Lower Semicontinuous Convex Relaxation in Optimization

We relate the argmin sets of a given function, not necessarily convex or lower semicontinuous, and its lower semicontinuous convex hull by means of explicit characterizations involving an appropriate concept of asymptotic functions. This question is connected to the subdifferential calculus of the Legendre–Fenchel conjugate function. The final expressions, which also involve a useful extension of the Fenchel subdifferential introduced in [R. Correa and A. Hantoute, Set-Valued Var. Anal., 18 (2010), pp. 405–422], are then written exclusively by means of primal objects relying on the initial function. This work extends to the infinite-dimensional setting of some related results given in [J. Benoist and J.-B. Hiriart-Urruty, SIAM J. Math. Anal., 27 (1996), pp. 1661–1679].