bv_cvxbook_extra_exercises

E f x depends only on x 2 show that f must have the

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Unformatted text preview: is any other convex function that satisfies these properties, then h(x) ≤ g (x) for all x. Thus, g is the maximum convex monotone underestimator of f . Remark. For simple functions (say, on R) it is easy to work out what g is, given f . On Rn , it can be very difficult to work out an explicit expression for g . However, systems such as CVX can immediately handle functions such as g , defined by partial minimization. 7 2.16 Circularly symmetric convex functions. Suppose f : Rn → R is symmetric with respect to rotations, i.e., f (x) depends only on x 2 . Show that f must have the form f (x) = φ( x 2 ), where φ : R → R is nondecreasing and convex, with dom f = R. (Conversely, any function of this form is symmetric and convex, so this form characterizes such functions.) 2.17 Infimal convolution. Let f1 , . . . , fm be convex functions on Rn . Their infimal convolution, denoted g = f1 ⋄ · · · ⋄ fm (several other notations are also used), is defined as g (x) = inf {f1 (x1 ) + · · · +...
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