bv_cvxbook_extra_exercises

# C from this show that f is convex 24 a quadratic over

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Unformatted text preview: using a relatively painless method, leveraging some composition rules and known convexity of a few other functions. (a) Explain why t − (1/t)uT u is a concave function on dom f . Hint. Use convexity of the quadratic over linear function. (b) From this, show that − log(t − (1/t)uT u) is a convex function on dom f . (c) From this, show that f is convex. 2.4 A quadratic-over-linear composition theorem. Suppose that f : Rn → R is nonnegative and convex, and g : Rn → R is positive and concave. Show that the function f 2 /g , with domain dom f ∩ dom g , is convex. 2.5 A perspective composition rule. [4] Let f : Rn → R be a convex function with f (0) ≤ 0. (a) Show that the perspective tf (x/t), with domain {(x, t) | t &gt; 0, x/t ∈ dom f }, is nonincreasing as a function of t. 5 (b) Let g be concave and positive on its domain. Show that the function dom h = {x ∈ dom g | x/g (x) ∈ dom f } h(x) = g (x)f (x/g (x)), is convex. (c) As an example, show that h(x) = xT x , ( n=1 xk )1/n k dom h = Rn ++ is convex. 2.6 Perspect...
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