| Issue |
ESAIM: COCV
Volume 22, Number 3, July-September 2016
|
|
|---|---|---|
| Page(s) | 728 - 742 | |
| DOI | https://doi.org/10.1051/cocv/2015023 | |
| Published online | 27 April 2016 | |
On the convexity of piecewise-defined functions∗,∗∗,∗∗∗
1
Mathematics, University of British Columbia
Okanagan, Kelowna,
B.C. V1V 1V7,
Canada.
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2
Computer Science, University of British Columbia
Okanagan, Kelowna,
B.C. V1V 1V7,
Canada.
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3
Department of Mathematical Sciences, University of Massachusetts
Lowell, Lowell,
M.A.
01854,
USA.
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Received:
21
August
2014
Revised:
27
March
2015
Abstract
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components – when can we conclude that the entire function is convex? In this paper we provide several convenient, verifiable conditions guaranteeing convexity (or the lack thereof). Several examples are presented to illustrate our results.
Mathematics Subject Classification: 26B25 / 52A41 / 65D17 / 90C25
Key words: Computer-aided convex analysis / convex function / convex interpolation / convex set / piecewise-defined function
HHB was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada under Discovery Grant #216877-2013 and by the Canada Research Chair Program.
YL was partially supported by NSERC Discovery grant #298145-2013.
HMP was partially supported by NSERC Discovery grant program – accelerator supplement grant #446219-2013 of HHB and an internal grant of University of Massachusetts Lowell.
© EDP Sciences, SMAI 2016
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