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Zbl 0982.90054
Ryoo, Hong Seo; Sahinidis, Nikolaos V.
Analysis of bounds for multilinear functions.
(English)
[J] J. Glob. Optim. 19, No.4, 403-424 (2001). ISSN 0925-5001; ISSN 1573-2916/e

The paper is concerned with the bounding of multilinear functions, which are defined as: $\sum_{i=1}^t a_i \prod_{j\in J_i} y_i$. Such functions are the building blocks of a variety of nonconvex optimization problems. One can bound such functions via bounds for monomial functions, which are defined as the product of $p$ variables, i.e. as $\varphi^p(y):= \prod_{j=1}^p y_j$. \par The present paper considers four bounding schemes: the arithmetic interval method, the logarithmic transformation method and a new scheme, which is a variant of the interval method, the exponent transformation method and a new scheme, which is a variant of the interval method. The tightness of the four bounding schemes is theoretically compared. Further, it is proved that one of the four bounding schemes provides the convex envelope and that two of the schemes provide the concave envelope for the monomial $\varphi^p(y)$ over $\bbfR_+^p$.
[Karel Zimmermann (Praha)]
MSC 2000:
*90C31 Sensitivity, etc.
90C26 Nonconvex programming

Keywords: multiplicative programs; bounding of multilinear functions; nonconvex optimization problems; arithmetic interval method; logarithmic transformation method; exponent transformation method; bounding schemes; convex envelope; concave envelope

Cited in: Zbl 1152.90610

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