W. Cook and S. Dash, On the matrix-cut rank of polyhedra, Mathematics of Operations Research 26 (2001), 19 -- 30.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem x : x # F , 1) The procedures in [SA90] LS91] L01b] and [BCC93] solve this problem by iteratively strengthening its continuous relaxation, until, after at most n iterations, the convex hull of is obtained. This bound on the number of iterations is tight ([CD01], L01] also see [GT01] and [CL01] for related topics) Nevertheless, a question of theoretical and practical interest is whether it is possible to modify the procedures so that the earlier iterations produce stronger relaxations. As shown in [L01] the methods used in [SA90] LS91] L01b] ....

....[LS91] Lemma 4.31 The set odd hole, set odd antihole, and set odd wheel inequalities are all guaranteed by the algorithm. 34 4. 6 Further remarks Consider the feasible system introduced in [CCH89] x j #S # 1, n (124) 125) This system was was analyzed in [CCH89] [CD01], GT01] L01] to show that either n or n 1 iterations of various procedures (Sherali Adams, the N operator) are required to prove that is empty. Here we will show that running the # algorithm proves the same result. We will denote by C(S) the inequality (124) corresponding to the set S. ....

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W. Cook and S. Dash, On the matrix-cut rank of polyhedra, Mathematics of Operations Research 26 (2001), 19 -- 30.

....the constraints of an operator that is exponentially stronger than the combination of N and S . In any case, a calculation shows that for fixed t 0, and n large, x # j 1 i.e. x # violates (53) by nearly a factor of 2. Related results concerning the N operator are given in [CD01] [GT01] L01] In particular, CD01] shows that starting from the system 1 2 , the N rank of the inequality 1 is n. An open issue concerns the rank of pitch k inequalities for (say) a set covering problem: is it bounded as a function of k We do not know the answer to this; but ....

....is exponentially stronger than the combination of N and S . In any case, a calculation shows that for fixed t 0, and n large, x # j 1 i.e. x # violates (53) by nearly a factor of 2. Related results concerning the N operator are given in [CD01] GT01] L01] In particular, [CD01] shows that starting from the system 1 2 , the N rank of the inequality 1 is n. An open issue concerns the rank of pitch k inequalities for (say) a set covering problem: is it bounded as a function of k We do not know the answer to this; but there is a positive (and perhaps not ....

W. Cook and S. Dash, On the matrix-cut rank of polyhedra, Mathematics of Operations Research 26 (2001), 19 -- 30.

....some of our results and especially extended them to polytopes in the 0=kcube [37] Cornujols and Li [18] proved that the Gomory mixed integer rank of polytopes in the n dimensional 0=1 cube is at least n. An upper bound of n follows from existing results in the literature. 9 Cook and Dash [17] presented a lower bound of n for the matrix cut rank of polytopes contained in the n dimensional 0=1 cube. Moreover, they adapted the proof of Proposition 2.4 to show that a polytope with empty integer hull and matrix cut rank n has to have at least 2 inequalities in any defining system of ....

W. J. Cook and S. Dash. On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26:19--30, 2001.

.... We show in Proposition 11 below that S n 1 (K) #= #, which implies that P S n 1 (K) The polytope K has been used earlier to show that n iterations are needed for the following procedures: taking Chvatal cuts [CCH89] the N operator [GT00] the N operator combined with taking Chvatal cuts [CD01], and the N operator combined with taking Gomory mixed integer cuts (equivalent to disjunctive cuts) CL01] The following (easy to verify) identities will be used in the proof: 1 K A (23) for any set A. For the second one, use the fact that k k k 1 . with ....

.... ) Therefore, # = 1 I ) since I U . By Lemma 2 (ii) this shows that MU (v R 0. Finally, M V (y) 0, since I#H ( 1) H I y H = 2 n 0. Example 2. Consider the polytope 2 , 24) i=1 x i 1 . This example was considered by Cook and Dash [CD01] as an example where the Lovasz Schrijver rank is n. The next result shows that the Sherali Adams rank is also equal to n. with zero entries except y : 1 and y i : i V ) Then, R n 1 (K) where K is defined by (24) Therefore, P S n 1 (K) Proof. One can easily verify (using ....

