R Giles, W. Pulleyblank (1979), Total dual integrality and integer polyhedra, Linear algebra and its applications, 25, 191-196.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Saddle Points B C first order second order conditions Local minima Constrained Local minima Fig. 2. Relationship among solution sets of Lagrangian methods for solving continuous problems. 2. 3 Lagrangian Relaxation There is a class of algorithms called Lagrangian relaxation [7, 8, 6, 24, 3] proposed in the literature that should not be confused with the Lagrange multiplier methods proposed in this paper. Lagrangian relaxation reformulates a linear integer minimization problem: z = minimize x Cx subject to Gx b where x is an integer vector of variables (7) 0 and C and G are ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....not perform well because lower bounds found using linearized constraints may be inaccurate when constraints are highly nonlinear, and inaccurate bounds may lead to incorrect pruning and infeasible solutions. 2.1. 3 Lagrangian relaxation There is a class of algorithms called Lagrangian relaxation [72, 74, 68, 142, 41] proposed in the literature that should not be confused with our proposed discrete constrained optimization method using Lagrange multipliers. Lagrangian relaxation reformulates a linear integer minimization problem: z = minimize Cx subject to Gx b where x is an integer vector of variables ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....bounds found through linearized constraints may be inaccurate when constraints are highly nonlinear, and inaccurate bounds may lead to incorrect pruning and infeasible solutions when the algorithm terminates. 2.1. 4 Lagrangian Relaxation There is a class of algorithms called Lagrangian relaxation [72, 74, 64, 180, 16] proposed in the literature that should not be confused with our proposed discrete constrained optimization method using Lagrange multipliers. Lagrangian relaxation reformulates a linear integer minimization problem: z = minimize x Cx subject to Gx b where x is an integer vector of variables ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....generalized covering problem. Constraint relaxation is often helpful for these methods because, after constraint relaxation, constraints become easy to satisfy, leading to easier to solve transformed 0 1 problems. 2.1. 3 Lagrangian relaxation A class of algorithms is called Lagrangian relaxation [69, 70, 65, 146, 33] that are often used in linear integer programming. Lagrangian relaxation transforms an linear integer minimization problem: z = minimize Cx subject to Gx b where x is an integer vector of variables (2.1) 0 and C and G are constant matrices into the following form: L(#) minimize (Cx ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....the previous section. Proof of Theorem 1.1. Let H(L;W ) as in Remark 3.1. Then we obviously have that P l is integral for all l 2 L if and only if P h is integral for all h 2 H(L;W ) For the proof of Theorem 1. 2 we need a well known characterization of TDI systems due to Giles and Pulleyblank [GP79]. Theorem 4.1. Let A 2 Z and b 2 Z . The system Ax b is TDI if and only if for every minimal face of P b the set of row vectors that are tight at this face determine a Hilbert basis of the cone that they generate. Proof of Theorem 1.2. Theorem 4.1 implies that for two strongly ....

R. Giles and W. R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Applications 25 (1979), 191-196.

....if there exists a hyperplane a x 0 such that f0g = fx 2 C : a x 0g. In the following we always consider rational polyhedral cones and call them cones for short. We are interested in a special subset of integral vectors in a cone, namely an integral generating set. De nition 3.3. [33] Let C be a rational polyhedral cone. A nite set H C Z Hilbert basis of C if every integral vector in C can be represented as a non negative integral combination of the elements of H. Example 3.1. Let C = fy 2 R : y 1 = 1 2 ; y 2 = 3 1 2 : 1 ; 2 0g: The set g does ....

....of A, let S be the set of shortest conic combinations with respect to the function b, i.e. S = fy 0 : A y = c such that b y is minimalg: Then Ax b is called TDI if there exists an integral vector in S. This geometric property can be expressed using Hilbert bases. Theorem 3.13. [33] Let A 2 Q and b 2 Q . The system Ax b is TDI if and only if for every face F of P = fx 2 R : Ax bg the set of row vectors that determine F is a Hilbert basis of the cone generated by these row vectors. In fact, the converse of Theorem 3.12 is also true. Theorem 3.14. 33] If a ....

