Exercise 1 Convexity (4 Points) Let X CR be a convex set. a) Let fi: X+R be convex functions for i = 1, ..., m. Show tha

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Exercise 1 Convexity 4 Points Let X Cr Be A Convex Set A Let Fi X R Be Convex Functions For I 1 M Show Tha 1 (39.28 KiB) Viewed 20 times

Exercise 1 Convexity (4 Points) Let X CR be a convex set. a) Let fi: X+R be convex functions for i = 1, ..., m. Show that f := -Qifi is convex if a; > 0 for all i = 1,..., m. Why is this requirement necessary? b) Let f,g: XR be two convex functions. Show that fog is in general not convex. Further, identify a simple additional requirement for f such that fog is convex and give a proof. c) Let f: X+R be convex and xo e X. Show that the level set Lo = {x EX : f(x) = f(x0)} = is a convex set.