On the convexity of piecewise-defined functions
Bauschke, Heinz H.1 ; Lucet, Yves2 ; Phan, Hung M.3
1 Mathematics, University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada.
2 Computer Science, University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada.
3 Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, M.A. 01854, USA.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 728-742.

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.

Reçu le :
DOI : 10.1051/cocv/2015023
Classification : 26B25, 52A41, 65D17, 90C25
Mots-clés : Computer-aided convex analysis, convex function, convex interpolation, convex set, piecewise-defined function
Bauschke, Heinz H. 1 ; Lucet, Yves 2 ; Phan, Hung M. 3

1 Mathematics, University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada.
2 Computer Science, University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada.
3 Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, M.A. 01854, USA. 
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Bauschke, Heinz H.; Lucet, Yves; Phan, Hung M. On the convexity of piecewise-defined functions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 728-742. doi : 10.1051/cocv/2015023. http://www.numdam.org/articles/10.1051/cocv/2015023/

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