[Article contains additional citation context not shown here]

F.R. Giles and W.R. Pulleyblank (1979), Total dual integrality and integer polyhedra, Linear Algebra and Applications 25, 191 - 196.

....vectors from K with integer coordinates. K is obviously a polyhedral cone and the vectors ; ff and fi in equation 2.3 are all from Z. To restate the existence of a subset C of S with the properties mentioned above we need the notion of an integral Hilbert basis. This notion was first introduced by [GP]. The following definition is a slight reformulation of the one in Chapter 16 of [Sc] Definition 2.1. Given a polyhedral cone K, let Z be the set of all the integer vectors in K. A finite set of vectors fa 1 ; a 2 ; a t g from Z is called an integral Hilbert basis if each integral vector ....

F.R.Giles, W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Its Applications, 25, pp. 191--196, 1979.

....a generalized covering problem. Constraint relaxation is often helpful for these methods because, after constraint relaxation, constraints become easy to satisfy, leading to easier to solve transformed 0 1 problems. 2.1. 3 Lagrangian relaxation A class of algorithms is called Lagrangian relaxation [69, 70, 65, 146, 33] that are often used in linear integer programming. Lagrangian relaxation transforms an linear integer minimization problem: z = minimize Cx subject to Gx # b where x is an integer vector of variables (2.1) x # 0 and C and G are constant matrices into the following form: L(#) minimize (Cx ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....the cone C and it is denoted by H(C) Supported by a Leibniz Preis of the German Science Foundation (DFG) awarded to M. Grotschel. y Supported by a Gerhard Hess Forschungsforderpreis of the German Science Foundation (DFG) 1 The name Hilbert basis was introduced by Giles and Pulleyblank [GP79] in the context of totally dual integral systems. It was shown by Gordan [G1873] that every rational polyhedral cone has an integral basis and for pointed cones we have the following result due to van der Corput [Cor31] The integral basis H(C) of a rational, pointed cone C R n is uniquely ....

F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Lineare Algebra Appl. 25, 191 - 196 (1979).

....not perform well because lower bounds found using linearized constraints may be inaccurate when constraints are highly nonlinear, and inaccurate bounds may lead to incorrect pruning and infeasible solutions. 2.1. 3 Lagrangian relaxation There is a class of algorithms called Lagrangian relaxation [72, 74, 68, 142, 41] proposed in the literature that should not be confused with our proposed discrete constrained optimization method using Lagrange multipliers. Lagrangian relaxation reformulates a linear integer 18 minimization problem: z = minimize Cx subject to Gx b where x is an integer vector of variables ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....cone. A finite subset H = fh 1 ; h t g C Z n is a Hilbert basis of C if every z 2 C Z n has a representation of the form z = t X i=1 i h i ; with non negative integral multipliers 1 ; t . The name Hilbert basis was introduced by Giles and Pulleyblank [GP79] in the context of totally dual integral systems. Essential is (see [G1873] C31] Theorem 1.1. Every rational polyhedral cone has a Hilbert basis. If it is pointed, then there exists a unique Hilbert basis that is minimal w.r.t. inclusion. In the following by a cone we always mean a rational ....

F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Lineare Algebra Appl. 25, 191 - 196 (1979).

....previous section. Proof of Theorem 1.1. Let H(L;W ) as in Remark 3.1. Then we obviously have that P l is integral for all l 2 L if and only if P h is integral for all h 2 H(L;W ) For the proof of Theorem 1. 2 we need a well known characterization of TDI systems due to Giles and Pulleyblank [GP79]. Theorem 4.1. Let A 2 Z m n and b 2 Z m . The system Ax b is TDI if and only if for every minimal face of P b the set of row vectors that are tight at this face determine a Hilbert basis of the cone that they generate. Proof of Theorem 1.2. Theorem 4.1 implies that for two strongly ....

