13dualmethods

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

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Unformatted text preview: 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 Dual feasibility ( λ, ν ) is dual feasible if λ followsequal and L ( x, λ, ν ) is bounded over x ∈ dom f : − G T λ − A T ν ∈ dom f * , λ followsequal • 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 int dom f * weak subgradient rule if ˆ x maximizes x T ˆ y − f ( x ) over x ∈ dom f , then ˆ x ∈ ∂f * (ˆ y ) f * ( y ) = sup x ∈ dom f ( x T y − f ( x )) ≥ ˆ x T y − f (ˆ x ) = ˆ x T ˆ y − f (ˆ x ) + ˆ x T ( y − ˆ y ) = f * (ˆ y ) + ˆ x T ( y − ˆ y ) Dual methods 13–4 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 (ˆ x ) , then x T ˆ y − f ( x ) reaches its maximum over x at ˆ x ; therefore ˆ x ∈ ∂f * (ˆ y ) ; this shows ˆ...
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This note was uploaded on 01/25/2010 for the course EE 236 taught by Professor Staff during the Spring '08 term at UCLA.

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

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