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Given α > 0, define a set-valued mapping as S α ( x ) = { y ∈ S ∣ f ( y ) + α ∥ x - y ∥ ≤ f ( x ) }. Clearly, this set is nonempty, since x ∈ S α (x) for all α > 0. Let f ̄ stand for the optimal value of f over S. Some equivalent expressions of weak sharpness in terms of S α are given below.
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