McMaster University Avance Optimization Laboratory Title: The Central Path Visits all the Vertices of the Klee-Minty Cube Authors: Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky AvOl-Report No. 2004/ August 2004, Hamilton, Ontario, Canaa
The Central Path Visits all the Vertices of the Klee-Minty Cube Antoine Deza Eissa Nematollahi Reza Peyghami Tamás Terlaky August 30, 2004 Abstract The Klee-Minty cube is a well-known worst case example for which the simplex metho takes an exponential number of iterations as the algorithm visits all the 2 n vertices of the n-imensional cube. While such behavior is exclue by polynomial interior point methos, we show that, by aing an exponential number of reunant inequalities, the central path can be bent along the eges of the Klee-Minty cube. More precisely, for an arbitrarily small, the central path takes 2 n 2 turns as it passes through the -neighborhoo of all the vertices of the Klee-Minty cube in the same orer as the simplex metho oes. Key wors: Linear programming, central path, Klee-Minty cube. MSC2000 Subject Classification: Primary: 90C05; Seconary: 90C5, 90C27, 52B2 Introuction While the simplex metho, introuce by Dantzig [], works very well in practice for linear optimization problems, Klee an Minty [5] gave an example in 972 for which the simplex metho takes an exponential number of iterations. More precisely, they consiere a maximization problem over an n-imensional squashe cube an prove that a variant of the simplex metho visits all its 2 n vertices; that is, the time complexity is not polynomial for the worst case, as 2 n iterations are necessary for this n-imensional linear optimization problem. The pivot rule use in the Klee-Minty example was the most negative reuce cost but variants of the Klee-Minty n-cube showing an exponential running time exist for most pivot rules; see [9] an the references therein. The Klee-Minty worst-case example partially stimulate the search for a polynomial algorithm an, in 979, Khachiyan s [4] ellipsoi metho prove that linear programming is inee polynomially solvable. In 984, Karmarkar [3] propose a more efficient polynomial algorithm that sparke the research on polynomial interior point methos. In short, while the simplex metho goes along the eges of the polyheron corresponing to the feasible region, interior point methos pass through the interior of this polyheron. Starting at the analytic center, most interior point methos follow the so-calle central path an converge to the analytic center of the optimal face; see e.g., [6, 7, 8,, 2]. 2
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 3 In this paper, following the Klee-Minty approach, we show that, by carefully aing an exponential number of reunant constraints to the Klee-Minty n-cube, the central path can be bent along its eges. In other wors, we give an example where, for an arbitrarily small, starting from the -neighborhoo of a vertex ajacent to the optimal solution, the central path takes 2 n 2 turns as, before converging to the optimal solution, it passes through the -neighborhoo of all the vertices of the Klee-Minty cube in the same orer as the simplex metho oes. Before stating the main result in Section 2 an giving its proof in Section 3, we illustrate the bening of the central path in the 2 an 3 imensional cases. Fig. an Fig. 2 show the trajectory of the central path starting from the highest vertex an converging to the origin after visiting each vertex of the Klee-Minty cube. The reunant constraints correspon to hyperplanes parallel to the facets of the cube containing the origin. More precisely, in imension 2, the reunant inequality 6 + x 0 is ae 5 360 times an the reunant inequality 6 +x 2 4 x 0 is ae 40 960 times. Starting from the highest vertex an with = 0., the central path visits the -neighborhoo of each vertex of the Klee-Minty cube in the same orer as the simplex algorithm oes before converging to the optimal solution; that is, the origin. In imension 3, the reunant inequality 48 + x 0 (resp. 48 + x 2 4 x 0 an 48 + x 3 4 x 2 0) is ae 6 280 (resp. 552 960, an 474 560 times). v {2} χ h χ h (v {} 2 ) v {,2} v {} v Figure : Central path nearing all the vertices of the Klee-Minty 2-cube. v {3} χ h v {,3} v {,2,3} χ h (v {} 3 ) v {2,3} v {2} v v {} v {,2} Figure 2: Central path nearing all the vertices of the Klee-Minty 3-cube.
