Dual-Feasible Functions for Integer Programming and Combinatorial Optimization by Claudio Alves Francois Clautiaux José Valerio de Carvalho & Jurgen Rietz

Author:Claudio Alves, Francois Clautiaux, José Valerio de Carvalho & Jurgen Rietz
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof

Define the auxiliary function as

(3.8)

Then, we have

First, some properties of h are derived, which will be used later to prove the sufficient conditions of Theorem 3.1. Clearly, one obtains h(x) = 0 for x ≤ 0 and h(1) = 1. Moreover, h rises monotonely in the closed interval [0, 1], because h is piecewise constant and one gets for any the estimations

Additionally, we have

(3.9)

due to the definition of h, because either or h(x) = x. In the latter case, it holds that , such that both inequalities become equivalent to x ≥ 0 or x ≤ 1, respectively. The function h is also symmetric inside the interval [0, 1], i.e.

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