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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ∈...
View Full Document

{[ snackBarMessage ]}

Page1 / 13

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online