4 The central path visits all the vertices of the Klee-Minty cube 2 Notations an the Main Result We consier the following Klee-Minty variant where is a small positive factor by which the unit cube [0,] n is squashe. min x n subject to 0 x x k x k x k for k = 2,...,n. We enote this linear optimization problem by KM. This minimization problem has 2n constraints, n variables an the feasible region is an n-imensional cube enote by C. Some variants of the simplex metho take 2 n iterations to solve KM as they visit all the vertices orere by the ecreasing value of the last coorinate x n starting from v {n} = (0,...,0,) till the optimal value x n = 0 is reache at the origin v. If an interior point metho is use to solve KM, the central path starts from the analytic center χ of C an converges to the origin as it is shown in Fig. 3. v {2} χ v {,2} v {} v Figure 3: The central path in the non-reunant Klee-Minty 2-cube. While aing a set h of reunant inequalities oes not change the feasible region of KM, the analytic center χ h an the central path are affecte by the aition of reunant constraints. We consier reunant inequalities inuce by hyperplanes parallel to the n facets of C containing the origin. To ease the analysis, we consier that all reunant hyperplanes are put at the same istance to the corresponing parallel facet of C. The constraint parallel to H : x = 0 is ae h times an the constraint parallel to H k : x k = x k is ae h k times for k = 2,...,n. By abuse of notation, the set h is enote by the integer-vector h = (h,...,h n ). With these notations, the reunant linear optimization problem KM h is efine by
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 5 min x n subject to 0 x x k x k x k for k = 2,...,n 0 + x repeate h times x + x 2 repeate h 2 times. x n + x n. repeate h n times. To give a flavor of the main result, we first present Lemma 2. stating that, by aing + n times the reunant inequality x n + x n to the original KM formulation, the analytic center χ h can be pushe arbitrarily close to the vertex v {n} = (0,...,0,). To warranty, without loss of generality, that h is integer value, we assume that both an are positive integers. Lemma 2.. Given 4, positive integer an h = (0,...,0, + ), the analytic center n χ h satisfies χ h v {n}. While Lemma 2. sets the starting point of the central path in the -neighborhoo of v {n}, Proposition 2.2 states that, for a careful choice of an h, the central path of the cube takes 2 n 2 turns before converging to the origin as it passes through the -neighborhoo of all the 2 n vertices of the Klee-Minty n-cube. Proposition 2.2. Given 4, < n, choose integer n2 n+ an h = 4n (2n,..., 2 n 2 k,..., 2n k ), then for each vertex v S of the Klee-Minty n-cube, there is a point χ h (v S n n ) of the central path satisfying χ h (vn) S v S. 3 Proofs of Lemma 2. an Proposition 2.2 3. Proof of Lemma 2. The analytic center χ h = (ξ h,...,ξh n) of KM h is the solution to the problem consisting of maximizing the prouct of the slack variables s = x s k = x k x k for k = 2,...,n s = x s k = x k x k for k = 2,...,n s = + s repeate h times. s n = + s n. repeate h n times.
