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Non polyhedral convex envelopes for 1-convex functions

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Abstract

In this paper we discuss how to derive the non polyhedral convex envelopes for some functions, called 1-convex throughout the paper, over boxes. The main result is about n-dimensional 1-convex functions, but we get to it by first discussing in detail some special cases, namely functions \(xyz^\delta \), \(\delta >1\), and, next, more general trivariate functions. The relation between the class of functions investigated in this paper and other classes investigated in the existing literature is discussed.

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Acknowledgments

The author is extremely grateful to two anonymous reviewers, whose comments and critics allowed to improve previous versions of this work.

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  1. Dipartimento di Ingegneria dell’Informazione, Università di Parma, Via G.P. Usberti, 181/A, 43124, Parma, Italy

    Marco Locatelli

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  1. Marco Locatelli
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Correspondence to Marco Locatelli.

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Locatelli, M. Non polyhedral convex envelopes for 1-convex functions. J Glob Optim 65, 637–655 (2016). https://doi.org/10.1007/s10898-016-0409-5

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