bv_cvxbook_extra_exercises

Hints draw a picture for the n 2 case rst for the

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Unformatted text preview: x 2≤1 2 fcshub (x) = fhub ( x ) = x 2 − 1/2 x 2 > 1. (The subscript stands for ‘circularly symmetric Huber function’.) Show that fcshub is convex. Find ∗ the conjugate function fcshub . 2.14 Reverse Jensen inequality. Suppose f is convex, λ1 > 0, λi ≤ 0, i = 2, . . . , k , and λ1 + · · · + λn = 1, and let x1 , . . . , xn ∈ dom f . Show that the inequality f ( λ1 x 1 + · · · + λn x n ) ≥ λ1 f ( x 1 ) + · · · + λn f ( x n ) always holds. Hints. Draw a picture for the n = 2 case first. For the general case, express x1 as a convex combination of λ1 x1 + · · · + λn xn and x2 , . . . , xn , and use Jensen’s inequality. 2.15 Monotone extension of a convex function. Suppose f : Rn → R is convex. Recall that a function h : Rn → R is monotone nondecreasing if h(x) ≥ h(y ) whenever x y . The monotone extension of f is defined as g (x) = inf f (x + z ). z0 (We will assume that g (x) > −∞.) Show that g is convex and monotone nondecreasing, and satisfies g (x) ≤ f (x) for all x. Show that if h...
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