By Jean-Bernard Lasserre

ISBN-10: 038709413X

ISBN-13: 9780387094137

In this publication 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 recognized non-stop duality options, and the reason in the back of them. additionally, the process highlights the variation among the discrete and non-stop situations. significant within the research are the continual and discrete Brion and Vergne's formulae for linear integration and counting. This technique presents a few new insights on duality techniques for integer courses, and likewise 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 publication will serve graduate scholars and researchers in utilized arithmetic, optimization, operations examine and desktop technological know-how. as a result huge functional significance of a few offered difficulties, researchers in different parts also will locate this booklet useful.

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During this booklet the writer analyzes and compares 4 heavily comparable difficulties, particularly linear programming, integer programming, linear integration, linear summation (or counting). the point of interest is on duality and the strategy is very novel because it places integer programming in point of view with 3 linked difficulties, and allows one to outline discrete analogues of famous non-stop duality recommendations, and the reason at the back of them.

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Extra resources for Linear and Integer Programming vs Linear Integration and Counting: A Duality Viewpoint

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Otherwise, we just need to multiply each constraint (Ax)i = yi by y1 = 0 when i = 2, 3, . . , m, so that the new matrix A and vector y still have entries in Z. Hence, there exists a vector D ∈ Zm with first entry D1 = 1 and such that y = Dy1 . Notice that D may have entries equal to zero or even negative, but not its first entry. 10), em := (1, 1, . ) is the vector of ones in Rm , and the real (fixed) vector w ∈ Rm + satisfies A ln(w) > c. 16). Consider the following simple change of variables.

Roughly speaking, we inductively compute real constants Qσ ,β and a fixed positive integer M, all of them completely independent of y, such that the counting function fd is given by the finite sum fd (y, c) = ∑ Aσ ∑ β ∈Zm , β ≤M Qσ ,β × ecσ x 0 m if x := A−1 σ [y − β ] ∈ N otherwise, where the first finite sum is computed over all invertible [m × m]-square submatrices Aσ of A. Crucial in our algorithm is an explicit decomposition in closed form (and thus an explicit formula for fd (y, c)) for the case n = m + 1, which we then repeatedly use for the general case n > m + 1.

P2 − z42 )(z2 − 1)(z32 − p)(p3 − z52 ) Next, we integrate I1 along the circle |z2 | = z∗2 . Taking p := p∗ as a constant, the first term of I1 has poles on circles of radii |p| > z∗2 , 1 < z∗2 , |p|1/3 < z∗2 , and |p|−1 < z∗2 . We consider poles outside the circle |z2 | = z∗2 in order to avoid considering the pole z2 := p1/3 , and we obtain − p3 . 4 Inversion of the Z-transform by residues 51 The second term of I1 has poles on circles of radii |p|1/2 > z∗2 , 1 < z∗2 , |p|1/3 < z∗2 , and |p|3/5 > z∗2 .

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