By Jean-Bernard Lasserre

ISBN-10: 038709413X

ISBN-13: 9780387094137

In this ebook the writer analyzes and compares 4 heavily similar difficulties, specifically linear programming, integer programming, linear integration, linear summation (or counting). the point of interest is on duality and the procedure is very novel because it places integer programming in viewpoint with 3 linked difficulties, and allows one to outline discrete analogues of famous non-stop duality innovations, and the explanation in the back of them. additionally, the technique highlights the adaptation among the discrete and non-stop instances. principal within the research are the continual and discrete Brion and Vergne's formulae for linear integration and counting. This process offers a few new insights on duality recommendations for integer courses, and in addition allows to retrieve and shed new mild on a few famous effects. for example, Gomory relaxations and the summary superadditive twin of integer courses are re-interpreted during this algebraic approach.

This e-book will serve graduate scholars and researchers in utilized arithmetic, optimization, operations learn and laptop technological know-how. a result of tremendous functional significance of a few provided difficulties, researchers in different parts also will locate this publication useful.

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10). 12) is obtained by analyzing the integral |z|=r zw dz with r > 0. This integral is equal to 2π i only if w = −1, whereas if w is any integer different than A c −1, then the integral is equal to zero. It remains to show that the domain {v ∈ Rm + |v > e } m (equivalently, {v ∈ R+ |A ln v > c}) is not empty. 1. , evaluating the complex integral I∗d : fd (y, c) = A where w ∈ Rm + satisfies w > c. 3 A dual approach 47 Exactly like the inversion problem I∗ is a dual of I, the inversion problem I∗d is a dual problem of Id .

M − 1. t. s1 , s2 , . , sm−1 , we get that H(p) is analytic on the domain of all points p ∈ C such that ℜ(p) = β , b for some real vector β = (γ1 , . . , γm−1 , βm ) > 0 with A β > 0. t. s1 , s2 , . , sk , 1 ≤ k ≤ m − 2, we get a function which is analytic on the domain of all points (sk+1 , . . , sm−1 , p) ∈ Cm−k such that |2β j s j − 1| = 1, s j = 0, j = k + 1, . . , m − 1, and ℜ(p) = β , y for some real vector β = (γ1 , . . , γk , βk+1 , . . , βm ) > 0 with A β > 0. Thus, there is no pole on the next integration path {|2γk+1 sk+1 − 1| = 1} or the final one {ℜ(p) = d}.

29] and Gritzmann and Klee [62]. In particular, improved versions of some of the above algorithms are also described in [29], and the software package VINCI developed at ETH in Z¨urich offers several alternative methods working with the half-space description or the vertex description (or both) of Ω . html. Most of the material in this chapter is from Barvinok [13], B¨ueler et al. [29], Brion and Vergne [27], and Lasserre and Zeron [95]. The residue method described in this chapter does not require one to work with a vector c ∈ Rn having some regularity property, as in Lawrence [102] and Brion and Vergne [27].

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