How good are interior point methods? Klee Minty cubes tighten iteration-complexity bounds

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1 Math Program, Ser A ( :1 14 DOI /s x FULL LENGTH PAPER How good are iterior poit methods? Klee Mity cubes tighte iteratio-complexity bouds Atoie Deza Eissa Nematollahi Tamás Terlaky Received: 7 Jauary 2005 / Revised: 16 August 2006 / Published olie: 21 October 2006 Spriger-Verlag 2006 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 path-followig 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 Keywords Liear programmig Iterior poit method Worst-case iteratio-complexity Mathematics Subject Classificatio (2000 Primary 90C05; Secodary 90C51 90C27 52B12 1 Itroductio While thesimplex method, itroduced by Datzig [1], works very well i practice for liear optimizatio problems, i 1972 Klee ad Mity [6] gave a example for which the simplex method takes a expoetial umber of iteratios Dedicated to Professor Emil Klafszky o the occasio of his 70th birthday A Deza E Nematollahi T Terlaky (B Advaced Optimizatio Laboratory, Departmet of Computig ad Software, McMaster Uiversity, Hamilto, ON, Caada terlaky@mcmasterca A Deza deza@mcmasterca E Nematollahi ematoe@mcmasterca

2 2 A Deza et al 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 [11] ad the refereces therei The Klee Mity worstcase example partially stimulated the search for a polyomial algorithm ad, i 1979, Khachiya s [5] ellipsoid method proved that liear programmig is ideed polyomially solvable I 1984, Karmarkar [4] 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 eg [7,9,10,14,15] I 2004, Deza et al [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 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 (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] iso(2 3 L = O(2 9 4,asN = O(2 6 2 ad L = O(2 6 3 for this example Therefore, these 2 1 sharp turs yield a ( 6 N l 2 N iteratio-complexity lower boud I this paper we show that a refied problem with the same (2 iteratio-complexity lower boud exhibits a early worst-case iteratiocomplexity as the complexity upper boud is O(2 5 2 I other words, this ew example, with N = O(2 2 3, essetially closes the iteratio-complexity gap with a ( 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, ε x k 1 x k 1 ε x k 1 for k = 2,,

3 How good are iterior poit methods? 3 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,, Theseth 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 ε x k 1 x k 1 ε x k 1 for k = 2,, 0 + x 1 repeated h 1 times ε x 1 d 2 + x 2 repeated h 2 times ε x 1 d + x repeated h times 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,, }, seefig1 ForS {1,, }, a vertex v S of C is defied by { v S 1, if 1 S 1 = 0, otherwise { v S k = 1 εv S k 1, ifk S εv S k 1, 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, ifk S x k εx k 1 ε k 1 δ, otherwise I this paper we focus o the followig problem C δ defied by 2(+1, ε = d = (2 +4,,2 k+5,,2 5, h = ( 22+8 (+1 δ (+1 δ,, 22+8 (+1 +k 1 δ +k 2 } k = 1,,

4 4 A Deza et al Fig 1 The δ-eighborhoods of the four vertices of the Klee Mity 2-cube v {2} v {1,2} v {1} where 0 <δ 1 4(+1 2+k+6 (+1 2k 1 δ 2k 2 v,, (+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 21 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 Sect 32 for the proof Note that the proof give i Sect 32 yields a slightly stroger result tha Propositio 21: 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 23 ad 24 i order to exhibit the sharpest bouds The correspodig liear optimizatio problem C 1/4(+1 depeds oly o the dimesio Propositio 21 The cetral path P of C δ vertex of the -cube itersects the δ-eighborhood of each Sice the umber of iteratios required by path-followig iterior poit methods is at least the umber of sharp turs, Propositio 21 yields a theoretical lower boud for the iteratio-complexity for solvig this -dimesioal liear optimizatio problem Corollary 22 For C δ, the iteratio-complexity lower boud of path-followig iterior poit methods is (2 Sice the theoretical iteratio-complexity ( upper boud for path-followig NL iterior poit methods is O, where N ad L respectively deote the umber of costraits ad the bit-legth of the iput-data, we have: Propositio 23 For C 1/4(+1, the iteratio-complexity upper boud of pathfollowig iterior poit methods is O(2 3 2 L; that is, O(

