Journal of Convex Analysis 08 (2001), No. 1, 109--126
Copyright Heldermann Verlag 2001
A Higher-Order Smoothing Technique for Polyhedral Convex Functions: Geometric and Probabilistic Considerations
Sophie Guillaume
Dept. of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France
Albert Seeger
Dept. of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France
[Abstract-pdf]
\def\R{\mathbb R} Let $\R^n$ denote the usual n-dimensional Euclidean space. A polyhedral convex function $f \colon \R^n \to \R\cup\{+\infty\}$ can always be seen as the pointwise limit of a certain family $\{f^t\}_{t>0}$ of $C^{\infty}$ convex functions. An explicit construction of this family $\{f^t\}_{t>0}$ can be found in a previous paper by the second author [A. Seeger, Smoothing a polyhedral convex function via cumulant transformation and homogenization, Annales Polinici Mathematici 67 (1997) 259--268]. The aim of the present work is to further explore this $C^{\infty}$-approximation scheme. In particular, one shows how the family $\{f^t\}_{t>0}$ yields first and second-order information on the behavior of $f$. Links to linear programming and Legendre-Fenchel duality theory are also discussed.
Keywords: Polyhedral convex function, smooth approximation, subgradient, linear programming.
MSC: 41A30; 52B70, 60E10.
[ Fulltext-pdf (416 KB)]