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Let Φ(·) be a concave criterion function defined on some set M⊂M, e.g., M = M ≥. The definition of Φ(·) can be extended to any p×p symmetric matrix in M by setting Φ(M) =− ∞ for M / ∈ M. This extension is then concave on M; its effective domain is the set dom(Φ) = {M ∈ M: Φ(M)> −∞}. Note that M ≥ ⊂ dom(Φ) whenΦ(·) positively homogeneous and isotonic; see Lemma 5.4-(iii). A concave function Φ(·) is called proper when dom(Φ) = ∅ and Φ(M) < ∞ for all M ∈ M. As a rule all the criteria we consider are proper. When Φ(·):M − → R is non-differentiable, the notion of gradient can be generalized as follows. A matrix ˜ M is called a subgradient of Φ(·) atM if Φ(A) ≤ Φ(M) + trace [ ˜ M(A − M)] , ∀A ∈ M. (A.1) Here trace(A, B) is the usual scalar product between A and B in M. Theset of all subgradients of Φ(·) atMis called the subdifferential1 of Φ(·) atM and is denoted by ∂Φ(M). The fact that these notions generalize that of gradient is due to the property ∂Φ(M) ={∇MΦ(M)} when Φ(·) is differentiable at M. In other situations ∂Φ(M) is not reduced to that singleton; it defines a convex set, closed if bounded, empty when M / ∈ dom(Φ), and satisfies the following properties. For any Φ(·) concave on M,

Publisher: 2013-09-21

Year: 2013

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