148 5.6 Nondifferentiable Convex Functions and Subdifferentials Similarly, if a1 < a2 < ··· < a2k and 2k g(x)= ∑ |x − ai|. i=1 Then 0 ∈ ∂ g(x0) if and only if x0 ∈ [ak,ak+1]. Thus, g has an absolute minimum at any point of [ak,ak+1]. The following theorem is a version of the Mean Value Theorem (Theorem 4.2.3) for nondifferentiable functions. Figure 5.9: Subdifferential mean value theorem. Theorem 5.6.8 Let f : R → R be a convex function and let a < b. Then there exists c ∈ (a,b) such that f (b) − f (a) ∈ ∂ f (c). (5.11) b − a Proof: Defne f (b) − f (a) g(x)= f (x) − (x − a)+ f (a) . b − a Then g is a convex function and g(a)= g(b). Thus, g has a local minimum at some c ∈ (a,b) and, hence, g also has an absolute minimum at c. Observe that the function f (b) − f (a) h(x)= − (x − a)+ f (a) b− a
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