W. Cook, A.M.H. Gerards, A. Schrijver, E. Tardos, Sensitivity theorems in integer linear programming, Math. Programming 34 (1986) 251--264.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....test set for (P ) c;b if it contains a test set for every cost vector c 2 R n . In contrast to the common de nition of test sets we do not impose niteness on T . This allows a treatment of test sets for mixed integer programs which need not be nite in general (see for example Cook et. al [5]) Once a nite (universal) test set T for (P ) c;b is available the following augmentation algorithm can be applied in order to solve the above optimization problem. This work was supported by the Schwerpunktprogramm Echtzeit Optimierung gro er Systeme of the Deutsche ....

....of the above augmentation procedure. We demonstrate that particularly in the LP case some caution is appropriate to avoid zig zagging (even to non optimal solutions) and we present a strategy which ensures termination. For the MIP case we present an example similar to the one by Cook et. al [5] which implies that MIP test sets need not be nite in general. In [5] the existence of nite MIP test sets is shown if considerations are restricted to a smaller family of optimization problems. In contrast to this approach we introduce a nite set of integer vectors which is no test set by the ....

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W. Cook, A. M. H. Gerards, A. Schrijver, and  E. Tardos. Sensitivity theorems in integer linear programming. Mathematical Programming, 34:251-264, 1986.

....length at most (n d 1 1) n 1) Chv atal (1973) showed that every bounded polyhedron P R n has nite rank. Schrijver (1980) extended this result to possibly unbounded, but rational polyhedra P = fx 2 R n j Ax bg, where A 2 Q m n ; b 2 Q m . Both results are implicit in (Gomory 1958) Cook, Gerards, Schrijver Tardos (1986) and Gerards (1990) proved that for every matrix A 2 Z m n there exists t 2 N such that for all right hand sides b 2 Z m , the Chv atal rank of P b = fx 2 R n j Ax bg is bounded by t. Already in dimension 2, there exist rational polyhedra of arbitrarily large Chv atal rank (Chv atal ....

Cook, W., Gerards, A. M. H., Schrijver, A. & Tardos, E. (1986), `Sensitivity theorems in integer linear programming', Mathematical Programming 34, 251 - 264.

....fl = 1 and ae = 1. The latter are two characteristics of Hoffman s bound that are particularly worth noting; namely, it holds globally for all vectors in n and with the residual itself as the upper bound. Hoffman s result has been strengthened and refined by many researchers; see, for example, [3, 4, 8, 12, 21, 34]. Among other things, some tight estimates of the multiplicative constant have been obtained. For general systems of convex differentiable inequalities, Robinson [33] established the error bound (1.2) under a Slater condition and the boundedness assumption of the solution set. This result was ....

W. Cook, A.M.H. Gerards, A. Schrijver and ' E. Tardos, "Sensitivity theorems in integer linear programming," Mathematical Programming 34 (1986) 251--264.

....the convex hull of P Z n . Chv atal [Chv73] showed that every bounded polyhedron P R n has finite rank. Schrijver [Sch80] extended this result to possibly unbounded but rational polyhedra P = fx 2 R n j Ax bg, where A 2 Q m Thetan ; b 2 Q m . Cook, Gerards, Schrijver, and Tardos [CGST86, Ger90] proved that for every matrix A 2 Z m Thetan there exists t 2 N such that for all righthand sides b 2 Z m , the Chv atal rank of P b = fx 2 R n j Ax bg is bounded by t. More details on these results can also be found in [Sch86, NW88] Already in dimension 2, there exist rational polyhedra ....

W. Cook, A. M. H. Gerards, A. Schrijver, and E. Tardos. Sensitivity theorems in integer linear programming. Mathematical Programming, 34:251 -- 264, 1986.

....of size OE. Suppose b is any point in R m with b 0. So, 0 is in K b . Define b 0 = b 0 1 ; b 0 2 ; b 0 m ) by : b 0 i = minfb i ; n2 3OE g: Then, K b Z n = K b 0 Z n : Proof : The proof is based on the following fact due to Cook, Gerards, Schrijver and Tardos [3] : Let Delta be the maximum absolute value of any subdeterminant of A. If a point p belongs to K b , and if K b contains some point in Z n , then there is a point q 2 Z n K b with jp i Gamma q i j n Delta for i = 1; 2; n. This fact is true for any right hand side b. It is clear ....

W.Cook, A.M.H.Gerards, A.Schrijver and E.Tardos, Sensitivity theorems in integer linear programming , Mathematical Programming 34 (1986) pp 251-264

....LPA , and these are precisely the set of edge directions of Sigma(A) Theorem 1.8) In Section 2 we examine the reduced Grobner basis G c and the universal Grobner basis UGBA . The set UGBA is contained in the Graver basis of A, which is the well known test set introduced in [Gra] see also [BJ] [CGST]) Example 2.11 shows that the Graver basis can be much larger than the universal Grobner basis. The integer analogue of the regular triangulation Delta c is the (initial) monomial ideal in c (I A ) We show how the two are related (Theorem 2.4) The main result in Section 3 is the structure ....

....showed that H : oe H oe nf0g is a universal test set for A. We call H the Graver basis of A. It is equivalent to the universal test set of IPA due to Blair and Jeroslow in [BJ] and under an appropriate transformation, to the universal test set due to Cook, Gerards, Schrijver and Tardos in [CGST] (see also Section 17.4 in [Schr] The following theorem relates the Graver basis and the universal Grobner basis of A. Theorem 2.7 [Th] The Graver basis of A contains the universal Grobner basis UGBA . Corollary 2.8 There exists only finitely many distinct reduced Grobner bases associated with ....

