by ε i 1 for i = j...,k 2, we have, for k = 2,..., 1, 2h k 1ε k 2 s + h jε j 1 k 1 s + h kε k 1 j s + εj 1 k s + εk 1 j s + εk 1 k s 2εk 2 k s + εj 1 k 3 ε i k 1 s + 2 j s = 0, i+1 which implies, sice s k 1 < d k 1 + 1, s j > d j, s k > d k ad s k εk 1 δ ad s k εk 1 δ as x C k δ, 2h k 1 ε k 2 d k 1 + 1 h jε j 1 h kε k 1 d j d k implyig, sice h 1 h jε j 1 d j by Corollary 3.3, εj 1 s j + 2 δ, i=j h 1 + 2h k 1ε k 2 d k 1 + 1 h kε k 1 d k εj 1 s j + 2 δ, that is, as 3 δ h 1 + 2h k 1ε k 2 d k 1 +1 h kε k 1 d k equatio of (2), we have by Corollary 3.2: s j εj 1 δ. Cosiderig the (k 1)-st h k 1 ε k 2 s k 1 h kε k 1 s k = εk 2 s + εk 1 k 1 s + εk 1 k s k εk 2 s, k 1
10 How good are iterior poit methods? which implies, sice s k 1 < d k 1 + 1, s k > d k ad s k εk 1 δ ad s k εk 1 δ as x C k δ, h k 1 ε k 2 d k 1 + 1 h kε k 1 d k which implies, sice 3 δ h k 1ε k 2 d k 1 +1 h kε k 1 d k x Ĉk δ. εk 2 s + 2 k 1 δ, by Corollary 3.3, that s k 1 εk 2 δ ad, therefore, 3.2 Proof of Propositio 2.1 Figure 2: The set P δ for the Klee-Mity 3-cube. For k = 2,...,, while Cδ k, defied i Lemma 3.5, ca be see as the cetral part of the cube C, the sets Tδ k = {x C : s k ε k 1 δ} ad Bδ k = {x C : s k ε k 1 δ}, ca be see, respectively, as the top ad bottom part of C. Clearly, we have C = Tδ k Ck δ Bk δ for each k = 2,...,. Usig the set Ĉk δ defied i Lemma 3.5, we cosider the set Ak δ = T δ k Ĉk δ Bk δ for k = 2...,, ad, for 0 < δ 1 4(+1), we show that the set P δ = k=2 Ak δ, see Figure 2, cotais the cetral path P. By Lemma 3.4, the startig poit χ of P belogs to N δ (v {} ). Sice P C ad C = k=2 (T δ k Ck δ Bk δ ), we have: P = C P = (Tδ k Ck δ Bk δ ) P = (Tδ k (Ck δ P) Bk δ ) P, k=2 k=2 that is, by Lemma 3.5, P (Tδ k Ĉk δ Bk δ ) = A k δ = P δ k=2 k=2
Atoie Deza, Eissa Nematollahi ad Tamás Terlaky 11 T 3 0 B 2 0 Ĉ 2 0 T 2 0 Ĉ 3 0 B 3 0 A 2 0 A 3 0 Figure 3: The sets A 2 0 ad A3 0 for the Klee-Mity 3-cube. Remark that the sets Cδ k,ĉk δ,t δ k,bk δ ad Ak δ ca be defied for δ = 0, see Figure 3, ad that the correspodig set P 0 = k=2 Ak 0 is precisely the path followed by the simplex method o the origial Klee-Mity problem as it pivots alog the edges of C. The set P δ is a δ-sized (cross sectio) tube alog the path P 0. See Figure 4 illustratig how P 0 starts at v {}, decreases with respect to the last coordiate x ad eds at v. v {3} v {1,3} v {1,2,3} v {2,3} v {2} v v {1} v {1,2} Figure 4: The path P 0 followed by the simplex method for the Klee-Mity 3-cube. 3.3 Proof of Propositio 2.4 We cosider the poit x of the cetral path which lies o the boudary of the δ-eighborhood of the highest vertex v {}. This poit is defied by: s 1 = δ,s k ε k 1 δ for k = 2,..., 1 ad s 2 ε δ. Note that the otatio s k for the cetral path (perturbed complemetarity) coditios, y k s k = µ for k = 1,...,p, is cosistet with the slacks itroduced after Corollary 3.3 with s +k = s k for k = 1,..., ad s pi +k = s k for k = 1,...,h i+1 ad i = 0,..., 1. Let µ deote the cetral path parameter correspodig to x. I the followig, we prove that µ ε 1 δ which implies that ay poit of the cetral path with correspodig parameter µ µ belog to the iterior of the δ-eighborhood of the highest vertex v {}. I particular, it implies
