Subdifferential estimate of the directional derivative, optimality criterion and separation principles

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Subdifferential estimate of the directional derivative, optimality criterion and separation principles

In this article (DOI: 10.1080/02331934.2011.645034), the formulation of the (ϵ-CS) property in Section 6 is incomplete: the assumption ‘ϕ inf-compact’ should be replaced by ‘ϕ − x* inf-compact for every x* ∈ X*’.

With this new formulation, the proof of (ϵ-Sdiff) ⇒ (ϵ-CS) in Theorem 6.1 is as follows. Let x* ∈ ∂(f + ϕ)(x). Then, (f + ϕ − x*)(x) = inf(f + ϕ − x*) = (f ▿ (ϕ − x*)⁻)(0). From (ϵ-Sdiff) we know that ∂_ϵ f(z) is not empty at some (any) point z ∈ dom f, hence there exists z* ∈ X* such that f − z* is bounded from below. Then, the inf-convolution (f − z*) ▿ (ϕ − x* + z*)⁻ is convex lower semicontinuous, so also is f ▿ (ϕ − x*)⁻. Therefore, by (ϵ-Sdiff), for every ϵ > 0, the set ∂_ϵ(f ▿ (ϕ − x*)⁻)(0) is not empty. As in the article, we conclude that 0 ∈ ∂_ϵ  f(x) + ∂_ϵ(ϕ − x*)(x), that is, x* ∈ ∂_ϵ  f(x) + ∂_ϵϕ(x).

To prove (ϵ-DD) from this corrected version of (ϵ-CS), we have to check that the function ϕ considered in the proof satisfies ‘ϕ − x* inf-compact for every x* ∈ X*’. This is evident since ϕ − x* is lower semicontinuous and is a compact subset of X.

Remark

The above proof shows that for convex f and ϕ, the formula

holds as soon as f ▿ (ϕ − x*)⁻ is lower semicontinuous at 0 for every x* ∈ ∂(f + ϕ)(x). The converse is also true. Indeed, assume that the formula holds and let x* ∈ ∂(f + ϕ)(x). Then (f ▿ (ϕ − x*)⁻)(0) = (f + ϕ − x*)(x) is finite and x* ∈ ∂_ϵ  f(x) + ∂_ϵϕ(x) for every ϵ > 0, so ∂_ϵ(f ▿ (ϕ − x*)⁻)(0) is not empty for every ϵ > 0, that is, f ▿ (ϕ − x*)⁻ is lower semicontinuous at 0. In practice, the point-wise version of this statement is more convenient: for every convex f, ϕ: X → (−∞, +∞] and x ∈ dom f ∩ dom ϕ, the following are equivalent:

i.	f ▿ ϕ− is lower semicontinuous at 0 and (f + ϕ)(x) = inf(f + ϕ),
ii.	.