5 How good are iterior poit methods? 5 Proof We have N = 2+ k=1 h k = 2+ ( ( +k k= k+8 ( 2k +1 ad, sice ( +k +1 k=1 e 2, we have N = O(2 2 3 ad L N l = O(2 2 4 Noticig that the oly two vertices with last coordiates smaller tha or equal to ε 1 are v ad v {1},withv = 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 µ 0 = 1, we obtai a startig poit which belogs to the iterior of the δ-eighborhood of the highest vertex v {}, see Sect 33 for a detailed proof I other words, a path-followig algorithm usig a stadard ɛ-precisio as stoppig criterio ca stop whe the duality gap is smaller tha ε as the optimal vertex is idetified, see [9] The correspodig iteratio-complexity boud O( N log N ɛ yields, for our costructio, a precisio-idepedet iteratio-complexity O( N l Nµ0 Nµ = O( N ad Propositio 23 ca therefore be stregtheed to: Propositio 24 For C 1/4(+1, the iteratio-complexity upper boud of pathfollowig iterior poit methods is O(2 5 2 Remark 25 (i (ii (iii (iv For C 1/4(+1, by Corollary 22 ad Propositio 24, the order of the iteratio-complexity of path-followig iterior poit methods is betwee 2 ad or, equivaletly, betwee N l 3 N ad N l N The k-th coordiate of the vector d correspods to the scalar d defied i [2] for dimesio k + 3 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 (11, 22 i dimesio 2 Our results apply to path-followig iterior poit methods but ot to other iterior poit methods such as Karmarkar s origial projective algorithm [4] Remark 26 (i (ii (iii Megiddo ad Schub [8] 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 Todd ad Ye [12] gave a ( 3 N iteratio-complexity lower boud betwee two updates of the cetral path parameter µ Vavasis ad Ye [13] provided a O(N 2 upper boud for the umber of approximately straight segmets of the cetral path

6 6 A Deza et al (iv (v A referee poited out that a kapsack problem with proper objective fuctio yields a -dimesioal example with + 1 costraits ad sharp turs Deza et al [3] provided a o-redudat costructio with N costraits ad N 4 sharp turs 3 Proofs of Propositio 21 ad Propositio Prelimiary lemmas Lemma 31 With b= 4 δ (1,,1, ε= 2(+1,d=(2+4,,2 k+5,,2 5, h = ( ( (+1 δ 1 δ,, 22+8 (+1 +k 1 2+k+6 (+1 2k 1,, ( δ +k 2 δ 2k 2 δ 2 2 ad A = we have A h 3b ε d ε d 2 +1 ε 2 d d ε k 1 d k +1 ε k d k ε 2 d ε 1 d 2ε 1 d +1 Proof As ε = 2(+1 ad d = (2+4,,2 k+5,,2 5, h ca be rewritte as ( h = 4 δ ( ε 4 1 ε,, d k ε ( 4 k 1 ε 1 d,, 3 ε k ε 1 ε ad A h 3b 2 ca be rewritte as ( 4 4 δ + 1 ε 1 4 ( 4 ε δ ε 1 ε 2 6 δ 4 ( 4 δ ε ( 2d k 4 ε δ d k + 1 ε 1 ε k 4 ( 4 δ ε 1 ε k+1 6 for k=2,, 1 δ 4 ( 4 δ ε d 3 ε δ d + 1 ε 6 δ,, which is equivalet to ( 1 ε 2 1 ε 3 2 ( 1 ε k+1 2 ε k + 1 ε ε 1 ε , d k 8 ε 1 ε k+1 1 ε for k = 2,, 1,

