Introduction to Mathematical Analysis I - Second Edition

132 4.7 NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS Exercises 4.6.1 (a) Let I be an interval and let f , g : I → R be convex functions. Prove that c f , 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 such that f · g is not convex. 4.6.2 Let f : R → R be a convex function. Given a , b ∈ R , prove that the function defined by g ( x ) = f ( ax + b ) , for x ∈ R is also a convex function on R . 4.6.3 I 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 . 4.6.4 B Prove that each of the following functions is convex on the given domain: (a) f ( x ) = e bx , x ∈ R , where b is a constant. (b) f ( x ) = x k , x ∈ [ 0 , ∞ ) and k ≥ 1 is a constant. (c) f ( x ) = − ln ( 1 − x ) , x ∈ ( − ∞ , 1 ) . (d) f ( x ) = − ln e x 1 + e x , x ∈ R . (e) f ( x ) = x sin x , x ∈ ( − π 4 , π 4 ) . 4.6.5 B Prove the following: (a) If a , b are nonnegative real numbers, then a + b 2 ≥ √ ab . (b) If a 1 , a 2 , . . . , a n , where n ≥ 2, are nonnegative real numbers, then a 1 + a 2 + · · · + a n n ≥ ( a 1 · a 2 · · · a n ) 1 / n . 4.7 NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS In this section, we introduce a new concept that is helpful in the study of optimization problems in which the objective function may fail to be differentiable. Definition 4.7.1 Let f : R → R be a convex function. A number u ∈ R is called a subderivative of the function f at ¯ x if u · ( x − ¯ x ) ≤ f ( x ) − f ( ¯ x ) for all x ∈ R . (4.16) The set of all subderivatives of f at ¯ x is called the subdifferential of f at ¯ x and is denoted by ∂ f ( ¯ x ) .