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.

DOI: 10.1051/cocv/2015023

Keywords: Computer-aided convex analysis, convex function, convex interpolation, convex set, piecewise-defined function

^{1}; Lucet, Yves

^{2}; Phan, Hung M.

^{3}

@article{COCV_2016__22_3_728_0, author = {Bauschke, Heinz H. and Lucet, Yves and Phan, Hung M.}, title = {On the convexity of piecewise-defined functions}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {728--742}, publisher = {EDP-Sciences}, volume = {22}, number = {3}, year = {2016}, doi = {10.1051/cocv/2015023}, zbl = {1356.26005}, mrnumber = {3527941}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015023/} }

TY - JOUR AU - Bauschke, Heinz H. AU - Lucet, Yves AU - Phan, Hung M. TI - On the convexity of piecewise-defined functions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 728 EP - 742 VL - 22 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015023/ DO - 10.1051/cocv/2015023 LA - en ID - COCV_2016__22_3_728_0 ER -

%0 Journal Article %A Bauschke, Heinz H. %A Lucet, Yves %A Phan, Hung M. %T On the convexity of piecewise-defined functions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 728-742 %V 22 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015023/ %R 10.1051/cocv/2015023 %G en %F COCV_2016__22_3_728_0

Bauschke, Heinz H.; Lucet, Yves; Phan, Hung M. On the convexity of piecewise-defined functions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 3, pp. 728-742. doi : 10.1051/cocv/2015023. http://archive.numdam.org/articles/10.1051/cocv/2015023/

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