Transcription of Lecture 3 Convex Functions - University of Illinois Urbana ...
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Lecture 3 Convex Functions - University of Illinois Urbana ...
Lecture 3. Convex Functions September 2, 2008. Lecture 3. Outline Convex Functions Examples Verifying Convexity of a Function Operations on Functions Preserving Convexity Convex Optimization 1. Lecture 3. Convex Functions Informally: f is Convex when for every segment [x1, x2], as x =. x1 +(1 )x2 varies over the line segment [x1, x2], the points (x , f (x )). lie below the segment connecting (x1, f (x1)) and (x2, f (x2)). Let f be a function from Rn to R, f : Rn R. The domain of f is a set in Rn defined by dom(f ) = {x Rn | f (x) is well defined (finite)}. Def. A function f is Convex if (1) Its domain dom(f ) is a Convex set in Rn and (2) For all x1, x2 dom(f ) and (0, 1).
Lecture 3 First-Order Condition f is differentiable if dom(f) is open and the gradient ∇f(x) = ∂f(x) ∂x1 ∂f(x) ∂x2 ∂f(x) ∂x n! exists at each x ∈ domf 1st-order condition: differentiable f is convex if and only if its domain is convex and f(x) + ∇f(x)T(z − x) ≤ f(z) for all x,z ∈ dom(f) A first order approximation is a global underestimate of f ...
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