LECTURE 8
LECTURE OUTLINE
Review of conjugate convex functions • Min common/max crossing duality • Weak duality • Special cases • Reading: Sections 1.6, 4.1, 4.2
All figures are courtesy of Athena Scientific, and are used with permission.
1 CONJUGACY THEOREM
n f (y) = sup x�y f(x) ,y x n − ⌘� ⌦� ⇤ ⌅ n f (x) = sup y�x f (y) ,x y n − ⌘� ⌦� ⇤ ⌅ If f is closed convex proper, then f = f. •
f(x) ( y,1) −
Slope = y
Hyperplane f (x) = sup y x f (y) � H = (x, w) w x�y = f (y) y n − ⇥⇤ | − − � ⇥ � ⇥ x 0