Abstract

Let Rn\mathbb{R}^n denote the usual n-dimensional Euclidean space. A polyhedral convex function f ⁣:Rn→R∪{+∞}f \colon \mathbb{R}^n \to \mathbb{R}\cup\{+\infty\} can always be seen as the pointwise limit of a certain family {ft}t>0\{f^t\}_{t>0} of C∞C^{\infty} convex functions. An explicit construction of this family {ft}t>0\{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∞C^{\infty}-approximation scheme. In particular, one shows how the family {ft}t>0\{f^t\}_{t>0} yields first and second-order information on the behavior of ff. Links to linear programming and Legendre-Fenchel duality theory are also discussed.

Author information

Contact details are reproduced from the original publication and may be historical.

Sophie Guillaume

Dept. of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France

Alberto Seeger

Dept. of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France

Keywords

  • Polyhedral convex function
  • smooth approximation
  • subgradient
  • linear programming

Mathematics Subject Classification

41A30; 52B70, 60E10

References

  1. R. Cominetti, J. San Martin: Asymptotic analysis of the exponential penalty trajectory in linear programming, Math. Programming 67 (1994) 169–187.https://doi.org/10.1007/BF01582220Google Scholar
  2. S. C. Fang, X. S. Li: On the entropic regularization method for solving min-max problems with applications, Math. Methods of Operations Research 46 (1997) 119–130.https://doi.org/10.1007/BF01199466Google Scholar
  3. J. B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms, Vol. II, Grundlehren der Mathematischen Wissenschaften 306, Springer-Verlag, Berlin, 1993.https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  4. S. S. Kutateladze: Convex operators, Russian Math. Surveys 34 (1979) 181–214.https://doi.org/10.1070/RM1979v034n01ABEH002874Google Scholar
  5. J. J. Moreau: Fonctionelles Convexes, Lecture Notes, Séminaire “Equations aux Dérivées Partielles”, Collège de France, Paris, 1966.Google Scholar
  6. R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton, N. J., 1970.https://doi.org/10.1515/9781400873173Google Scholar
  7. A. Seeger: Smoothing a polyhedral convex function via cumulant transformation and homogenization, Annales Polinici Mathematici 67 (1997) 259–268.https://doi.org/10.4064/ap-67-3-259-268Google Scholar
  8. R. L. Sheu, S. Y. Wu: Combined entropic regularization and path-following method for solving finite convex min-max problems subject to infinitely many linear constraints, J. Optim. Theory and Appl. 101 (1999) 167–190.https://doi.org/10.1023/A:1021727228957Google Scholar

Suggested citation

S. Guillaume, A. Seeger. “A Higher-Order Smoothing Technique for Polyhedral Convex Functions: Geometric and Probabilistic Considerations.” Journal of Convex Analysis 8 (2001), No. 1, 109–126. https://doi.org/10.68381/jca08005

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Published by Heldermann Verlag, 2001. Rights now held by Banach Press LLC.