Abstract
In this work we give an extension of the Brøndsted–Rockafellar Theorem, and some of its important consequences, to proper convex lower-semicontinuous epi-pointed functions defined in locally convex spaces. We use a new approach based on a simple variational principle, which also allows recovering the classical results in a natural way.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The arguments used in the proof of this result (Theorem 4.4) follows the suggestion made by one of the referees.
References
Benoist, J., Daniilidis, A.: Integration of Fenchel subdifferentials of epi-pointed functions. SIAM J. Optim. 12(3), 575–582 (2002)
Benoist, J., Hiriart-Urruty, J.-B.: What is the subdifferential of the closed convex hull of a function? SIAM J. Math. Anal. 27(6), 1661–1679 (1996)
Borwein, J.M.: A note on \(\varepsilon \)-subgradients and maximal monotonicity. Pacif. J. Math. 103(2), 307–314 (1982)
Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)
Correa, R., García, Y., Hantoute, A.: Integration formulas via the Fenchel subdifferential of nonconvex functions. Nonlinear Anal. 75(3), 1188–1201 (2012)
Correa, R., Hantoute, A.: New formulas for the Fenchel subdifferential of the conjugate function. Set-Valued Var. Anal. 18(3–4), 405–422 (2010)
Correa, R., Hantoute, A.: Subdifferential of the conjugate function in general Banach spaces. Top 20(2), 328–346 (2012)
Correa, R., Hantoute, A.: Lower semicontinuous convex relaxation in optimization. SIAM J. Optim. 23(1), 54–73 (2013)
Correa, R., Hantoute, A., Pérez-Aros, P.: On the Klee–Saint Raymond’s characterization of convexity. SIAM J. Optim. 26(2), 1312–1321 (2016)
Correa, R., Hantoute, A., Salas, D.: Integration of nonconvex epi-pointed functions in locally convex spaces. J. Convex Anal. 23(2), 511–530 (2016)
Debreu, Gerard: Theory of value: an axiomatic analysis of economic equilibrium. Cowles Foundation for Research in Economics at Yale University, Monograph 17. John Wiley, Inc., New York (1959)
Hiriart-Urruty, J.-B., Phelps, R.R.: Subdifferential calculus using \(\epsilon \)-subdifferentials. J. Funct. Anal. 118(1), 154–166 (1993)
Laurent, Pierre-Jean: Approximation et optimisation. Hermann, Paris (1972)
Pérez-Aros, P.: Propiedades variacionales de funciones convexas desde el análisis epi-puntado. Engineering thesis, Universidad de Chile (2014)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
Thibault, Lionel: A generalized sequential formula for subdifferentials of sums of convex functions defined on Banach spaces. In: Recent developments in optimization (Dijon, 1994), vol. 429 of Lecture Notes in Econom. and Math. Systems, pp. 340–345. Springer, Berlin, (1995)
Acknowledgements
We are very grateful to both referees for their suggestions and comments. We also thank one of them for drawing our attention to the new proof of Theorem 4.4.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor R. Tyrrell Rockafellar on the occasion of his 80th birthday.
This work is partially supported by CONICYT Grants: Fondecyt 1151003, Fondecyt 1150909, Basal PFB -03 and Basal FB0003, CONICYT-PCHA/doctorado Nacional/2014-21140621.
Rights and permissions
About this article
Cite this article
Correa, R., Hantoute, A. & Pérez-Aros, P. On Brøndsted–Rockafellar’s Theorem for convex lower semicontinuous epi-pointed functions in locally convex spaces. Math. Program. 168, 631–643 (2018). https://doi.org/10.1007/s10107-017-1110-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-017-1110-2
Keywords
- Convex and epi-pointed functions
- Locally convex spaces
- Fenchel subdifferential
- Brøndsted–Rockafellar’s Theorem