Smoothing a polyhedral convex function via cumulant transformation and homogenization

Alberto Seeger

Annales Polonici Mathematici (1997)

  • Volume: 67, Issue: 3, page 259-268
  • ISSN: 0066-2216

Abstract

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Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family g ₜ t > 0 which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family g ₜ t > 0 involves the concept of cumulant transformation and a standard homogenization procedure.

How to cite

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Alberto Seeger. "Smoothing a polyhedral convex function via cumulant transformation and homogenization." Annales Polonici Mathematici 67.3 (1997): 259-268. <http://eudml.org/doc/270164>.

@article{AlbertoSeeger1997,
abstract = {Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family $\{gₜ\}_\{t>0\}$ which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family $\{gₜ\}_\{t>0\}$ involves the concept of cumulant transformation and a standard homogenization procedure.},
author = {Alberto Seeger},
journal = {Annales Polonici Mathematici},
keywords = {polyhedral convex function; smooth approximation; Laplace transformation; cumulant transformation; homogenization; recession function},
language = {eng},
number = {3},
pages = {259-268},
title = {Smoothing a polyhedral convex function via cumulant transformation and homogenization},
url = {http://eudml.org/doc/270164},
volume = {67},
year = {1997},
}

TY - JOUR
AU - Alberto Seeger
TI - Smoothing a polyhedral convex function via cumulant transformation and homogenization
JO - Annales Polonici Mathematici
PY - 1997
VL - 67
IS - 3
SP - 259
EP - 268
AB - Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family ${gₜ}_{t>0}$ which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family ${gₜ}_{t>0}$ involves the concept of cumulant transformation and a standard homogenization procedure.
LA - eng
KW - polyhedral convex function; smooth approximation; Laplace transformation; cumulant transformation; homogenization; recession function
UR - http://eudml.org/doc/270164
ER -

References

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  8. [8] A. S. Lewis, Convex analysis on the Hermitian matrices, SIAM J. Optim. 6 (1996), 164-177. Zbl0849.15013
  9. [9] J. E. Martinez-Legaz, On convex and quasiconvex spectral functions, in: Proc. 2nd Catalan Days on Appl. Math., M. Sofonea and J. N. Corvellec (eds.), Presses Univ. de Perpignan, Perpignan, 1995, 199-208. Zbl0911.90277
  10. [10] M. L. Overton and R. S. Womersley, Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices, Math. Programming 62 (1993), 321-357. Zbl0806.90114
  11. [11] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0193.18401
  12. [12] A. Seeger, Smoothing a nondifferentiable convex function: the technique of the rolling ball, Technical Report 165, Dep. of Mathematical Sciences, King Fahd Univ. of Petroleum and Minerals, Dhahran, Saudi Arabia, October 1994. Zbl0921.49008
  13. [13] A. Seeger, Convex analysis of spectrally defined matrix functions, SIAM J. Optim. 7 (1997), 679-696. Zbl0890.15018

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