W. Cook and S. Dash. On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26:19--30, 2001.

....also cases where the N operator does not help. This is the case, for instance, for the polytope 1 P = fx 2 R d j P d i=1 x i 1g if we start from its relaxation K = fx 2 R d j P d i=1 x i 1 2 g; then the same number d of iterations is needed for finding P using the N or the N operator [8]. Other examples are given in [8] 15] Moreover, geometric conditions are studied in [15] under which the N operator yields a tighter relaxation than the N operator. If we apply the Lov asz Schrijver construction to the pair P = CUT(G) K = MET(G) we obtain the sequence of linear and ....

....does not help. This is the case, for instance, for the polytope 1 P = fx 2 R d j P d i=1 x i 1g if we start from its relaxation K = fx 2 R d j P d i=1 x i 1 2 g; then the same number d of iterations is needed for finding P using the N or the N operator [8] Other examples are given in [8], 15] Moreover, geometric conditions are studied in [15] under which the N operator yields a tighter relaxation than the N operator. If we apply the Lov asz Schrijver construction to the pair P = CUT(G) K = MET(G) we obtain the sequence of linear and semidefinite relaxations N t (MET(G) ....

W. Cook and S. Dash. On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26:19--30, 2001.

.... 11 below that S n Gamma1 (K) 6= which implies that P 6= S n Gamma1 (K) The polytope K has been used earlier to show that n iterations are needed for the following procedures: 11 taking Chv atal cuts [CCH89] the N operator [GT00] the N operator combined with taking Chv atal cuts [CD01], and the N operator combined with taking Gomory mixed integer cuts (equivalent to disjunctive cuts) CL01] The following (easy to verify) identities will be used in the proof: X K A ( Gamma1) jKj 2 jKj = 1 2 jAj ; X K A jKj ( Gamma1) jKj 2 jKj = Gamma jAj 2 jAj (23) for any ....

..... By Lemma 2 (ii) this shows that MU (v R y) 0. Finally, M V (y) 0, since P I H ( Gamma1) jHnIj y H = 1 2 n 0. Example 2. Consider the polytope K : fx 2 [0; 1] n j n X i=1 x i 1 2 g; 24) then P = fx 2 [0; 1] n j P n i=1 x i 1g. This example was considered by Cook and Dash [CD01] as an example where the Lov asz Schrijver rank is n. The next result shows that the Sherali Adams rank is also equal to n. 12 Proposition 12. Let y 2 P(V ) with zero entries except y ; 1 and y i : 1 n 1 (i 2 V ) Then, y 2 R n Gamma1 (K) where K is defined by (24) Therefore, P ae S ....

W. Cook and S. Dash. On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26:19--30, 2001.

.... the Dvoretzky Kiefer Wolfowitz framework including the case where the probability distribution of demand is not completely known is developed by Karl Hinderer (see [41] His monograph was basic for a special direction of mathematical modelling in inventory research in Germany (see [35] 81] [46], 54] 9. Conclusions Kenneth J. Arrow wrote as President of the International Society for Inventory Research in a message to members: The process of inventory accumulation, holding, and decumulation is in itself a significant economic problem . it manifests elements of some of the ....

Kalin, D., 1980. On the optimality of (oe; S) policies. Mathematics of Operations Research 5: 293 - 307.

.... which in turn generalizes several works dealing with the convergence behavior of the central path and the continuous trajectories of various interior point algorithms for linear and convex programming (e.g. see McLinden [12, 11] Megiddo [13] Kojima et al. 10] Adler and Monteiro [1] Monteiro [14, 15] and Monteiro and Zhou [16] Our paper is organized as follows. In Section 2 we study the behavior the associated dual path of solutions of the family of problems (3) and develop a convergence result for dual sequences that asymptotically behave like the dual path. Using this asymptotic result ....

, On the continuous trajectories for a potential reduction algorithm for linear programming, Mathematics of Operations Research, 17 (1992), pp. 225--253.

