141 Exercises 5.5.1 (a) Let I be an interval in R and let f,g: I → R be convex functions. Prove that cf , f + g, and max{ f,g} are convex functions on I, where c ≥ 0 is a constant. (b) Find two convex functions f and g on an interval I in R such that f · g is not convex. 5.5.2 Let f : R → R be a convex function. Given a,b ∈ R, prove that the function defned by g(x)= f (ax + b), for x ∈ R is also a convex function on R. 5.5.3 ▶ Let I be an interval and let f : I → R be a convex function. Suppose that φ is a convex, increasing function on an interval J that contains f (I). Prove that φ ◦ f is convex on I. 5.5.4 ▷ Prove that each of the following functions is convex on the given domain: (a) f (x)= ebx ,x ∈ R, where b is a constant. k (b) f (x)= x , x ∈ [0,∞) and k ≥ 1 is a constant. (c) f (x)= −ln(1− x), x ∈ (−∞,1). ex (d) f (x)= −ln , x ∈ R. 1 + ex π (e) f (x)= xsinx, x ∈ (− π 4 , 4 ). 5.5.5 ▷ Prove the following: (a) If a,b are nonnegative real numbers, then √ a + b ≥ ab. 2 (b) If a1,a2,...,an, where n ≥ 2, are nonnegative real numbers, then a1 + a2 + ··· an) 1/n + an ≥ (a1 · a2 ··· . n 5.6 Nondifferentiable Convex Functions and Subdifferentials In this section, we introduce a generalized differentiation concept that is useful in the study of optimization problems in which the objective functions may fail to be differentiable. Defnition 5.6.1 Let f : R → R be a convex function. A number u ∈ R is called a subderivative of the function f at x0 if u· (x − x0) ≤ f (x) − f (x0) for all x ∈ R. (5.8) The set of all subderivatives of f at x0 is called the subdifferential of f at x0 and is denoted by ∂ f (x0).
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