R. Giles and W. R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Applications 25 (1979), 191-196.

....rational generators. A nite set H = fh 1 ; h t g C Z d is a Hilbert basis of C if every z 2 C Z d has a representation of the form z = t X i=1 i h i ; with non negative integral multipliers 1 ; t . The name Hilbert basis was introduced by Giles and Pulleyblank [12] in the context of totally dual integral systems. Note that every pointed, rational cone has a unique Hilbert basis that is minimal with respect to inclusion ( 28, 35] Let O j be the j th orthant of Z d and H j (A) be the unique minimal Hilbert basis of the pointed rational cone fv 2 R d : ....

F. R. Giles and W. R. Pulleyblank. Total dual integrality and integer polyhedra. Linear Algebra and its Applications, 25:191-196, 1979.

....there exists a hyperplane a T x # 0 such that 0 = x # C : a T x # 0 . In the following we always consider rational polyhedral cones and call them cones for short. We are interested in a special subset of integral vectors in a cone, namely an integral generating set. Definition 3.3. [33] Let C be a rational polyhedral cone. A finite set H # C # Z n is a Hilbert basis of C if every integral vector in C can be represented as a non negative integral combination of the elements of H. Example 3.1. Let C = y # R 2 : y 1 = # 1 # 2 , y 2 = 3# 1 # 2 : # 1 , # 2 # 0 . The ....

....of A, let S be the set of shortest conic combinations with respect to the function b, i.e. S = y # 0 : A T y = c such that b T y is minimal . Then Ax # b is called TDI if there exists an integral vector in S. This geometric property can be expressed using Hilbert bases. Theorem 3.13. [33] Let A # Q mn and b # Q m . The system Ax # b is TDI if and only if for every face F of P = x # R n : Ax # b the set of row vectors that determine F is a Hilbert basis of the cone generated by these row vectors. In fact, the converse of Theorem 3.12 is also true. Theorem 3.14. ....

[Article contains additional citation context not shown here]

F.R. Giles and W.R. Pulleyblank (1979), Total dual integrality and integer polyhedra, Linear Algebra and Applications 25, 191 - 196.

.... m and call S(a; b) the submonoid of N n m of all its solutions, i.e. S(a; b) y; z) 2 N n m P i a i y i P j b j z j = 0 ; the Hilbert basis S min (a; b) S(a; b) is the set of all minimal nontrivial solutions of the equation, with respect to the componentwise ordering (see [1]) Lemma 4 ( 4] The monoid S(a; b) is generated by S min (a; b) which is of finite cardinality; for any solution (y; z) 2 S min (a; b) kyk 1 max j b j and kzk 1 max i a i . As a consequence, it is possible to prove the following Theorem 4 Let x 2 hc 1 ; c n i; then (x) ....

GILES, F. R., AND PULLEYBLANK, W. R. Total dual integrality and integer polyhedra. Linear Algebra Appl. 25 (1979), 191--196.

.... S(a, b) the submonoid of N n m of all its solutions, i.e. S(a, b) # (y, z) # N n m # # # i a i y i # j b j z j = 0 # ; the Hilbert basis S min (a, b) # S(a, b) is the set of all minimal nontrivial solutions of the equation, with respect to the componentwise ordering (see [GP79] Lemma 4 ( Lam87] The monoid S(a, b) is generated by S min (a, b) which is of finite cardinality; for any solution (y, z) # S min (a, b) #y# 1 # max j b j and #z# 1 # max i a i . As a consequence, it is possible to prove the following Theorem 4 Let x # #c 1 , c n #; ....

F. R. Giles and W. R. Pulleyblank. Total dual integrality and integer polyhedra. Linear Algebra Appl., 25:191--196, 1979.

