Convexity and Well-Posed Problems by Roberto Lucchetti

By Roberto Lucchetti

"In this publication the writer makes a speciality of the examine of convex services and their homes less than perturbations of knowledge. specifically, he illustrates the information of balance and well-posedness and the connections among them.

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Next, let X d be a (norm one) fixed direction. Let us consider the linear functional, defined on span {d}, ld (h) = af (x; d) if h = ad. Then ld (h) ≤ f (x; h) for all h in span {d}. The equality holds for h = ad and a > 0, while ld (−d) = −f (x; d) ≤ f (x; −d). 1), there is a linear functional x∗d ∈ X ∗ agreeing with ld on span {d}, and such that x∗d , h ≤ f (x; h) ∀h ∈ X. Then x∗d ∈ ∂f (x), so that x∗d = x∗ . As by construction x∗ , d = f (x; d) ∀d ∈ X, it follows that f is Gˆ ateaux differentiable at x and x∗ = ∇f (x).

5. The approximate subdifferential ∂1 (| · |)(0). The following result is easy and provides useful information. 5 Let f ∈ Γ (X). Then 0∗ ∈ ∂ε f (x0 ) if and only if inf f ≥ f (x0 ) − ε. Thus, whenever an algorithm is used to minimize a convex function, if we look for an ε-solution, it is enough that 0 ∈ ∂ε f (x), a much weaker condition than 0 ∈ ∂f (x). 14). 6 Let f ∈ Γ (X), x ∈ dom f . Then, ∀d ∈ X, f (x; d) = lim+ sup{ x∗ , d : x∗ ∈ ∂ε f (x)}. ε→0 Proof. Observe at first that, for monotonicity reasons, the limit in the above formula always exists.

This implies that limt→0+ g(t; d) always exists and lim g(t; d) = inf g(t). t→0+ t>0 If there is t¯ > 0 such that g(t¯) ∈ R and if g is lower bounded, then the limit must be finite. (x) = ∞ if and only if f (x + td) = ∞ for all Of course, limt→0+ f (x+td)−f t t > 0. Note that we shall use the word directional derivative, even if d is not a unit vector. The next estimate for the directional derivative is immediate. 3 Let f ∈ Γ (X) be Lipschitz with constant k in a neighborhood V of x. Then |f (x; d)| ≤ k, ∀d ∈ X : d = 1.

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