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By Stephen Boyd,Lieven Vandenberghe
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Extra info for Convex Optimization, Solutions Manual
3. Prove the following. 21) if and only if there exists a σ such that ∇2 f (x) + σ∇f (x)∇f (x)T 0. 22) for all y = 0 if and only if there exists a σ such ∇2 f (x) + σ∇f (x)∇f (x)T 0. 27) Hint. We can assume without loss of generality that ∇2 f (x) is diagonal. 21) if and only if either ∇f (x) = 0 and ∇2 f (x) or ∇f (x) = 0 and the matrix H(x) = ∇2 f (x) ∇f (x)T 0, ∇f (x) 0 has exactly one negative eigenvalue. 22) for all y = 0 if and only if H(x) has exactly one nonpositive eigenvalue. Hint. You can use the result of part (a).
Therefore the affine function h(x) = (c − aT x)/b lies between f and g. 13 Kullback-Leibler divergence and the information inequality. 17). Prove the information inequality: Dkl (u, v) ≥ 0 for all u, v ∈ Rn ++ . Also show that Dkl (u, v) = 0 if and only if u = v. Hint. The Kullback-Leibler divergence can be expressed as Dkl (u, v) = f (u) − f (v) − ∇f (v)T (u − v), 3 Convex functions n where f (v) = i=1 vi log vi is the negative entropy of v. Solution. The negative entropy is strictly convex and differentiable on Rn ++ , hence f (u) > f (v) + ∇f (v)T (u − v) for all u, v ∈ Rn ++ with u = v.
Let f be a convex function. Define the function g as f (αx) . g(x) = inf α>0 α (a) Show that g is homogeneous (g(tx) = tg(x) for all t ≥ 0). (b) Show that g is the largest homogeneous underestimator of f : If h is homogeneous and h(x) ≤ f (x) for all x, then we have h(x) ≤ g(x) for all x. (c) Show that g is convex. Solution. (a) If t > 0, f (αtx) f (αtx) = t inf = tg(x). α>0 α tα For t = 0, we have g(tx) = g(0) = 0. g(tx) = inf α>0 Exercises (b) If h is a homogeneous underestimator, then h(x) = h(αx) f (αx) ≤ α α for all α > 0.