Revista Matemática Iberoamericana


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Volume 27, Issue 2, 2011, pp. 449–474
DOI: 10.4171/RMI/643

Published online: 2011-08-31

The $\varepsilon$-strategy in variational analysis: illustration with the closed convexification of a function

Jean-Baptiste Hiriart-Urruty[1], Marco A. López[2] and Michel Volle[3]

(1) Université Paul Sabatier, Toulouse, France
(2) Alicante University, Spain
(3) Université d'Avignon, France

In this work, we concentrate our interest and efforts on general variational (or optimization) problems which do not have solutions necessarily, but which do have approximate solutions (or solutions within $\varepsilon > 0$). We shall see how to recover all the (exact) minimizers of the relaxed version of the original problem (by closed-convexification of the objective function) in terms of the $\varepsilon $-minimizers of the original problem. Applications to two approximation problems in a Hilbert space setting will be shown.

Keywords: $\varepsilon$-solutions in variational problems, relaxation, closed-convexification, approximate projections

Hiriart-Urruty Jean-Baptiste, López Marco, Volle Michel: The $\varepsilon$-strategy in variational analysis: illustration with the closed convexification of a function. Rev. Mat. Iberoamericana 27 (2011), 449-474. doi: 10.4171/RMI/643