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
Mots clés : Computer-aided convex analysis, convex function, convex interpolation, convex set, piecewise-defined function
@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://www.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://www.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://www.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, Tome 22 (2016) no. 3, pp. 728-742. doi : 10.1051/cocv/2015023. http://www.numdam.org/articles/10.1051/cocv/2015023/
H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011). | MR | Zbl
W. Dahmen, Convexity and Bernstein-Bézier polynomials, in Curves and surfaces (Chamonix-Mont-Blanc, 1990). Academic Press (1991) 107–134. | MR | Zbl
Computing the conjugate of convex piecewise linear-quadratic bivariate functions. Math. Program. (Series B) 139 (2013) 161–184. | DOI | MR | Zbl
and ,Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions. Comput. Optim. Appl. 58 (2014) 249–272. | DOI | MR | Zbl
, and ,On convexity of piecewise polynomial functions on triangulations. Comput. Aid. Geom. Des. 6 (1989) 181–187. | DOI | MR | Zbl
,Convexity preserving interpolation. Comput. Aid. Geom. Des. 16 (1999) 127–147. | DOI | MR | Zbl
,What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52 (2010) 505–542. | DOI | MR | Zbl
,Y. Lucet, Techniques and Open Questions in Computational Convex Analysis, in Comput. Anal. Math. Springer (2013) 485–500. | MR | Zbl
A simple proof of the Jensen-type inequality of Fink and Jodeit. Mediter. J. Math. 13 (2016) 119–126. | DOI | MR | Zbl
and ,B.S. Mordukhovich and N.M. Nam, An Easy Path to Convex Analysis and Applications. Morgan & Claypool (2014). | MR | Zbl
Relative convexity and its applications. Aequationes Math. 89 (2015) 1389–1400. | DOI | MR | Zbl
and ,The convex envelope is the solution of a nonlinear obstacle problem. Proc. of the AMS 135 (2007) 1689–1694. | DOI | MR | Zbl
,R.T. Rockafellar, Convex Analysis. Princeton University Press (1970). | MR | Zbl
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer (1998). | MR | Zbl
Convexity preserving splines over triangulations. Computer Aid. Geom. Des. 28 (2011) 270–284. | DOI | MR | Zbl
and ,Convexity preserving splines. Adv. Comput. Math. 40 (2014) 117–135. | DOI | MR | Zbl
and ,On the structure of convex piecewise quadratic functions. J. Optim. Theor. Appl. 72 (1992) 499–510. | DOI | MR | Zbl
,Classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems. SIAM J. Control Optim. 52 (2014) 2857–2876. | DOI | MR | Zbl
and ,C. Zălinescu, Convex Analysis in General Vector Spaces. World Scientific (2002). | MR | Zbl
Cité par Sources :