12 How good are iterior poit methods? that by settig the cetral path parameter to µ 0 = 1, we obtai a startig poit which belogs to the iterior of the δ-eighborhood of the vertex v {}. Estimatio of the cetral path parameter µ: The formulatio of the dual problem of C δ is: max z = 2 p k y k d k y i k=+1 k=1 i=p k 1 +1 subject to y k εy k+1 y +k εy +k+1 + p k i=p k 1 +1 y i ε y y 2 + where p 0 = 2 ad p k = 2 + h 1 + + h k for k = 1,...,. p k+1 i=p k +1 p i=p 1 +1 y i = 0 for k = 1,..., 1 y i = 1 y k 0 for k = 1,...,p, For k = 1,...,, multiplyig by ε k 1 the k-th equatio of the above dual costraits ad summig the up, we have: which implies y 1 y +1 2 ( 2+h εy +2 + ε 2 y +3 + + ε 1 ) 1 y 2 + 2ε 1 y 2 y 1 + 2+h 1 i=2+1 y i i=2+1 implyig, sice for i = 2 + 1,...,2 + h 1, s i yields y i µ, that 2ε 1 y 2 y 1 + µh 1 = µ δ + µh 1. y i = ε 1 Sice for i = p 1 + 1,...,p, s i = d + x ε x 1 d + 1 yields y i µ costrait implies p y 2 y i 1 µh d + 1 1 i=p 1 +1 ( ) which, combied with the previously obtaied iequality, gives µ 2hε 1 d +1 h 1 1 δ ad, sice Corollary 3.2 gives 2hε 1 d +1 h 1 1 δ 2 δ, we have µ ε 1 δ. d +1, the last dual 2ε 1, Ackowledgmets. We would like to thak a associate editor ad the referees for poitig out the paper [7] ad for helpful commets ad correctios. May thaks to Yiyu Ye for
Atoie Deza, Eissa Nematollahi ad Tamás Terlaky 13 precious suggestios ad hits which triggered this work ad for iformig us about the papers [11, 12]. Research supported by a NSERC Discovery grat ad a MITACS grat for the last two authors, by the Caada Research Chair program for the first ad last authors ad by a NSERC Discovery grat for the first author. Refereces [1] G. B. Datzig: Maximizatio of a liear fuctio of variables subject to liear iequalities. I: T. C. Koopmas (ed.) Activity Aalysis of Productio ad Allocatio. Joh Wiley (1951) 339 347. [2] A. Deza, E. Nematollahi, R. Peyghami ad T. Terlaky: The cetral path visits all the vertices of the Klee-Mity cube. Optimizatio Methods ad Software (to appear). [3] N. K. Karmarkar: A ew polyomial-time algorithm for liear programmig. Combiatorica 4 (1984) 373 395. [4] L. G. Khachiya: A polyomial algorithm i liear programmig. Soviet Mathematics Doklady 20 (1979) 191 194. [5] V. Klee ad G. J. Mity: How good is the simplex algorithm? I: O. Shisha (ed.) Iequalities III, Academic Press (1972) 159 175. [6] N. Megiddo: Pathways to the optimal set i liear programmig. I: N. Megiddo (ed.) Progress i Mathematical Programmig: Iterior-Poit ad Related Methods, Spriger- Verlag (1988) 131 158; also i: Proceedigs of the 7th Mathematical Programmig Symposium of Japa, Nagoya, Japa (1986) 1 35. [7] N. Megiddo ad M. Shub: Boudary behavior of iterior poit algorithms i liear programmig. Mathematics of Operatios Research 14-1 (1989) 97 146. [8] C. Roos, T. Terlaky ad J-Ph. Vial: Theory ad Algorithms for Liear Optimizatio: A Iterior Poit Approach. Wiley-Itersciece Series i Discrete Mathematics ad Optimizatio. Joh Wiley (1997). [9] G. Soeved: A aalytical cetre for polyhedros ad ew classes of global algorithms for liear (smooth, covex) programmig. I: A. Prékopa, J. Szelezsá, ad B. Strazicky (eds.) System Modellig ad Optimizatio: Proceedigs of the 12th IFIP-Coferece, Budapest 1985. Lecture Notes i Cotrol ad Iformatio Scieces 84 Spriger Verlag (1986) 866 876. [10] T. Terlaky ad S. Zhag: Pivot rules for liear programmig - a survey. Aals of Operatios Research 46 (1993) 203 233. [11] M. Todd ad Y. Ye: A lower boud o the umber of iteratios of log-step ad polyomial iterior-poit liear programmig algorithms. Aals of Operatios Research 62 (1996) 233 252. [12] S. Vavasis ad Y. Ye: A primal-dual iterior-poit method whose ruig time depeds oly o the costrait matrix. Mathematical Programmig 74 (1996) 79 120.
14 How good are iterior poit methods? [13] S. J. Wright: Primal-Dual Iterior-Poit Methods. SIAM Publicatios (1997). [14] Y. Ye: Iterior-Poit Algorithms: Theory ad Aalysis. Wiley-Itersciece Series i Discrete Mathematics ad Optimizatio. Joh Wiley (1997). Atoie Deza, Eissa Nematollahi, Tamás Terlaky Advaced Optimizatio Laboratory, Departmet of Computig ad Software, McMaster Uiversity, Hamilto, Otario, Caada. Email: deza, ematoe, terlaky@mcmaster.ca