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Unformatted text preview: x 2≤1
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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- Fall '13
- The Aeneid