Unique Minimizers and the Representation of Convex Envelopes in Locally Convex Vector Spaces
T. Ruf, Bernd G Schmidt · Journal of convex analysis · 2022
It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let X X be a locally convex Hausdorff space and f\colon X\to (-\infty, \infty] f : X → ( − ∞ , ∞ ] a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem \displaystyle \min_{x \in X} f(x) - \langle x', x \rangle_X min x ∈ X f ( x ) − ⟨ x ′ , x ⟩ X admits at most one minimizer as x' x ′ ranges over \text{\rm dom}\, \left( \partial f^* \right) dom ( ∂ f ∗ ) if and only if the biconjugate f^{**} f ∗ ∗ is essentially strictly convex and agrees with f f at all points where f^{**} f ∗ ∗ is subdifferentiable. We prove this via a representation formula for f^{**} f ∗ ∗ that might be of independent interest.