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Unformatted text preview: Clarke derivatives and generalized gradient Let f be Lipschitz near x ∈ R n . The Clarke generalized devivative of f at x in the direction v ∈ R n is f ◦ ( x ; v ) = lim sup y → x, t ↓ f ( y + tv ) f ( y ) t . The generalized gradient of f at x is defined to be ∂f ( x ) = { ζ ∈ R n : f ◦ ( x ; v ) ≥ v T ζ for every v ∈ R n } = co { lim ∇ f ( x i ) : x i → x and ∇ f ( x i ) exists } . Audet and Vicente (SIOPT 2008) Optimization under general constraints 73/109 Clarke derivatives and generalized gradient Let f be Lipschitz near x ∈ R n . The Clarke generalized devivative of f at x in the direction v ∈ R n is f ◦ ( x ; v ) = lim sup y → x, t ↓ f ( y + tv ) f ( y ) t . The generalized gradient of f at x is defined to be ∂f ( x ) = { ζ ∈ R n : f ◦ ( x ; v ) ≥ v T ζ for every v ∈ R n } = co { lim ∇ f ( x i ) : x i → x and ∇ f ( x i ) exists } ....
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This document was uploaded on 10/30/2011 for the course FIN 3403 at University of Florida.
 Spring '06
 Tapley
 Finance, Derivatives

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