136 5.5 Convex Functions and Derivatives 5.4.3 Let f ,g: D→Rbe lower semicontinuous functions and let k >0 be a constant. Prove that f +gandk f are lower semicontinous functions on D. 5.4.4 ▶Let f : R→Rbe a lower semicontinuous function such that lim x→∞ f (x) = lim x→−∞ f (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. Definition 5.5.1 Let I be an interval in Rand let f : I →R. We say that f is convex on I if f (λu+(1−λ)v) ≤λf (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. (b) f : R→R, f (x) =x2. Here note first that 2xy ≤x2 +y2 for all real numbers x,y. Then, if 0<λ <1 and x,y ∈R, we get f (λx+(1−λ)y) = (λx+(1−λ)y)2 = λ2x2 +2λ(1−λ)xy+(1−λ)2y2 ≤ λ2x2 +λ(1−λ)(x2 +y2)+(1−λ)2y2 = λ(λx2 +(1−λ)x2)+(1−λ)(λy2 +(1−λ)y2) = λx2 +(1−λ)y2 = λf (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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