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Closed convex cones are self dual with respect to

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Unformatted text preview: a finite set of extreme points and extreme directions • A real-valued convex function is continuous and has nice differentiability properties • Closed convex cones are self-dual with respect to polarity • Convex, lower semicontinuous functions are selfdual with respect to conjugacy 5 DUALITY • Two different views of the same object. • Example: Dual description of signals. Frequency domain Time domain • Dual description of closed convex sets A union of points An intersection of halfspaces 6 DUAL DESCRIPTION OF CONVEX FUNCTIONS • Define a closed convex function by its epigraph. • Describe the epigraph by hyperplanes. • Associate hyperplanes with crossing points (the conjugate function). (−y, 1) f (x ) Slope = y 0 x infn {f (x) − x⇥ y } = −f (y ) x⇤⌅ Dual Description Primal Description Crossing points f ∗ (y ) Values f (x) 7 FENCHEL PRIMAL AND DUAL PROBLEMS f1 (x) ∗ −f1 (y ) Slope y ∗ ∗ f1 (y ) + f2 (−y ) ∗ f2 (−y ) −f2 (x) x∗ Primal Problem Description Vertical Distances x Dual Problem Description Crossing Point Differentials • Primal problem: ⌅ ⇤ min f1 (x) + f2 (x) x • Dual problem: ⇤ max − y ⇤ f1 (y ) − ⇤ ⇤ where f1 and f2 are the conjugates 8 ⌅ ⇤ f2 (−y ) FENCHEL DUALITY f1 (x) Slope y ∗ ∗ −f1 (y ) Slope y ∗ ∗ f1 (y ) + f2 (−y ) ∗ f2 (−y ) −f2 (x) x∗ � x � ⇥ ⇥ min f1 (x) + f2 (x) = max − f1 (y ) − f2 (−y ) x y • Under favorable conditions (convexity): − The optimal primal and dual values are equal − The optimal primal and dual solutions are related 9 A MORE ABSTRACT VIEW OF DUALITY • Despite its elegance, the Fenchel framework is somewhat indirect. • From duality of set descriptions, to − duality of functional descriptions, to − duality of problem descriptions. • A more direct approach: − Start with a set, then − Define two simple prototype problems dual to each other. • Avoid functional descriptions (a simpler, less constrained framework). 10 MIN COMMON...
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