....call S(a, b) the submonoid of N n m of all its solutions, that is, S(a, b) # ( y, z) # N n m # # # i a i y i # j b j z j = 0 # . The Hilbert basis S min (a, b) # S(a, b) is the set of all nontrivial solutions that are minimal with respect to the componentwise ordering (see [1]) Lemma 7 ( 4] The monoid S(a, b) is generated by S min (a, b) which is of finite cardinality; for every solution ( y, z) # S min (a, b) #y# 1 # max j b j and #z# 1 # max i a i . As a consequence, it is possible to prove the following Theorem 8 Let x # #c 1 , c n #; ....

....# 2 . PROOF. Since t n # F # log # ( # 5n) # , by the previous lemma the set of points t# p p # Z, t n = t# p p # Z, t n determines at least one interval of length greater than 1 2 # 2 # log # ( # 5n) # 2 # 1 2 (# # 5n) 2 in [0, 1]. Choosing x as its middle point, we have the thesis. 2 Now we can state a lower bound for # : Theorem 18 For every # # (0, 1) there exists an x # (0, 1) such that # (x) # 2# 1 2 # 5# # 3. PROOF. Let n = # # (2 # 5#) # ; then, by the previous lemma, a simple substitution ....

[Article contains additional citation context not shown here]

F. R. Giles and W. R. Pulleyblank. Total dual integrality and integer polyhedra. Linear Algebra Appl., 25:191--196, 1979.

....integer vectors of 1 the form T A; 2 [0; 1] m . This upper bound is exponential in the encoding length of A, even in xed dimension. One can further restrict the cutting planes c T x bc to those corresponding to a totally dual integral (TDI) system de ning P (Edmonds Giles 1977, Giles Pulleyblank 1979, Schrijver 1980) The number of inequalities of a minimal TDI system for a polyhedron P can still be exponential in the size of P , even in xed dimension (Schrijver 1986, p. 317) The contributions of this paper are twofold. In the rst part, we prove that in xed dimension the number of ....

Giles, F. R. & Pulleyblank, W. R. (1979), `Total dual integrality and integer polyhedra', Linear Algebra and its Applications 25, 191 - 196.

....if there exists a hyperplane a T x 0 such that f0g = fx 2 C : a T x 0g. In the following we always consider rational polyhedral cones and call them cones for short. We are interested in a special subset of integral vectors in a cone, namely an integral generating set. De nition 3.3. [33] Let C be a rational polyhedral cone. A nite set H C Z n is a Hilbert basis of C if every integral vector in C can be represented as a non negative integral combination of the elements of H. Example 3.1. Let C = fy 2 R 2 : y 1 = 1 2 ; y 2 = 3 1 2 : 1 ; 2 0g: The set f(1; ....

....of A, let S be the set of shortest conic combinations with respect to the function b, i.e. S = fy 0 : A T y = c such that b T y is minimalg: Then Ax b is called TDI if there exists an integral vector in S. This geometric property can be expressed using Hilbert bases. Theorem 3.13. [33] Let A 2 Q m n and b 2 Q m . The system Ax b is TDI if and only if for every face F of P = fx 2 R n : Ax bg the set of row vectors that determine F is a Hilbert basis of the cone generated by these row vectors. In fact, the converse of Theorem 3.12 is also true. Theorem 3.14. 33] If a ....

[Article contains additional citation context not shown here]

F.R. Giles and W.R. Pulleyblank (1979), Total dual integrality and integer polyhedra, Linear Algebra and Applications 25, 191 - 196.

.... like combinatorial convexity and Supported by a Gerhard Hess Forschungsforderpreis of the German Science Foundation (DFG) toric varieties (cf. e.g. DHH98] Ewa96] Oda88] Stu96] polynomial rings and ideals (cf. e.g. BG98] BGT97] or in integer programming (cf. e.g. Gra75] [GP79], Sch80] Seb90] Wei98] their structure is not very well understood yet. A first systematic study was given by Sebo [Seb90] In particular, the following three conjectures about the nice geometrical structure of Hilbert bases of an integral pointed polyhedral cone C ae R n are due to ....

....8; 8; 8; 8) We are grateful to T. Hibi and A. Sebo for asking us about the existence of a 0=1 embedding. 3 Remarks Finally, we want to remark that the name Hilbert basis was introduced by Giles Pulleyblank in their investigations of so called TDI systems in integer linear programming [GP79]. An integral linear system Ax b, A 2 Z m Thetan , b 2 Z m , is called TDI (totally dual integral) if the minimum in the linear programming duality equation minfb y : A y = c; y 0g = maxfc x : Ax bg (3.1) can be achieved by an integer vector y 2 Z m for each integer ....

[Article contains additional citation context not shown here]

F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra Appl. 25 (1979), 191--196.

....Saddle Points A B C satisfying the first order second order conditions Local minima Constrained Local minima Fig. 2. Relationship among solution sets of Lagrangian methods for solving continuous problems. 2. 3 Lagrangian Relaxation There is a class of algorithms called Lagrangian relaxation [7, 8, 6, 24, 3] proposed in the literature that should not be confused with the Lagrange multiplier methods proposed in this paper. Lagrangian relaxation reformulates a linear integer minimization problem: z = minimize x Cx subject to Gx b where x is an integer vector of variables (7) x 0 and C and G are ....

F. R. Giles and W. R. Pulleyblank. Total Dual Integrality and Integer Polyhedra, volume 25. Elsevier North Holland, Inc., 1979.

....the results on Hilbert bases of cut cones, section 5 explores Hilbert bases for other lattices, which could lead to an algorithmic approach and section 6 describes the implications of these results on a related object of quadric programming. 2 Hilbert bases Hilbert bases were introduced in [GP 79] after the work of Hilbert (in relation with the Nullenstellensatz) and Gordan. A clear description of these notions can be found for example in [Sc 86] 2.1 Definition and fundamental theorems Let C be a polyhedral cone, integral over a lattice L (C is a generated as a cone, by a finite number ....

F.R. Giles, W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and its Applications 25, p.191-196, 1979

....are integral. TDI systems were introduced by Edmonds and Giles (1977) Each iteration of Schrijver s procedure consists of the following two steps. 1. Given a rational polyhedron P , find a TDI system Ax b defining P , with A integral. 2. Round down the right hand side b. It has been proved by Giles and Pulleyblank (1979) and Schrijver (1981) that there exists a TDI system as in step 1 of Schrijver s procedure for every rational polyhedron P , and that the TDI system is unique if P is full dimensional. Finding such a TDI system can be done in finite time. After one iteration of the above procedure we get a ....

R. Giles and W.R. Pulleyblank (1979) "Total dual integrality and integer polyhedra", Linear Algebra and Its Applications 25 191--196.

....the submonoid of N nCm of all its solutions, i.e. S.a; b D Phi . y; z 2 N nCm fi fi P i a i y i Gamma P j b j z j D 0 Psi I the Hilbert basis S min .a; b ae S. a; b is the set of all minimal nontrivial solutions of the equation, with respect to the componentwise ordering (see [GP79] Lemma 4 ( Lam87] The monoid S.a; b is generated by S min .a; b , which is of Thetanite cardinality; for any solution . y; z 2 S min .a; b , kyk 1 max j b j and kzk 1 max i a i . As a consequence, it is possible to prove the following Theorem 4 Let x 2 hc 1 ; c n i; then .x ....

F. R. Giles and W. R. Pulleyblank. Total dual integrality and integer polyhedra. Linear Algebra Appl., 25:191#196, 1979.

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R Giles, W. Pulleyblank (1979), Total dual integrality and integer polyhedra, Linear algebra and its applications, 25, 191-196.

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