6 The central path visits all the vertices of the Klee-Minty cube Equivalently, χ h is the solution of the following maximization problem max x i.e., with the convention x 0 = 0, max x n k= n (log s k + log s k + h k log s k ), k= ( ) log(x k x k ) + log( x k x k ) + h k log( + x k x k ). The optimality conitions (the graient is equal to zero at optimality) for this concave maximization problem give where σ h k σk+ h σ k h + h σ k+ h k h k+ σ k h σ k+ h + hn σn h σ n h σ n h = 0 for k =,...,n = 0 () σk h > 0, σh k > 0, σh k > 0 for k =,...,n, σ h = ξh σ h k = ξh k ξh k for k = 2,...,n σ h = ξh σ h k = ξh k ξh k for k = 2,...,n σ h k = + σh k for k =,...,n. The following lemma states that, for h n large enough relatively to the other h k values, the analytic center χ h is pushe to the neighborhoo of the vertex v {n} = (0,...,0,). Lemma 3.. Given 4 an h,...,h n, we have χ h v {n} for h n + η n where { ( hk = max η n k n 2 n k + 2 )}. Proof. The analytic center χ h = (ξ h,...,ξh n) is the solution of (). Let us consier the n-th equation of (). Since σ n h +, we have: h n +. Thus, as h σ n h n + η n, we have σ n h η n, which implies σ n h 2. Let us then consier the (n )-th equation. We have σ h n = σ n h + σn h + σ n h h n σ n h + h n σ n h σ n h h n σ n h + h n σ n h. Since σ h n η n, σ h n an σh n +, this implies σ h n 2 h n η n 2,
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 7 i.e., σn h 2. The first n equations of () can be rewritten as ( ) σk h + h k σ k h = σ k h + σk+ h + h k+ σ k+ h + σ k+ h for k =,...,n. For k n 2, forwar substitutions for the k-th, (k + )-th,... (n )-th equations give which implies σ h k + h k σ h k = σ h k + 2 σ h k n j=k+ n k σ h n j k σ h j h k σ h k Since σ n h η n, σ k h an σh n +, this implies + n k σ h n + h n n k σ n h. + h n n k σ h n + n k σ n h, σ h k 2n k η n h k 2, i.e., σk h 2 for k =,...,n 2. Therefore, for h n + η n, we have: ξk h for k =,...,n an ξn h. Lemma 2. is a irect corollary of Lemma 3. with h = (0,...,0,h n ). 3.2 Proof of Proposition 2.2 3.2. Preliminary Lemmas Lemma 3.2. Given 4 an integer n2n+, then h = 4n (2n,..., 2n 2 k,..., 2n k ) n is a positive an integer solution of Ah b, where b = 4n (,...,) an 2 0 0 0... 0 n + + 0 0... 0 0 2 2 + 0... 0 0 2 2 3 A = (+) +... 0 0.......... 2 3 (+) (+) (+)... n 2 0 (+) 2 (+) 3 2 (+)... n 2 + Proof. Multiplying both sies of Ah b by (+) 4n, we have ( ) 2 n + ( ) 2 n 2 + ( ) 2 n+ 5 + ( ) 2 n 2 k + k j= n ( 2 n 2 j ) + for k = 4,...,n,
8 The central path visits all the vertices of the Klee-Minty cube which, since ( ) n2 n, is implie by the obvious conitions n2 n 2 n + n2 n 2 n 2 + n2 n 2 n+ 5 + n2 n k2 n 3 2 k + + for k = 4,...,n. To ease the notations, we efine, for, k =,...,n l k = u k = h k + h k+ h k h k+ +. Lemma 3.3. The system Ah b is equivalent to 2h n n + h 4n l 4n l k k k j= u j j 4n for k = 2,...,n. Proof. The first 2 inequalities are irect reformulations of the first 2 inequalities of the system Ah b. For k = 2,...,n, the inequality: l k k k j= u j j 4n can be 2h rewritten as: k k + h k+ k h k (+) j=2 h j j 4n. Corollary 3.4. For h satisfying the last n inequalities of Ah b, we have l k k 2k+ n for k =,...,n. Proof. The proof is by inuction on k. We have l 4n u k l k in Lemma 3.3. an