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ArticleOriginal scientific text

Title

Smoothing a polyhedral convex function via cumulant transformation and homogenization

Authors 1

Affiliations

  1. Department of Mathematics, University of Avignon, 33, Rue Louis Pasteur, 84000 Avignon, France

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.

Keywords

ENG
polyhedral convex function, smooth approximation, Laplace transformation, cumulant transformation, homogenization, recession function

Bibliography

  1. O. Barndorff-Nielsen, Exponential families: exact theory, Various Publ. Ser. 19, Inst. of Math., Univ. of Aarhus, Denmark, 1970.
  2. A. Ben-Tal and M. Teboulle, A smoothing technique for nondifferentiable optimization problems, in: Lecture Notes in Math. 1405, S. Dolecki (ed.), Springer, Berlin, 1989, 1-11.
  3. D. Bertsekas, Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, New York, 1982.
  4. C. Davis, All convex invariant functions of hermitian matrices, Arch. Math. (Basel) 8 (1957), 276-278.
  5. R. A. El-Attar, M. Vidyasagar, and S. R. K. Dutta, An algorithm for l₁-norm minimization with application to nonlinear l₁-approximation, SIAM J. Numer. Anal. 16 (1979), 70-86.
  6. R. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer, Berlin, 1985.
  7. C. Lemaréchal and C. Sagastizábal, Practical aspects of the Moreau-Yosida regularization: theoretical preliminaries, SIAM J. Optim. 7 (1997), 367-385.
  8. A. S. Lewis, Convex analysis on the Hermitian matrices, SIAM J. Optim. 6 (1996), 164-177.
  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.
  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.
  11. R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970.
  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.
  13. A. Seeger, Convex analysis of spectrally defined matrix functions, SIAM J. Optim. 7 (1997), 679-696.
Annales Polonici Mathematici
1997/67/3
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Pages:
259-268
Main language of publication
English
Received
1996-04-02
Accepted
1996-10-17
Published
1997
Exact and natural sciences