....the running time of Dinic s algorithm improves from O(n 2 m) to O(n 2 1 m) Column 3 of Table 1.1 summarizes these improvements for several network flow algorithms. We obtain further running time improvements by modifying the algorithms. This modification applies only to preflow push algorithms [2, 3, 14, 15, 16, 17]; we call it the two edge push rule. According to this rule, we always push flow from a vertex in V 1 and push flow on two edges at a time, in a step called a bipush, so that no excess accumulates at vertices in V 2 . This rule allows us to charge all computations to examinations of vertices in V ....

.... m n ) n 1 n log U log( m n ) n 2 1 log U log( m n 1 ) dm=ne processors dm=ne processors dm=n 1 e processors Parametric Flows GGT[14] n 3 n 1 n 2 n 2 1 n GGT w nm log( n 2 m ) n 1 m log( n 2 m ) n 1 m log( n 2 1 m 2) dynamic trees [14] Min Cost Flows Cost scaling [17] n 3 log(nC) n 2 1 n log(n 1 C) n 1 m n 3 1 log(n 1 C) Cost scaling w nm log( n 2 m ) n 1 m log( n 2 m ) n 1 m log( n 2 1 m 2) dynamic trees [17] Delta log(nC) Delta log(n 1 C) Delta log(n 1 C) Table 1.1 A summary of the results discussed in this paper. Column 2 contains ....

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, Solving minimum-cost flow problems by successive approximation, Mathematics of Operations Research, 15 (1990), pp. 430--466.

....models there is. For completeness the basic results for contraction models were also derived. The main results of this chapter are published in [72] To the author s best knowledge there has been no work in ADPs concerning general policies. Some recent related work has been done by Waldmann [83] who developed a highly general model of dynamic programming problems, with a focus on deriving approximation bounds. Heger [28, 29] extended many of the standard mdp results to cover the risk sensitive model. Although his work derives many of the important theorems, it does not present these ....

K.-H. Waldmann. On bounds for dynamic programs. Mathematics of Operations Research, 10(2):220--232, May 1985.

....Markov chain with states X and transition probabilities p(x; y) P a (x; a)P (x; a; y) infinite state and action spaces. His model can be viewed as the continuation of the work of Bertsekas [5] and Bertsekas and Shreve [6] who proved similar statements under different assumptions. Waldmann [58] developed a highly general model of dynamic programming problems, with a focus on deriving approximation bounds. Heger [15, 16] extended many of the standard mdp results to cover the risk sensitive model. Although his work derives many of the important theorems, it does not present these ....

K.-H. Waldmann. On bounds for dynamic programs. Mathematics of Operations Research, 10(2):220--232, May 1985.

.... x such that x 2 P : fx : l A T x ug: 1) For convenience we denote r : l u 2 and s : u Gammal 2 . We also denote by e the all one vector and e j the j th column of the identity matrix. In [5] Burrell and Todd proposed a parallel cut ellipsoid algorithm based on the results of Todd [29]. Note that P can be alternately written as P = fx 2 R n : a T i x Gamma l i ) a T i x Gamma u i ) 0; i = 1; mg (2) where a i is the i th column of A and l i , u i are the corresponding components of l, u respectively. Now choose a nonnegative diagonal matrix D = diag(d) diag(d ....

....shows that E = fx 2 R n : x Gamma x c ) T ADA T (x Gamma x c ) x T c (ADA T )x c Gamma l T Dug (4) where x c : x c (d) ADA T ) Gamma1 ADr (5) is the center of E. If the current center violates some constraint, say, l i a T i x u i , by the results of Todd [29] we can construct a new ellipsoid that contains that part of the previous one between the parallel hyperplanes a T i x = l i and a T i x = u i , and the volume of the ellipsoid decreases by a factor which is, at worst, exp( Gamma 1 2(n 1) Consider the problem min v(d) f(d) Delta h(d) ....

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, On minimum volume ellipsoids containing part of a given ellipsoid, Mathematics of Operations Research, 7 (1980), pp. 253--261.

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W. Cook and S. Dash, On the matrix-cut rank of polyhedra, Mathematics of Operations Research 26 (2001), 19 -- 30.

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W. Cook and S. Dash, On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26:19-30, 2001.

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