Exercise 6.4.3

Answers

Multiply (7) by M and (8) by m, and subtract to remove ∥∇ ⁡f(xk)∥2, we have

Mf(xk+1) − mf(x∗) ≤ Mf(x k) − mf(xk) M (f(xk+1) − f(x∗)) + (M − m)f(x∗) ≤ (M − m)f(x k) M (f(xk+1) − f(x∗)) ≤ (M − m) (f(x k) − f(x∗)) f(xk+1) − f(x∗) ≤ M − m M (f(xk) − f(x∗)) f(xk+1) − f(x∗) ≤ (1 − m M) (f(xk) − f(x∗))
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2020-03-20 00:00
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