the result follows by using Corollary 3.5. Given 4 an a positive integer h satisfying the last n inequalities of Ah b, we have χ h v {n} 2h n n for h + + 2. Proof. Corollary ( ) 3.4 implies l k 0 for k =,...,n. Hence, in Lemma 3. we have η n = h 2 n + 2 which gives the result. The central path of KM h can be efine as the set of analytic centers χ h (α) = (x h,...,xh n,α) of the intersection of the hyperplane H α : x n = α with the feasible region of KM h where 0 α ξ h n, see [7]. These intersections are calle the α-level sets an χh (α) is the solution of the following system { s h k s s h k+ h k s h k+ + h k s h k h k+ s h k+ = 0 for k =,...,n s h k > 0, sh k > 0, sh k > 0 for k =,...,n, (2)
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 9 where s h = x h s h k = x h k xh k for k = 2,...,n s h n = α x n s h = x h s h k = x h k xh k for k = 2,...,n s h n = α x h n s h k = + s h k for k =,...,n. In the rest of the paper, we assume that n an that a positive integer h satisfying Ah b is given. Corollary 3.5 implies that ξ h n an therefore we can consier the α-level set for α n as it implies α ξ h n. Lemma 3.6. Given 4, < n, integer n2 n+ an a positive integer h satisfying Ah b; for 0 α n an k {,n}, if the k-th coorinate x h k of the analytic center χ h (α) satisfies ] x h k [ k t k k, k + t k k, where then, x hˆk for some ˆk smaller than or equal to k. t = t 2 = 2 4n t k+ = t k n for k = 2,...,n, Proof. Assume to the contrary that the statement is false, i.e., x hˆk < for ˆk =,...,k. Consiering the first equation of (2) an successively using x h <, < an Lemma 3.3, we have s h 2 + s h 2 = s h which implies either or s h x h 2 xh + + h s h h 2 s h 2 2 < ( ) + 2 4n x h 2 x h + h + h 2 + h + h 2 4n ( ) + 2 4n 4n = t 2, 2 > ( ) 2 4n 4n >. Since x h 2 <, this implies xh 2 < t 2. Similarly, consiering the ˆk-th equation of (2) for ˆk = 2,...,k 2, we have: x hˆk < ˆk tˆkˆk. Consiering the (k )-st equation an successively using x h k < k, x h k 2 < an Corollary 3.4, we have, s h k + s h k = s h k s h + h k k s h h k k s h k k 2 k 2 + h k + h k l k 2k n k 2,
0 The central path visits all the vertices of the Klee-Minty cube which implies either or x h k xh k + k 2 k n < (k 2 t k k 2 ) + k 2 k n k t k k, x h k xh k k 2 k n > (k 2 t k k 2 ) k 2 k n k + t k k. This is impossible as x h k [ k t k k, k + t k k ]. Lemma 3.7. Given 4, < n, n2 n+, a positive integer h satisfying Ah b an t,...,t n as specifie in Lemma 3.6; for 0 α n an k {,n}, if the k-th coorinate x h k of the analytic center χh (α) satisfies ] x h k [t k k, t k k, then s hˆk 2ˆkn ˆk for some ˆk smaller than or equal to k. Proof. Assume to the contrary that the statement is false, i.e., s hˆk < 2ˆkn ˆk for ˆk =,...,k. This implies x hˆk < ˆk 4n ˆk j= 2 j 2 for ˆk =,...,k. Consiering the (k )-st equation of (2) an using s h k = 2xh k 2 sh k, we have s h k + s h k = s h k s h + h k k s h h k k s h 2k n k k 2 2 2 k n k 2 + h k + h k l k. By Corollary 3.4, this implies which further implies either, since n t k, or x h k xh k + k 2 k n < k k 2 4n j= x h k xh k k 2 k n > k k 2 4n j= s h k + s h k 2k n k 2, 2 j 2 + k 2 k n = k k 4n j= This is impossible because x h k [ t k k, t k k ]. 2 j 2 k n 2 j 2 k 2 k n = k k 4n j= t k k, 2 j 2 t k k.
