{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

15cvxccv

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

This preview shows pages 1–4. Sign up to view the full content.

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

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

View Full Document
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
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

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

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

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern