1. Let ƒ : Rd → R be a twice differentiable and µ-strongly-convex function. Prove that f is bounded below. That is, there is a value CER (that depends on µ) such that for all x € Rª, we have f(x) ≥ C. 2. Suppose that f: Rd → R is ẞ-smooth, i.e., Vf is ẞ-Lipschitz continuous. Show that g(x) = f(Ax+b) with A = Rmxd and b = Rm is ẞ||A||2-smooth. Here, ||A|| = ||AT || is the operator norm of A (and AT) that satisfies ||Ax2≤A|X2 for all x. 3. Given a family of convex functions f; : Rd → R such that j = {1,...,d}. Let us define f(x) Prove that f(x) is convex. := maxje{1,.,d}{ƒj (x)}. 4. For each of the following functions, prove whether or not they are convex and whether or not they have a L-Lipschitz- continuous gradient. i) f(x) = ex², x = R. ii) f(x, y) = (x² + y²) ½, such that (x, y) = R².
1. Let ƒ : Rd → R be a twice differentiable and µ-strongly-convex function. Prove that f is bounded below. That is, there is a value CER (that depends on µ) such that for all x € Rª, we have f(x) ≥ C. 2. Suppose that f: Rd → R is ẞ-smooth, i.e., Vf is ẞ-Lipschitz continuous. Show that g(x) = f(Ax+b) with A = Rmxd and b = Rm is ẞ||A||2-smooth. Here, ||A|| = ||AT || is the operator norm of A (and AT) that satisfies ||Ax2≤A|X2 for all x. 3. Given a family of convex functions f; : Rd → R such that j = {1,...,d}. Let us define f(x) Prove that f(x) is convex. := maxje{1,.,d}{ƒj (x)}. 4. For each of the following functions, prove whether or not they are convex and whether or not they have a L-Lipschitz- continuous gradient. i) f(x) = ex², x = R. ii) f(x, y) = (x² + y²) ½, such that (x, y) = R².
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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Transcribed Image Text:1. Let ƒ : Rd → R be a twice differentiable and µ-strongly-convex function. Prove that f is bounded below. That is, there
is a value CER (that depends on µ) such that for all x € Rª, we have f(x) ≥ C.
2. Suppose that f: Rd → R is ẞ-smooth, i.e., Vf is ẞ-Lipschitz continuous. Show that g(x) = f(Ax+b) with A = Rmxd
and b = Rm is ẞ||A||2-smooth. Here, ||A|| = ||AT || is the operator norm of A (and AT) that satisfies ||Ax2≤A|X2
for all x.
3. Given a family of convex functions f; : Rd → R such that j = {1,...,d}. Let us define f(x)
Prove that f(x) is convex.
:= maxje{1,.,d}{ƒj (x)}.
4. For each of the following functions, prove whether or not they are convex and whether or not they have a L-Lipschitz-
continuous gradient.
i) f(x) = ex², x = R.
ii) f(x, y) = (x² + y²) ½, such that (x, y) = R².
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