13dualmethods

13dualmethods - EE236C(Spring 2008-09 13 Dual methods dual...

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EE236C (Spring 2008-09) 13. Dual methods dual of convex problem with linear constraints differentiability of dual function dual decomposition rate control 13–1 Convex problem with linear constraints minimize f ( x ) subject to Gx precedesequal h Ax = b ( G R m × n , A R p × n ) dual function g ( λ, ν ) = inf x dom f L ( x, λ, ν ) = inf x dom f ( f ( x ) + ( G T λ + A T ν ) T x h T λ b T ν ) = h T λ b T ν f * ( G T λ A T ν ) f * ( y ) = sup x dom f ( x T y f ( x )) is the conjugate of f this lecture: methods that solve dual by a first-order method Dual methods 13–2
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Dual feasibility ( λ, ν ) is dual feasible if λ followsequal 0 and L ( x, λ, ν ) is bounded over x dom f : G T λ A T ν dom f * , λ followsequal 0 to solve dual problem when feasible set is not R m + × R p make dual domain explicit ( e.g. , when using gradient projection) minimize L with a method that detects unboundedness, and use unbounded direction to generate cutting-plane for dual domain ( e.g. , when using ACCPM) dual methods simplify if feasible set is R m + × R p some sufficient conditions: dom f is bounded ( e.g. , after addition of variable bounds) f strongly convex ( e.g. , after addition of a regularization term) Dual methods 13–3 Subgradients of conjugate function f * ( y ) = sup x dom f ( x T y f ( x )) from page 2–7, f * is subdifferentiable at all points in intdom f * weak subgradient rule if ˆ x maximizes x T ˆ y f ( x ) over x dom f , then ˆ x ∂f * y ) f * ( y ) = sup x dom f ( x T y f ( x )) ˆ x T y f x ) = ˆ x T ˆ y f x ) + ˆ x T ( y ˆ y ) = f * y ) + ˆ x T ( y ˆ y ) Dual methods 13–4
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Conjugate of closed functions a convex function is closed if epi f is a closed set (see BV § A.3.3) properties: for f closed and convex f ** = f : conjugate of conjugate of f is f (BV exercise 3.39) y ∂f ( x ) if and only if x ∂f * ( y ) proof: if ˆ y ∂f x ) , then x T ˆ y f ( x ) reaches its maximum over x at ˆ x ; therefore ˆ x ∂f * y ) ; this shows ˆ y ∂f x ) = ˆ x ∂f * y ) reverse implication follows from f ** = f Dual methods 13–5 Differentiability of conjugate if f is closed and strictly convex, then f *
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