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15cvxccv - EE236C(Spring 2008-09 15 Saddle-point problems...

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EE236C (Spring 2008-09) 15. Saddle-point problems definition convex-concave games primal-dual decomposition 15–1 Min-max inequality the inequalities inf x X f ( x, ˆ y ) f x, ˆ y ) sup y Y f x, y ) hold for any function f , any sets X , Y , any point x, ˆ y ) X × Y as a consequence, the min-max inequality sup y Y inf x X f ( x, y ) inf x X sup y Y f ( x, y ) holds without exception Saddle-point problems 15–2
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Saddle-point ( x , y ) X × Y is a saddle-point of f on X × Y if inf x X f ( x, y ) = f ( x , y ) = sup y Y f ( x , y ) min-max equality: if f has a saddle-point ( x , y ) , then sup y Y inf x X f ( x, y ) = inf x X sup y Y f ( x, y ) and this quantity is equal to the saddle-point value f ( x , y ) Saddle-point problems 15–3 Lagrangian duality an important example is the Lagrangian of an optimization problem f ( x, y ) = f 0 ( x ) + m summationdisplay i =1 y i f i ( x ) , X = intersectiondisplay i =0 ,...,m dom f i , Y = R m + supremum of f over y is the primal objective: sup y Y f ( x, y ) = braceleftbigg f 0 ( x ) f i ( x ) 0 , i = 1 , . . . , m + otherwise infimum of f over x is the dual objective min-max inequality corresponds to weak duality Saddle-point problems 15–4
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Zero-sum game f is payoff function of zero-sum two-player game player 1 chooses a strategy x X ; player 2 chooses a strategy y Y players pick strategies without knowledge of opponent’s choice if f ( x, y ) > 0 , P1 pays f ( x, y ) to P2; otherwise, P2 pays f ( x, y ) to P1 min-max inequality if y = ˆ y is known to P1, P1 selects x = argmin x X f ( x, y ) if x = ˆ x is known to P2, P2 selects y = argmax y y f ( x, y ) the min-max inequality inf x X f ( x, ˆ y ) sup y Y f x, y ) x, ˆ y ) X × Y states the obvious fact that it helps to know your opponent’s strategy Saddle-point problems 15–5 saddle-point
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