MIT6_079F09_lec03

MIT6_079F09_lec03 - Convex Optimization — Boyd&...

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Unformatted text preview: Convex Optimization — Boyd & Vandenberghe 3. Convex functions • basic properties and examples • operations that preserve convexity • the conjugate function • quasiconvex functions • log-concave and log-convex functions • convexity with respect to generalized inequalities 3–1 Definition f : R n R is convex if dom f is a convex set and → f ( θx + (1 − θ ) y ) ≤ θf ( x ) + (1 − θ ) f ( y ) for all x, y ∈ dom f , 0 ≤ θ ≤ 1 ( x, f ( x )) ( y, f ( y )) • f is concave if − f is convex • f is strictly convex if dom f is convex and f ( θx + (1 − θ ) y ) < θf ( x ) + (1 − θ ) f ( y ) for x, y ∈ dom f , x = y , 0 < θ < 1 Convex functions 3–2 Examples on R convex: • aﬃne: ax + b on R , for any a, b ∈ R • exponential: e ax , for any a ∈ R powers: x α on R ++ , for α ≥ 1 or α ≤ • • powers of absolute value: | x | p on R , for p ≥ 1 • negative entropy: x log x on R ++ concave: • aﬃne: ax + b on R , for any a, b ∈ R • powers: x α on R ++ , for 0 ≤ α ≤ 1 • logarithm: log x on R ++ Convex functions 3–3 Examples on R n and R m × n aﬃne functions are convex and concave; all norms are convex examples on R n aﬃne function f ( x ) = a T x + b • • norms: x p = ( n | x i | p ) 1 /p for p ≥ 1 ; x ∞ = max k | x k | i =1 examples on R m × n ( m n matrices) × aﬃne function • m n f ( X ) = tr ( A T X ) + b = A ij X ij + b i =1 j =1 • spectral (maximum singular value) norm f ( X ) = X 2 = σ max ( X ) = ( λ max ( X T X )) 1 / 2 Convex functions 3–4 Restriction of a convex function to a line f : R n R is convex if and only if the function g : R R , → → g ( t ) = f ( x + tv ) , dom g = { t | x + tv ∈ dom f } is convex (in t ) for any x ∈ dom f , v ∈ R n can check convexity of f by checking convexity of functions of one variable example. f : S n R with f ( X ) = log det X , dom f = S ++ n → g ( t ) = log det( X + tV ) = log det X + log det( I + tX − 1 / 2 V X − 1 / 2 ) n = log det X + log(1 + tλ i ) i =1 where λ i are the eigenvalues of X − 1 / 2 V X − 1 / 2 g is concave in t (for any choice of X ≻ , V ); hence f is concave Convex functions 3–5 Extended-value extension extended-value extension f ˜ of f is f ˜ ( x ) = f ( x ) , dom f, f ˜ ( x ) = dom f x ∈ ∞ , x ∈ often simplifies notation; for example, the condition 0 ≤ θ ≤ 1 = f ˜ ( θx + (1 − θ ) y ) ≤ θf ˜ ( x ) + (1 − θ ) f ˜ ( y ) ⇒ (as an inequality in R ∪ {∞} ), means the same as the two conditions dom f is convex • • for x, y ∈ dom f , 0 ≤ θ ≤ 1 = ⇒ f ( θx + (1 − θ ) y ) ≤ θf ( x ) + (1 − θ ) f ( y ) Convex functions 3–6 First-order condition f is differentiable if dom f is open and the gradient ∂f ( x ) ∂f ( x ) ∂f ( x ) = , , . . . , ∇ f ( x ) ∂x 1 ∂x 2 ∂x n exists at each x ∈ dom f 1st-order condition: differentiable f with convex domain is convex iff f ( y ) ≥ f ( x ) + ∇ f ( x ) T ( y − x ) for all...
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This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.

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MIT6_079F09_lec03 - Convex Optimization — Boyd&...

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