136 5.5 Convex Functions and Derivatives 5.4.3 Let f,g: D → R be lower semicontinuous functions and let k > 0 be a constant. Prove that f + g and kf are lower semicontinous functions on D. 5.4.4 ▶ Let f : R → R be a lower semicontinuous function such that lim f (x)= lim f (x)= ∞. x→∞ x→−∞ Prove that f has an absolute minimum at some x0 ∈ R. 5.5 Convex Functions and Derivatives We discuss in this section an interesting class of functions that plays an important role in convex optimization problems. Throughout this section, we assume that I is a nonempty interval in R. Defnition 5.5.1 Let I be an interval in R and let f : I → R. We say that f is convex on I if f (λ u +(1− λ )v) ≤ λ f (u)+(1− λ ) f (v) for all u,v ∈ I and for all λ ∈ (0,1). Figure 5.3: A Convex Function. ■ Example 5.5.1 The following functions are convex. (a) f : R → R, f (x)= x. This is straightforward. 2 2 (b) f : R → R, f (x)= x . Here note frst that 2xy ≤ x + y2 for all real numbers x,y. Then, if 0 < λ < 1 and x,y ∈ R, we get f (λ x +(1 − λ )y) = (λ x +(1 − λ )y)2 2 2 = λ 2x + 2λ (1 − λ )xy+(1 − λ )2y 2 2 2 ≤ λ 2x + λ (1 − λ )(x + y2)+(1 − λ )2y = λ (λ x2 +(1 − λ )x2)+(1 − λ )(λ y2 +(1 − λ )y2) 2 = λ x2 +(1− λ )y = λ f (x)+(1 − λ ) f (y). (c) f : R → R, f (x)= |x|. This follows from the triangle inequality and other basic properties of absolute value.
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