How can I show that a twice continuously differentiable function with a lipschitz continuous hessian with all eigenvalues≥mu is mu strongly convex? Definition 2.2. A function f : R" → R is said to be u-strongly convex if38Vx, y E R" :f (x) + (Vf(x), y – x) + y – x||² < f(y)(2.4) Definition (Lipschitz continuity)Suppose that• X and y open sets• F: X →Y• || · ||x and || · ||y are normsThen• F is Lipschitz continuous at x E X if 3 y(x) such that|F(z) - F(x) ||ly < Y(x) ||z-제1xfor all z E X.• F is Lipschitz continuous throughout/in X if 3 y such that||F(2) – F(x)||» < l|z – x|| xfor all x and z E X.Theorem (Taylor approximations for real-valued functions)Let S be an open subset of R", s E R", and suppose that f : S → R is continuouslydifferentiable throughout S and g = Vf is Lipschitz continuous at x with Lipschitzconstant y (x) for some appropriate vector norm. It follows that if the segment[x, x + s] E S, then\F(x + s) – m“ (x + s)| < (x)|||°,wherem" (x + s) = f(x) +g(x)*s.If in addition, f is twice continuously differentiable throughout S and H =Lipschitz continuous at x, with Lipschitz constant y (x), then= V²f isF(x + s) – mº (x + s)| < e (x)||s||°,wherem° (x + s) = f(x) + 8(x)"s + }s'H(x)s.

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How can I show that a twice continuously differentiable function with a lipschitz continuous hessian with all eigenvalues≥mu is mu strongly convex?

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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Question

How can I show that a twice continuously differentiable function with a lipschitz continuous hessian with all eigenvalues≥mu is mu strongly convex?

Definition 2.2. A function f : R" → R is said to be u-strongly convex if
38
Vx, y E R" :
f (x) + (Vf(x), y – x) + y – x||² < f(y)
(2.4)

Definition (Lipschitz continuity)
Suppose that
• X and y open sets
• F: X →Y
• || · ||x and || · ||y are norms
Then
• F is Lipschitz continuous at x E X if 3 y(x) such that
|F(z) - F(x) ||ly < Y(x) ||z-제1x
for all z E X.
• F is Lipschitz continuous throughout/in X if 3 y such that
||F(2) – F(x)||» < l|z – x|| x
for all x and z E X.
Theorem (Taylor approximations for real-valued functions)
Let S be an open subset of R", s E R", and suppose that f : S → R is continuously
differentiable throughout S and g = Vf is Lipschitz continuous at x with Lipschitz
constant y (x) for some appropriate vector norm. It follows that if the segment
[x, x + s] E S, then
\F(x + s) – m“ (x + s)| < (x)|||°,
where
m" (x + s) = f(x) +g(x)*s.
If in addition, f is twice continuously differentiable throughout S and H =
Lipschitz continuous at x, with Lipschitz constant y (x), then
= V²f is
F(x + s) – mº (x + s)| < e (x)||s||°,
where
m° (x + s) = f(x) + 8(x)"s + }s'H(x)s.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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