16vi

# 16vi - EE236C(Spring 2008-09 16 Variational inequalities...

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Unformatted text preview: EE236C (Spring 2008-09) 16. Variational inequalities • variational inequality • monotonicity • examples • linear complementarity problem • analytic center cutting-plane method • extragradient method 16–1 Variational inequality given closed convex set C , mapping F : R n → R n , find ˆ x ∈ C such that F (ˆ x ) T ( x − ˆ x ) ≥ ∀ x ∈ C C ˆ x − F (ˆ x ) equivalently, solve ˆ x = P C (ˆ x − F (ˆ x )) where P C is projection on C Variational inequalities 16–2 Monotonicity we will focus on variational inequalities with monotone F : ( F ( u ) − F ( v )) T ( u − v ) ≥ ∀ u, v • F is strictly monotone if ( F ( u ) − F ( v )) T ( u − v ) > ∀ u, v, u negationslash = v • F is strongly monotone if there exists a σ > such that ( F ( u ) − F ( v )) T ( u − v ) ≥ σ 2 bardbl u − v bardbl 2 2 ∀ u, v Variational inequalities 16–3 Convex optimization minimize f ( x ) subject to x ∈ C with f convex, C a convex set optimality condition (236B lecture 4-9) ˆ x ∈ C is optimal if ∇ f (ˆ x ) T ( x − ˆ x ) ≥ ∀ x ∈ C this is a variational inequality with F ( x ) = ∇ f ( x ) Variational inequalities 16–4 monotonicity F ( x ) = ∇ f ( x ) is monotone if and only if f is convex • if f is convex, then for all u, v ∈ dom f , ( ∇ f ( u ) − ∇ f ( v )) T ( u − v ) = −∇ f ( u ) T ( v − u ) − ∇ f ( v ) T ( u − v ) ≥ ( f ( u ) − f ( v )) + ( f ( v ) − f ( u )) = • if ∇ f ( x ) is monotone, then for all x, y ∈ dom f , f ( y ) = f ( x ) + integraldisplay 1 ∇ f ( x + t ( y − x )) T ( y − x ) dt ≥ f ( x ) + ∇ f ( x ) T ( y − x ) Variational inequalities 16–5 Convex-concave saddle-point problems suppose f ( u, v ) is convex-concave and U and V are convex saddle-point: (ˆ u, ˆ v ) ∈ U × V is a saddle-point f (ˆ u, v ) ≤ f (ˆ u, ˆ v ) ≤ f ( u, ˆ v ) ∀ ( u, v ) ∈ U × V (1) variational inequality formulation (ˆ u, ˆ v ) is a saddle-point iff bracketleftbigg ∇ u f (ˆ u, ˆ v ) −∇ v f (ˆ u, ˆ v ) bracketrightbigg T bracketleftbigg u − ˆ u v − ˆ v bracketrightbigg ≥ ∀ ( u,v ) ∈ U × V (2) this is a variational inequality with F ( u, v ) = ( ∇ u f ( u, v ) , −∇ v f ( u, v )) Variational inequalities 16–6 proof • if (ˆ u, ˆ v ) satisfies the variational inequality, then for all ( u, v ) ∈ U × V f (ˆ u, ˆ v ) ≤ f (ˆ u, ˆ v ) + ∇ f u (ˆ u, ˆ v ) T ( u − ˆ u ) ≤ f ( u, ˆ v ) f (ˆ u, ˆ v ) ≥ f (ˆ u, ˆ v ) + ∇ f v (ˆ u, ˆ v ) T ( v − ˆ v ) ≥ f (ˆ u, v ) therefore (1) holds • if (ˆ u, ˆ v ) is a saddle-point, then ˆ u minimizes f ( u, ˆ v ) over u ∈...
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16vi - EE236C(Spring 2008-09 16 Variational inequalities...

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