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 3.2.2 Proof of Proposition 2.2 By analogy with the unit cube [0,] n, we enote the vertices of the Klee-Minty cube C using a subset S of {,...,n}. For S {,...,n}, a vertex v S of C is efine by { v S, if S = 0, otherwise { vk S v S = k, if k S vk S, otherwise k = 2,...,n. Proposition 3.8. Given 4, < n, n2 n+ an a positive integer h satisfying Ah b, for k n, the (k + )-th an k-th coorinates of the analytic center χ h (vn S ) of the vn S -level set satisfy x h k+ vs k+ t k+ k x h k vs k t k k. Proof. Assume to the contrary that the statement is false, i.e., for at least one k smaller than or equal to n, we have: x h k+ vs k+ t k+ k an x h k vs k > t k k. We consier a case by case analysis. Case : v S k = 0 The inequality x h k vs k > t k k implies x h k > t k k an, since x h k xh k+ xh k, we have t k k < x h k+ < t k k. Since t k+ < t k, this implies t k+ k < x h k+ < t k+ k. This contraicts the inequality x h k+ vs k+ t k+ k, where v S k+ = 0 or because vs k = 0. Case 2: 0 < v S k < The inequality x h k vs k > t k k implies x h k ] 0, v S k t k k [ or x h k ] v S k + t k k, [. By ]a,b[ we enote the open interval between a an b. Subcase 2.: x h k ] v S k + t k k, [ Since x h k xh k+ xh k, we have (vs k + t k k ) < x h k+ < (vs k + t k k ). Since t k+ < t k, this implies v S k + t k+ k < x h k+ < vs k t k+ k. This contraicts the inequality x h k+ vs k+ t k+ k, where v S k+ = vs k or vs k. Subcase 2.2: x h k ] 0, v S k t k k [ Subsubcase 2.2.: x h k < k t k k Consiering the k-th equation of (2) an successively using x k < k an Corollary 3.4, we have s h k+ + s h k+ = s h s k h k which implies either + h k s h h k+ k s h k+ k k + h k + h k+ l k 2k+ n k, x h k+ xh k + 2 k n k < ( k t k k ) + 2 k n k k t k+ k,
2 The central path visits all the vertices of the Klee-Minty cube or x h k+ xh k k 2 k n > (k t k k ) k 2 k n k + t k+ k. This contraicts the inequality x h k+ vs k+ t k+ k, where v S k+ k because v S k > 0. Subsubcase 2.2.2: k t k k x h k < vs k t k k (we have k since 0 < v S k < ) By Lemma 3.6, there is a ˆk smaller than or equal to k such that x hˆk, which implies: s hˆk 2. Consiering the ˆk-th equation of (2) an using s hˆk 2, we have ˆk ˆk ˆk s hˆk+ = s hˆk s hˆk ˆk s hˆk+ + hˆk s hˆk ˆk hˆk+ ˆk ˆk ˆk s hˆk+ 2 ˆk s hˆk+ + hˆk ˆk hˆk+ ˆk +, which implies ˆk s hˆk+ hˆk ˆk hˆk+ ˆk + = uˆkˆk. The previous inequality, which correspons to the case i = ˆk, can be generalize to: i s h i+ i u j j for i = ˆk,...,k, j=ˆk which clearly hols for k = ˆk + an, for k > ˆk +, is obtaine by multiplying the i-th equation of (2) by i for i = ˆk +,...,k an using successively a similar argument. Noticing that we coul have initially permute s hˆk+ an s hˆk+, gives k s h k k u j j. j=ˆk Together with the k-th equation of (2), this implies k s h k+ + k s h k+ = k s h k k s h k + h k k s h k h k+ k s h k+ h k k + h k+ k k u j j, j=ˆk i.e., k s h k+ + k s h k+ k l k k u j j. j=ˆk Using Lemma 3.3, Corollary 3.4 an u k l k, this implies k s h k+ + k s h k+ 4n ˆk + j= u j j 2ˆk+ n,
