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MIT6_079F09_lec05

# MIT6_079F09_lec05 - Convex Optimization — Boyd&...

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Unformatted text preview: Convex Optimization — Boyd & Vandenberghe 5. Duality • Lagrange dual problem • weak and strong duality • geometric interpretation • optimality conditions • perturbation and sensitivity analysis • examples • generalized inequalities 5–1 Lagrangian standard form problem (not necessarily convex) minimize f ( x ) subject to f i ( x ) ≤ , i = 1 , . . . , m h i ( x ) = 0 , i = 1 , . . . , p variable x ∈ R n , domain D , optimal value p ⋆ Lagrangian: L : R n × R m × R p → R , with dom L = D × R m × R p , m p L ( x, λ, ν ) = f ( x ) + λ i f i ( x ) + ν i h i ( x ) i =1 i =1 • weighted sum of objective and constraint functions • λ i is Lagrange multiplier associated with f i ( x ) ≤ • ν i is Lagrange multiplier associated with h i ( x ) = 0 Duality 5–2 Lagrange dual function Lagrange dual function: g : R m × R p → R , g ( λ, ν ) = inf L ( x, λ, ν ) x ∈D m p = inf f ( x ) + λ i f i ( x ) + ν i h i ( x ) x ∈D i =1 i =1 g is concave, can be −∞ for some λ , ν ⋆ lower bound property: if λ , then g ( λ, ν ) ≤ p proof: if x ˜ is feasible and λ , then f (˜ x ) ≥ L (˜ x, λ, ν ) ≥ inf L ( x, λ, ν ) = g ( λ, ν ) x ∈D minimizing over all feasible x ˜ gives p ⋆ ≥ g ( λ, ν ) Duality 5–3 Least-norm solution of linear equations minimize x T x subject to Ax = b dual function • Lagrangian is L ( x, ν ) = x T x + ν T ( Ax − b ) • to minimize L over x , set gradient equal to zero: ∇ x L ( x, ν ) = 2 x + A T ν = 0 = ⇒ x = − (1 / 2) A T ν • plug in in L to obtain g : g ( ν ) = L (( − 1 / 2) A T ν, ν ) = − 1 ν T AA T ν − b T ν 4 a concave function of ν lower bound property : p ⋆ ≥ − (1 / 4) ν T AA T ν − b T ν for all ν Duality 5–4 Standard form LP minimize c T x subject to Ax = b, x dual function • Lagrangian is L ( x, λ, ν ) = c T x + ν T ( Ax − b ) − λ T x = − b T ν + ( c + A T ν − λ ) T x • L is aﬃne in x , hence − b T ν A T ν − λ + c = 0 g ( λ, ν ) = inf L ( x, λ, ν ) = x −∞ otherwise g is linear on aﬃne domain { ( λ, ν ) | A T ν − λ + c = 0 } , hence concave lower bound property : p ⋆ ≥ − b T ν if A T ν + c Duality 5–5 Equality constrained norm minimization minimize x subject to Ax = b dual function g ( ν ) = inf ( x − ν T Ax + b T ν ) = b T ν A T ν ∗ ≤ 1 x −∞ otherwise where v ∗ = sup bardbl u bardbl≤ 1 u T v is dual norm of · proof: follows from inf x ( x − y T x ) = 0 if y ∗ ≤ 1 , −∞ otherwise • if y ∗ ≤ 1 , then x − y T x ≥ for all x , with equality if x = 0 • if y ∗ > 1 , choose x = tu where u ≤ 1 , u T y = y ∗ > 1 : x − y T x = t ( u − y ∗ ) → −∞ as t → ∞ lower bound property: p ⋆ ≥ b T ν if A T ν ∗ ≤ 1 Duality 5–6 Two-way partitioning minimize x T W x subject to x i 2 = 1 , i = 1 , . . . , n • a nonconvex problem; feasible set contains 2 n discrete points • interpretation: partition { 1 , . . . , n, ....
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MIT6_079F09_lec05 - Convex Optimization — Boyd&...

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