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2subgradients

# 2subgradients - EE236C(Spring 2008-09 2 Subgradients...

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EE236C (Spring 2008-09) 2. Subgradients definition subgradient calculus optimality conditions via subgradients directional derivative 2–1 Basic inequality recall basic inequality for convex differentiable f : f ( y ) f ( x ) + f ( x ) T ( y x ) first-order approximation of f at x is global lower bound • ∇ f ( x ) defines non-vertical supporting hyperplane to epi f at ( x, f ( x )) bracketleftbigg f ( x ) 1 bracketrightbigg T parenleftbiggbracketleftbigg y t bracketrightbigg bracketleftbigg x f ( x ) bracketrightbiggparenrightbigg 0 ( y, t ) epi f what if f is not differentiable? Subgradients 2–2

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Subgradient of a function definition: g is a subgradient of a convex function f at x dom f if f ( y ) f ( x ) + g T ( y x ) for all y dom f x 1 x 2 f ( x 1 ) + g T 1 ( x x 1 ) f ( x 2 ) + g T 2 ( x x 2 ) f ( x 2 ) + g T 3 ( x x 2 ) f ( x ) g 2 , g 3 are subgradients at x 2 ; g 1 is a subgradient at x 1 Subgradients 2–3 properties f ( x ) + g T ( y x ) is a global lower bound on f g defines non-vertical supporting hyperplane to epi f at ( x, f ( x )) bracketleftbigg g 1 bracketrightbigg T parenleftbiggbracketleftbigg y t bracketrightbigg bracketleftbigg x f ( x ) bracketrightbiggparenrightbigg 0 ( y, t ) epi f if f is convex and differentiable, then f ( x ) is a subgradient of f at x applications algorithms for nondifferentiable convex optimization optimality conditions, duality for nondifferentiable problems Subgradients 2–4
Example f ( x ) = max { f 1 ( x ) , f 2 ( x ) } f 1 , f 2 convex and differentiable x 0 f 1 ( x ) f 2 ( x ) subgradients at x 0 form line segment [ f 1 ( x 0 ) , f 2 ( x 0 )] if f 1 x ) > f 2 x ) , subgradient of f at ˆ x is f 1 x ) if f 1 x ) < f 2 x ) , subgradient of f at ˆ x is f 2 x ) Subgradients 2–5 Subgradients and sublevel sets if g is a subgradient of f at x , then f ( y ) f ( x ) = g T ( y x ) 0 f ( y ) f ( x 0 ) x 0 g ∂f ( x 0 ) x 1 f ( x 1 ) nonzero subgradients at x define supporting hyperplanes to sublevel set { y | f ( y ) f ( x ) } Subgradients 2–6

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Subdifferential subdifferential of f at x dom f is the set of all subgradients of f at x notation: ∂f ( x ) properties ∂f ( x ) is a closed convex set (possibly empty) proof: ∂f ( x ) is an intersection of halfspaces ∂f ( x ) = braceleftbig g | f ( x ) + g T ( y x ) f ( y ) y dom f bracerightbig if x intdom f then ∂f ( x ) is nonempty and bounded Subgradients 2–7 Examples absolute value f ( x ) = | x | f ( x ) = | x | ∂f ( x ) x x 1 1 Euclidean norm f ( x ) = bardbl x bardbl 2 ∂f ( x ) = 1 bardbl x bardbl 2 x if x negationslash = 0 , ∂f ( x ) = { g | bardbl g bardbl 2 1 } if x = 0 Subgradients 2–8
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2subgradients - EE236C(Spring 2008-09 2 Subgradients...

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