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 3 which implies either x h k+ xh k + k 2ˆkn < (vs k t k k ) + k 2ˆkn vs k t k+ k, or x h k+ xh k k 2ˆkn > (vs k t k k ) k 2ˆkn vs k + t k+ k. This contraicts the inequality x h k+ vs k+ t k+ k, where v S k+ = vs k or vs k. Case 3: v S k = The inequality x h k vs k > t k k implies x h k < t k k. Subcase 3.: x h k < t k k Consiering the k-th equation of (2) an successively using x h k < t k k, t k k an Corollary 3.4, we have s h k+ + s h k+ = s h s k h k which implies either + h k s h h k+ k s h k+ t k k t k k + h k + h k+ l k 2k+ n k, x h k+ xh k + k 2 k n < t k k + k 2 k n = (t k + 2 k n )k, or x h k+ xh k k 2 k n > t k k k 2 k n = (t k + 2 k n )k. This contraicts the inequality x h k+ vs k+ t k+ k, where v S k+ = or because vs k =. Subcase 3.2: t k k x h k < t k k Subsubcase 3.2.: k = From the first equation of (2), we have s h 2 + s h 2 = s h s h + h s h h 2 s h 2 + l 4n. which implies either x h 2 x h + 2 < ( ) + 2 4n 4n t 2, or x h 2 xh 2 > ( ) 2k 4n 4n + t 2. This contraicts x h 2 vs 2 t 2 where v S 2 = or since vs =.
4 The central path visits all the vertices of the Klee-Minty cube Subsubcase 3.2.2: k By Lemma 3.7, there is a ˆk smaller than or equal to k such that s hˆk ˆk 2ˆkn. Consiering the ˆk-th equation of (2), we have ˆk ˆk ˆk s hˆk+ = s hˆk s hˆk ˆk s hˆk+ + hˆk s hˆk ˆk hˆk+ ˆk s hˆk+ 2ˆk n + hˆkˆk hˆk+ ˆk + = uˆkˆk + 2ˆk n. This inequality, which correspons to the case i = ˆk, can be generalize to: i s h i+ i 2ˆkn u j j + j=ˆk for i = ˆk,...,k, which clearly hols for k = ˆk + an, for k > ˆk +, is obtaine by multiplying the i-th equation of (2) by i for i = ˆk +,...,k an using successively a similar argument. Noticing that we coul have initially permute s hˆk+ an s hˆk+, gives k s h k k 2ˆkn u j j +. j=ˆk Together with the k-th equation of (2), this implies k s h k+ + k s h k+ = k s h k k s h k + h k k s h k h k+ k s h k+ k 2ˆkn u i i Using Lemma 3.3, Corollary 3.4, an u k l k, the previous inequality gives i=ˆk + h k k + h k+ k. k s h k+ + k s h k+ k 2ˆkn l k k u i i i=ˆk 4n ˆk 2ˆkn + u i i i= 2ˆk+ n 2ˆkn = 2ˆkn, which implies either or x h k+ xh k + k 2ˆk n < ( t k k ) + k k 2ˆk n t k+ k, x h k+ xh k 2ˆk n > ( t k k ) 2ˆk n + t k+ k. This contraicts x h k+ vs k+ t k+ k where v S k+ = or since vs k =. Proposition 2.2 is a irect corollary of Proposition 3.8 since, for S an S {n}, we have x h n v S n = 0; implying x h k vs k t k k for k =,...,n. In other wors, χ h (v S n) v S. Furthermore, by Corollary 3.5 we have χ h v {n}, an the central path converges to the origin v. k
Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky 5 4 Remarks an Future Work We showe that, without changing the geometry of the feasible set of KM, the central path can be force to visit arbitrarily small neighborhoos of all the vertices of the Klee-Minty n-cube by carefully aing reunant constraints. This result highlights that, although the central path is a smooth analytical curve in the interior of the set of feasible solutions, it might be severely istracte by reunant constraints. In particular, exponentially many reunant constraints inter-playing with the geometry of the problem, may force the central path to take exponentially many an arbitrarily sharp turns. Our example leas to an Ω(2 n ) lower boun for the number of iterations neee for central path-following interior point methos. The theoretical iteration-complexity upper boun O( NL) = O(2 9n n 4 ) as, for this example, the number of constraints N = O(2 6n n 2 ) an the bit-length of the input-ata L = O(2 6n n 3 ). Therefore, the Ω(2 n ) lower boun