W. Cook, A.M.H. Gerards, A. Schrijver and ' E. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming 34 (1986) 251--264.

....R. THOMAS literature. In 1975, Graver [Gra] showed the existence of a finite set of vectors that solve all integer programs with coefficient matrix A. We call this test set the Graver basis of A and it will be discussed further in Section 4. Variants of the Graver basis appear in both [BJ] and [CGST]. In 1981, Scarf [Sca] introduced a test set for integer programs called the neighbors of the origin. Relationships among these test sets (including Grobner bases) are discussed in [Tho] Scarf s ideas have been used recently to provide new results on minimal resolutions of monomial ideals in ....

....since A 2 Z d Thetan . Graver showed that GrA : oe H oe nf0g, which we call the Graver basis of A, is a universal test set for IPA . It is equivalent to the universal test set of IPA due to Blair and Jeroslow [BJ] and to the universal test set due to Cook, Gerards, Schrijver and Tardos in [CGST] (see also Section 17.4 in [Sch] Let UGBA denote the union of all reduced Grobner bases G c as c varies over all generic cost vectors in R n . Clearly, UGBA is a uniquely defined universal test set for IPA and we call it the universal Grobner basis of A. The sets GrA and UGBA are related as ....

W. Cook, A.M.H. Gerards, A. Schrijver and ' E. Tardos,, Sensitivity theorems in integer linear programming, Mathematical Programming 34 (1986), 251-264.

....literature. In 1975, Graver [17] showed the existence of a finite set of vectors that solve integer programs of the form IP A;c (b) for all b and all c. We call this test set the Graver basis of A and it will be discussed further in Section 4. Variants of the Graver basis appear in both [6] and [9]. In 1981, Scarf [23] introduced a test set for integer programs called the neighbors of the origin. Relationships among these test sets (including Grobner bases) are discussed in [30] 4 Universal test sets for linear and integer programming Thus far we examined test sets for linear and integer ....

W. Cook, A.M.H. Gerards, A. Schrijver and ' E. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming Vol.34 (1986) pp. 251-264.

....in IP A;c as follows: Given a feasible solution ff to the integer program IP A;c (b) the optimal solution fi to IP A;c (b) is the exponent vector of the normal form of x ff with respect to G c . The reduced Grobner basis G c is a test set for the family IP A;c [Tho] in the sense of [BJ] [CGST], Gra] Sca] and [Sch] Proposition 1.1 [Tho] i) The exponent vectors of the monomials in the initial ideal in c (I A ) are precisely the non optimal solutions to the programs in IP A;c . ii) The monoid N(A) is in bijection with the set of exponent vectors of standard monomials of in c (I A ....

W. Cook, A.M.H. Gerards, A. Schrijver and ' E. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming 34 (1986) pp. 251--264.

....constraints can be altered in certain ways without affecting the proof. The classical duality based sensitivity analysis for linear programming has been generalized to nonlinear and integer programming, in the latter case by analyzing how the optimal value depends on the vector of right hand sides [1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 24, 25]. But the generalization suggested here takes a different direction. Rather than viewing the dual solution as a numerical function of right hand sides, it views the dual solution as encoding a proof of optimality. The classical linear programming dual can be seen as a special case of this, because ....

Cook, W., A. M. H. Gerards, A. Schrijver and E. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming 34 (1986) 251-264.

....in the accompanying Part II of this paper. Research carried out in the Netherlands involves both theoretical and more problem specific results. Gerards and Schrijver have considered several important theoretical issues, see e.g. Schrijver (1980,1981) Grotschel, Lov asz and Schrijver (1981) Cook, Gerards, Schrijver and Tardos (1986). Here we also want to mention the result of H.W. Lenstra (1983) that the integer programming problem (1) can be solved in polynomial time for a fixed number of variables. Although not specifically a result in polyhedral combinatorics, it is central in integer programming and combinatorial ....

W. Cook, A.M.H. Gerards, A. Schrijver, and ' E. Tardos (1986) "Sensitivity theorems in integer linear programming", Mathematical Programming 34 251--264.

....locality between the optima of the problems P and IP . The problem of determining upper bounds on the deviation between w an optimum of an integer linear problem and x an optimum of the associated linear problem has been addressed in a general context by Cook, Gerards, Schrijver and Tardos [5]. Unfortunately, their results are not applicable in our case, since their bound for kx Gamma w k1 is much larger than 2, the maximum deviation. By exploiting the peculiarities of the constraints matrix and of the objective function of P in (2) we had the hope to find a small value 2 ....

W. Cook, A. M. H. Gerards, A. Schrijver, and E. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming, 34 (1986), pp. 251--264.

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W. Cook, A.M.H. Gerards, A. Schrijver, E. Tardos, Sensitivity theorems in integer linear programming, Math. Programming 34 (1986) 251--264.

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Cook, W., Gerards, A.M.H., Scrijver, A. and Tardos, ' E. Sensitivity Theorems in Integer Linear Programming. Mathematical Programming 34 (1986), pp. 251-264.

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Cook, W., Gerards, A.M.H., Schrijver, A., Tardos, ' E. (1986): Sensitivity theorems in integer linear programming. Mathematical Programming, 34:251--264.

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