7 How good are iterior poit methods? 7 ( 2 ε + 1 ε 3 d 4 2 ε 1 ε As 1 1 ε 2 ε , 1 ε 3 2 0, 1 3 ε a ε + 1 k+1 ε 3 2 system is implied by 0, the above ε, ( 1 ε k+1 2 ε k d k 8 ε for k = 2,, 1, 2 ε d 4 ε, as 1 ε 2 = 2 k+1 ε k implied by ε k a ε k = 2 k ( k 2 k+2, the above system is 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 32 With the same assumptios as i Lemma 31 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 ( h 2 h 2 ε d 2 2 δ, ( h k+1 h k+1 ε k d k+1 + 2( h h ε 1 d δ 2 δ, for k = 2,, 1, thus, A( h h b 2, which implies, sice A h 3b 2 by Lemma 31, that Ah b Corollary 33 With the same assumptios as i Lemma 31 ad h = h, we have: h k+1ε k d + 4 k+1 δ for k = 1,, 1 h k ε k 1 d k +1 Proof For k = 1,, 1, oe ca easily check that the first k iequalities of Ah b imply h kε k 1 d k +1 h k+1ε k d + 4 k+1 δ

8 8 A Deza et al 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 s 1 = 1 x 1 s k = 1 εx k 1 x k s 1 = + s 1 s = d + s for k = 2,, for k = 2,, repeated h 1 times repeated h times Equivaletly, χ is the solutio of the followig maximizatio problem: max x ( l sk + l s k + h k l s k, k=1 ie, with the covetio x 0 = 0, max x 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 = 0 for k = 1,, 1, 1 σ 1 σ + h σ = 0 (1 σk > 0, σ k > 0, σ k > 0 for k = 1,,, h k+1ε σ k+1 σ1 = ξ 1 σk = 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 + σk 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 34 For C δ, we have: χ N δ (v {}

9 How good are iterior poit methods? 9 Proof Addig the th equatio of (1 multiplied by ε 1 to the jth 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 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 33, 2h ε 1 d + 1 h 1 εk 1, implyig, sice 2h ε 1 d +1 h 1 d 1 1 δ by Corollary 32, σ k εk 1 δ for k = 1,, 1 The -th equatio of (1 implies: h ε 1 h ε 1 d +1 σ σ k 1 δ by Corollary 32, we have: 1 δ h ε 1 d +1 ε 1 σ ; that is, sice σ < d + 1 ad 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[9] These itersectios (α are called the level sets ad χ (α is the solutio of the followig system: where { 1 s k ε s 1 k+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 35 For C δ,ck δ ={x C : s k ε k 1 δ, s k ε k 1 δ} ad Ĉδ k ={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,,

10 10 A Deza et al Proof Let x Cδ k P Addig the (k 1th equatio of (2 multiplied by εk 2 to the ith 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 k 1 2εk 2 s k 1 + h jε j 1 s j + εj 1 s j + h kε k 1 k i=j s k ε i s i+1 + εj 1 s j = 0, + εk 1 s k + εk 1 s k which implies, sice s k 1 < d k 1 + 1, s j s k εk 1 δ as x Cδ k, > d j, s k > d k ad s k εk 1 δ ad 2h k 1 ε k 2 d k h jε j 1 h kε k 1 εj 1 d j d k s + 2 j δ, implyig, sice h 1 h jε j 1 d j by Corollary 33, h 1 + 2h k 1ε k 2 d k h kε k 1 εj 1 d k s + 2 j δ, that is, as 3 δ h 1 d + 2h k 1ε k 2 1 d k 1 +1 h kε k 1 d by Corollary 32: s k j ε j 1 δ Cosiderig the (k 1th equatio of (2, we have 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 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 h kε k 1 εk 2 d k s + 2 k 1 δ, which implies, sice 3 δ h k 1ε k 2 d k 1 +1 ad, therefore, x Ĉδ k h kε k 1 d k by Corollary 33, that s k 1 εk 2 δ 32 Proof of Propositio 21 For k = 2,,, while C k δ, defied i Lemma 35, ca be see as the cetral part of the cube C,thesetsT k δ ={x C : s k ε k 1 δ} ad B k δ ={x C : s k ε k 1 δ},

