Introduction to Mathematical Analysis I - Second Edition

138 4.7 NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS This implies 0 · ( x − ¯ x ) = 0 ≤ f ( x ) − f ( ¯ x ) for all x ∈ R . It follows from ( 4.16 ) that 0 ∈ ∂ f ( ¯ x ) . Conversely, if 0 ∈ ∂ f ( ¯ x ) , again, by ( 4.16 ) , 0 · ( x − ¯ x ) = 0 ≤ f ( x ) − f ( ¯ x ) for all x ∈ R . Thus, f has an absolute minimum at ¯ x . Example 4.7.4 Let k be a positive integer and a 1 < a 2 < · · · < a 2 k − 1 . Define f ( x ) = 2 k − 1 ∑ i = 1 | x − a i | , for x ∈ R . It follows from the subdifferential formula in Example 4.7.3 that 0 ∈ ∂ f ( ¯ x ) if and only if ¯ x = a k . Thus, f has a unique absolute minimum at a k . Figure 4.10: Subdifferential of f ( x ) = ∑ 2 k − 1 i = 1 | x − a i | . Figure 4.11: Subdifferential of g ( x ) = ∑ 2 k i = 1 | x − a i | . Similarly, if a 1 < a 2 < · · · < a 2 k and g ( x ) = 2 k ∑ i = 1 | x − a i | . Then 0 ∈ ∂ g ( ¯ x ) if and only if ¯ x ∈ [ a k , a k + 1 ] . Thus, g has an absolute minimum at any point of [ a k , a k + 1 ] .