Abstract

It is shown that any convex function can be approximated by a family of differentiable with Lipschitz continuous gradient and strongly convex approximates in a "self-dual" way: the conjugate of each approximate is the approximate of the conjugate of the original function. The approximation technique extends to saddle functions, and is self-dual with respect to saddle function conjugacy and also partial conjugacy that relates saddle functions to convex functions.

Author information

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

Keywords

  • Convex functions
  • approximation
  • Moreau envelopes
  • duality
  • saddle functions

Mathematics Subject Classification

52A41, 90C25, 90C59, 90C46, 26B25

References

  1. H. Attouch: Variational Convergence for Functions and Operators, Pitman, Boston (1984).Google Scholar
  2. H. Attouch, D. Aze, R. J.-B. Wets: On continuity properties of the partial Legendre-Fenchel transform: convergence of sequences of augmented Lagrangian functions, Moreau-Yosida approximates and subdifferential operators, in: Fermat Days 85: Mathematics for Optimization, J.-B. Hiriart-Urruty (ed.), North-Holland, Amsterdam (1986) 1–42.https://doi.org/10.1016/S0304-0208(08)72392-2Google Scholar
  3. H. Attouch, D. Azé, R. J.-B. Wets: Convergence of convex-concave saddle functions: continuity properties of the Legendre-Fenchel transform and applications to convex programming and mechanics, Ann. Inst. H. Poincaré, Anal. Non Linéaire 5 (1988) 537–572.https://doi.org/10.1016/s0294-1449(16)30335-3Google Scholar
  4. H. Attouch, R. J.-B. Wets: A convergence theory for saddle functions, Trans. Amer. Math. Soc. 280(1) (1983) 1–41.https://doi.org/10.1090/S0002-9947-1983-0712247-XGoogle Scholar
  5. V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces, D. Reidel, Dordrecht (1986).Google Scholar
  6. A. Ben-Tal, M. Teboulle: A smoothing technique for nondifferentiable optimization problems, in: Optimization (Varetz, 1988), Lect. Notes Math. 1405, Springer, Berlin (1989) 1–11.https://doi.org/10.1007/BFb0083582Google Scholar
  7. A. Ben-Tal, M. Teboulle: A conjugate duality scheme generating a new class of differentiable duals, SIAM J. Optim. 6(3) (1996) 617–625.https://doi.org/10.1137/S1052623493254297Google Scholar
  8. M. Ghomi: The problem of optimal smoothing for convex functions, Proc. Amer. Math. Soc. 130(8) (2002) 2255–2259.https://doi.org/10.1090/S0002-9939-02-06743-6Google Scholar
  9. R. Goebel: Convexity, Convergence and Feedback in Optimal Control, PhD Thesis, University of Washington, Seattle (2000).Google Scholar
  10. R. Goebel: Regularity of the optimal feedback and the value function in convex problems of optimal control, Set-Valued Anal. 12(1-2) (2004) 127–145.https://doi.org/10.1023/B:SVAN.0000023404.46708.60Google Scholar
  11. R. Goebel: Convex optimal control problems with smooth Hamiltonians, SIAM J. Control Optim. 43(5) (2005) 1781–1811.https://doi.org/10.1137/S0363012902411581Google Scholar
  12. S. Guillaume, A. Seeger: A higher-order smoothing technique for polyhedral convex functions: geometric and probabilistic considerations, J. Convex Analysis 8(1) (2001) 109–126.https://doi.org/10.68381/jca08005Google Scholar
  13. C. Lemaréchal, C. Sagastizábal: Practical aspects of the Moreau-Yosida regularization: theoretical preliminaries, SIAM J. Optim. 7(2) (1997) 267–385.https://doi.org/10.1137/S1052623494267127Google Scholar
  14. J.-J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. Fr. 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  15. R. T. Rockafellar: A general correspondence between dual minimax problems and convex programs, Pacific J. Math. 25 (1968) 597–611.https://doi.org/10.2140/pjm.1968.25.597Google Scholar
  16. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  17. R. T. Rockafellar: Generalized Hamiltonian equations for convex problems of Lagrange, Pacific J. Math. 33(2) (1970) 411–427.https://doi.org/10.2140/pjm.1970.33.411Google Scholar
  18. R. T. Rockafellar: Saddle points of Hamiltonian systems in convex problems of Lagrange, J. Optimization Theory Appl. 12(4) (1973) 367–390.https://doi.org/10.1007/BF00940418Google Scholar
  19. R. T. Rockafellar: Conjugate Duality and Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics 16, SIAM, Philadelphia (1974).https://doi.org/10.1137/1.9781611970524.ch1Google Scholar
  20. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  21. R. T. Rockafellar, P. R. Wolenski: Convexity in Hamilton-Jacobi theory. 1: Dynamics and duality, SIAM J. Control Optim. 39(5) (2000) 1323–1350.https://doi.org/10.1137/S0363012998345366Google Scholar
  22. A. Seeger: Smoothing a nondifferentiable convex function: the technique of the rolling ball, Rev. Mat. Apl. 18(1) (1997) 45–60.Google Scholar
  23. A. Seeger: Smoothing a polyhedral convex function via cumulant transformation and homogenization, Ann. Polon. Math. 67 (1997) 259–268.https://doi.org/10.4064/ap-67-3-259-268Google Scholar
  24. S. Simons: Personal communication (2004).Google Scholar
  25. M. Teboulle: Entropic proximal mappings with applications to nonlinear programming, Math. Oper. Res. 17(3) (1992) 670–690.https://doi.org/10.1287/moor.17.3.670Google Scholar

Suggested citation

R. Goebel. “Self-Dual Smoothing of Convex and Saddle Functions.” Journal of Convex Analysis 15 (2008), No. 1, 179–190. https://doi.org/10.68381/jca15012

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press.