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.
"The $\varepsilon$-strategy in variational analysis: illustration with the closed convexification of a function." Rev. Mat. Iberoamericana 27 (2) 449 - 474, May, 2011.