yiels an Ω( 6 N ) iteration-complexity lower boun. Using a ifferent ln 2 N analysis, To an Ye [0] gave an Ω( 3 N) iteration-complexity lower boun between two upates of the barrier function. In a subsequent paper, Deza, Nematollahi an Terlaky [2] essentially close the gap between the lower an upper bouns. State-of-the-art preprocessing tools in moern linear optimization software woul eliminate the ae reunant inequalities. Therefore, interior point methos base coes woul solve the preprocesse KM h efficiently, just as commercial simplex coes o solve the KM in only one pivot. A challenging task woul be to esign a variant of KM h that cannot be easily simplifie by known preprocessing heuristics. Acknowlegments. Research supporte by the NSERC Discovery grant #48923 an a MITACS grant for the last three authors, by the Canaa Research Chair program for the first an last authors an by the NSERC Discovery grant #3969 for the first author. References [] G. B. Dantzig: Maximization of a linear function of variables subject to linear inequalities. In: T. C. Koopmans (e.) Activity Analysis of Prouction an Allocation. John Wiley (95) 339 347. [2] A. Deza, E. Nematollahi an T. Terlaky: How goo are interior point methos? Klee- Minty cubes tighten iteration-complexity bouns. AvOL-Report 2004/20, McMaster University (2004). [3] N. K. Karmarkar: A new polynomial-time algorithm for linear programming. Combinatorica 4 (984) 373 395. [4] L. G. Khachiyan: A polynomial algorithm in linear programming. Soviet Mathematics Doklay 20 (979) 9 94.
6 The central path visits all the vertices of the Klee-Minty cube [5] V. Klee an G. J. Minty: How goo is the simplex algorithm? In: O. Shisha (e.) Inequalities III, Acaemic Press (972) 59 75. [6] N. Megio: Pathways to the optimal set in linear programming. In: N. Megio (e.) Progress in Mathematical Programming: Interior-Point an Relate Methos, Springer- Verlag (988) 3 58; also in: Proceeings of the 7th Mathematical Programming Symposium of Japan, Nagoya, Japan (986) 35. [7] C. Roos, T. Terlaky an J-Ph. Vial: Theory an Algorithms for Linear Optimization: An Interior Point Approach. Wiley-Interscience Series in Discrete Mathematics an Optimization. John Wiley (997). [8] G. Sonneven: An analytical centre for polyherons an new classes of global algorithms for linear (smooth, convex) programming. In: A. Prékopa, J. Szelezsán, an B. Strazicky (es.) System Moelling an Optimization: Proceeings of the 2th IFIP- Conference, Buapest 985. Lecture Notes in Control an Information Sciences 84 Springer Verlag (986) 866 876. [9] T. Terlaky an S. Zhang: Pivot rules for linear programming - a survey. Annals of Operations Research 46 (993) 203 233. [0] M. To an Y. Ye: A lower boun on the number of iterations of long-step an polynomial interior-point linear programming algorithms. Annals of Operations Research 62 (996) 233 252. [] S. J. Wright: Primal-Dual Interior-Point Methos. SIAM Publications (997). [2] Y. Ye: Interior-Point Algorithms: Theory an Analysis. Wiley-Interscience Series in Discrete Mathematics an Optimization. John Wiley (997). Antoine Deza, Eissa Nematollahi, Reza Peyghami, Tamás Terlaky Avance Optimization Laboratory, Department of Computing an Software, McMaster University, Hamilton, Ontario, Canaa. Email: eza, nematoe, terlaky@mcmaster.ca, peyghami@optlab.mcmaster.ca