Lecture15

Lecture15 - C&O 355 Lecture 15 N Harvey Topics Subgradient...

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Lecture 15 N. Harvey

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Topics Subgradient Inequality Characterizations of Convex Functions Convex Minimization over a Polyhedron (Mini)-KKT Theorem Smallest Enclosing Ball Problem
Subgradient Inequality Prop: Suppose f : R ! R is differentiable. Then f is convex iff f(y) ¸ f(x) + f’(x)(y -x) 8 x,y 2 R Proof: ( : See Notes Section 3.2. ) : Exercise for Assignment 4. ¤

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Convexity and Second Derivative Prop: Suppose f : R ! R is twice-differentiable. Then f is convex iff f’’(x) ¸ 0 8 x 2 R . Proof: See Notes Section 3.2.
Subgradient Inequality in R n Prop: Suppose f : R n ! R is differentiable. Then f is convex iff f(y) ¸ f(x) + r f (x) T (y-x) 8 x,y 2 R Proof: ( : Exercise for Assignment 4. ) : See Notes Section 3.2. ¤

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Minimizing over a Convex Set Prop: Let C μ R n be a convex set. Let f : R n ! R be convex and differentiable. Then x minimizes f over C iff r f(x) T (z-x) ¸ 0 8 z 2 C. Proof: ( direction Direct from subgradient inequality. f(z) ¸ f(x) + r f(x) T (z-x) ¸ f(x) Subgradient inequality Our hypothesis
Minimizing over a Convex Set Prop: Let C μ R n be a convex set. Let f : R n ! R be convex and differentiable. Then x minimizes f over C iff r f(x) T (z-x) ¸ 0 8 z 2 C. Proof: ) direction Let x be a minimizer, let z 2 C and let y = z-x. Recall that r f(x) T y = f’( x;y) = lim t ! 0 f(x+ty)-f(x). If limit is negative then we have f(x+ty)<f(x) for some t 2 [0,1], contradicting that x is a minimizer. So the limit is non-negative, and r f(x) T y ¸ 0. ¥ t

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Positive Semidefinite Matrices (again) Assume M is symmetric Old definition: M is PSD if 9 V s.t. M = V T V. New definition: M is PSD if y T My ¸ 0 8 y 2 R n . Claim: Old ) New. Proof: y T My = y T V T Vy = k Vy k 2 ¸ 0. Claim: New ) Old. Proof: Based on spectral decomposition of M.
Convexity and Hessian Prop: Let f: R n ! R be a C 2 -function. Let H(x) denote the Hessian of f at point x.

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This note was uploaded on 06/16/2011 for the course C 355 taught by Professor Harvey during the Fall '09 term at Waterloo.

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Lecture15 - C&O 355 Lecture 15 N Harvey Topics Subgradient...

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