11 How good are iterior poit methods? 11 Fig 2 The set P δ for the Klee Mity 3-cube T 3 0 B 2 0 Ĉ 2 0 T 2 0 Ĉ 3 0 B 3 0 A 2 0 A 3 0 Fig 3 The sets A 2 0 ad A3 0 for the Klee Mity 3-cube 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 35, we cosider the set A k δ = Tk δ Ĉk δ Bk 1 δ for k = 2,, ad, for 0 <δ 4(+1, we show that the set P δ = k=2 A k δ,seefig2, cotais the cetral path P By Lemma 34, the startig poit χ of P belogs to N δ (v {} Sice P C ad C = k=2 (Tδ k Ck δ Bk δ, we have: P = C P = k=2 that is, by Lemma 35, ( Tδ k Ck δ Bk δ P = P k=2 k=2 ( Tδ k Ĉk δ Bk δ = (T δ (C k δ k P B k δ P, A k δ = P δ Remark that the sets Cδ k, Ĉk δ, Tk δ, Bk δ ad Ak δ ca be defied for δ = 0, see Fig 3, ad that the correspodig set P 0 = k=2 A k 0 is precisely the path followed by the simplex method o the origial Klee-Mity problem as it pivots alog the edges of C ThesetP δ is a δ-sized (cross sectio tube alog the path P 0 See Fig 4 illustratig how P 0 starts at v {}, decreases with respect to the last coordiate x ad eds at v k=2

12 12 A Deza et al Fig 4 The path P 0 followed by the simplex method for the Klee Mity 3-cube v {3} v {1,3} v {1,2,3} v {2,3} v {2} v v {1} v {1,2} 33 Proof of Propositio 24 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 33 with s +k = s k for k = 1,, ad s pi +k = s k for k = 1,, h i+1 Ê 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 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 {} 331 Estimatio of the cetral path parameter µ The formulatio of the dual problem of C δ is: max z = subject to 2 k=+1 y k p k d k y i k=1 i=p k 1 +1 y k εy k+1 y +k εy +k+1 + p k i=p k 1 +1 y i ε y y 2 + 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, where p 0 = 2 ad p k = 2 + h 1 + +h k for k = 1,,

13 How good are iterior poit methods? 13 For k = 1,,, multiplyig by ε k 1 the kth equatio of the above dual costraits ad summig the up, we have: 2+h 1 y 1 y +1 2 (εy +2 + ε 2 y ε 1 y 2 + which implies 2ε 1 y 2 y h 1 i=2+1 y i i=2+1 y i = ε 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 Sice for i = p 1 + 1,, p, s i = d + x ε x 1 d + 1 yields y i µ d +1, the last dual costrait implies y 2 p i=p 1 +1 y i 1 µh d ( which, combied with the previously obtaied iequality, gives µ 2h ε 1 1 δ µ ε 1 δ d +1 h 1 2ε 1, ad, sice Corollary 32 gives 2h ε 1 d +1 h 1 d 1 1 δ 2 δ, we have Ackowledgmets We would like to thak a associate editor ad the referees for poitig out the paper [8] ad for helpful commets ad correctios May thaks to Yiyu Ye for precious suggestios ad hits which triggered this work ad for iformig us about the papers [12,13] Research supported by a NSERC Discovery grat, by a MITACS grat ad by the Caada Research Chair program Refereces 1 Datzig, GB: Maximizatio of a liear fuctio of variables subject to liear iequalities I: Koopmas, TC (ed Activity Aalysis of Productio ad Allocatio, pp Wiley, New York ( Deza, A, Nematollahi, E, Peyghami, R, Terlaky, T: The cetral path visits all the vertices of the Klee Mity cube Optim Methods Softw 21 5, ( Deza, A, Terlaky, T, Zicheko, Y: Polytopes ad arragemets: diameter ad curvature AdvOL-Report 2006/09, McMaster Uiversity ( Karmarkar, NK: A ew polyomial-time algorithm for liear programmig Combiatorica 4, (1984

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