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 161 Variational inequality given closed convex set C , mapping F : R n R n , find x C such that F ( x ) T ( x x ) x C C x F ( x ) equivalently, solve x = P C ( x F ( x )) where P C is projection on C Variational inequalities 162 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 > such that ( F ( u ) F ( v )) T ( u v ) 2 bardbl u v bardbl 2 2 u, v Variational inequalities 163 Convex optimization minimize f ( x ) subject to x C with f convex, C a convex set optimality condition (236B lecture 4-9) x C is optimal if f ( x ) T ( x x ) x C this is a variational inequality with F ( x ) = f ( x ) Variational inequalities 164 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 165 Convex-concave saddle-point problems suppose f ( u, v ) is convex-concave and U and V are convex saddle-point: ( u, v ) U V is a saddle-point f ( u, v ) f ( u, v ) f ( u, v ) ( u, v ) U V (1) variational inequality formulation ( u, v ) is a saddle-point iff bracketleftbigg u f ( u, v ) v f ( u, v ) bracketrightbigg T bracketleftbigg u u v 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 166 proof if ( u, v ) satisfies the variational inequality, then for all ( u, v ) U V f ( u, v ) f ( u, v ) + f u ( u, v ) T ( u u ) f ( u, v ) f ( u, v ) f ( u, v ) + f v ( u, v ) T ( v v ) f ( u, v ) therefore (1) holds if ( u, v ) is a saddle-point, then u minimizes f ( u, v ) over u...
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16vi - EE236C (Spring 2008-09) 16. Variational inequalities...

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