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

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Unformatted text preview: 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 (ˆ x, ˆ y ) ≤ sup y ∈ Y f (ˆ x, y ) hold for any function f , any sets X , Y , any point (ˆ 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 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 ( 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 ( x ) f i ( x ) ≤ , 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 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 ) > , 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 (ˆ x, y ) ∀ (ˆ x, ˆ...
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This note was uploaded on 01/25/2010 for the course EE 236 taught by Professor Staff during the Spring '08 term at UCLA.

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

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