By Roberto Lucchetti

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

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Extra info for Convexity and Well-Posed Problems

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10 Let X be a Banach space and let int dom f = ∅. Let x ∈ dom f \ int dom f . Prove that the normal cone to dom f at the point x is unbounded. Hint. 5 by separating x from int dom f . We now see how to evaluate the subdifferential of the inf convolution, at least in a particular case. 11 Let X be a Banach space, let f, g ∈ Γ (X), let x ∈ X and let u, v be such that u + v = x and (f ∇g)(x) = f (u) + g(v). Then ∂(f ∇g)(x) = ∂f (u) ∩ ∂g(v). Proof. Let x∗ ∈ ∂f (u) ∩ ∂g(v). 3) g(z) ≥ g(v) + x∗ , z − v .

Thus the indicator function of A is not Gˆ ateaux differentiable at x, but ∂IA (x) = {0}. 9). Recall that, denoting by {e1 , . . , en } the canonical basis in Rn , the partial derivatives of f at x are defined as follows: f (x + tei ) − f (x) ∂f , (x) = lim t→0 ∂xi t whenever the limit exists and is finite. Then we have the following proposition. 8 Let f : Rn → R be convex. Then f is (Gˆ ateaux) differ∂f (x), i = 1, . . , n entiable at x ∈ Rn if and only if the partial derivatives ∂x i exist. Proof.

Let us start with two definitions. 1 Let X be a Banach space and f ∈ Γ (X). Suppose x ¯ ∈ int dom f . The subdifferential ∂f is said to be Lipschitz stable at x ¯ if ∂f (¯ x) = {¯ p} and there are ε > 0, K > 0 such that p − p¯ ≤ K x − x ¯ , provided x − x ¯ < ε, p ∈ ∂f (x). 2 Let X be a Banach space and f ∈ Γ (X). Suppose x ¯ ∈ int dom f . 9) provided p ∈ ∂f (x). 3 Let X be a Banach space and f ∈ Γ (X). Suppose x ¯ ∈ int dom f . We say that f is twice Fr´echet differentiable at x ¯ if ∂f (¯ x) = p¯ and there is a quadratic form Q(x) := Ax, x (A : X → X ∗ linear bounded operator) such that lim x→¯ x f (x) − p¯, x − x ¯ − (1/2)Q(x − x ¯) = 0.

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