Suppose f is differentiable everywhere. Show that if æ* is a local minimizer for f, Vf(x*) = 0. Note that we are not assuming the gradient map is continuous.
Suppose f is differentiable everywhere. Show that if æ* is a local minimizer for f, Vf(x*) = 0. Note that we are not assuming the gradient map is continuous.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Exercise 1.2 (
: Let f: R" → R.
(a) Suppose f is differentiable everywhere. Show that if a* is a local minimizer for f, Vf(x) = 0.
Note that we are not assuming the gradient map is continuous.
(b) Suppose f is twice continuously differentiable. By defining (n)
formula
(1) = (0) + - $¹(0) + ² * $"(t) dt da.
Hence, show that for any a, s € R"
f(x+ns), justify the
\ƒ(x + s) − ƒ(x) − g(x)¹s − ½s¹H(x)s| ≤ 79||$||2,
whenever f has a Lipschitz continuous Hessian with Lipschitz constant on S.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a6376af-076b-46db-ad44-654584889b97%2F246fa46d-ce17-4a31-ac54-52b1964302e2%2Fdx7jdjk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 1.2 (
: Let f: R" → R.
(a) Suppose f is differentiable everywhere. Show that if a* is a local minimizer for f, Vf(x) = 0.
Note that we are not assuming the gradient map is continuous.
(b) Suppose f is twice continuously differentiable. By defining (n)
formula
(1) = (0) + - $¹(0) + ² * $"(t) dt da.
Hence, show that for any a, s € R"
f(x+ns), justify the
\ƒ(x + s) − ƒ(x) − g(x)¹s − ½s¹H(x)s| ≤ 79||$||2,
whenever f has a Lipschitz continuous Hessian with Lipschitz constant on S.
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