2. Show that the following functions are convex by verifying the condition that ² f(x) 20 is satisfied for all a in the domain of f: (a) f(u₁, ₂) = n(e" +e"), (b) f(u₁, U2, U3, U4) = ln(1-₁-₂-3-4) over the domain {ue Rª|u₁ + 1₂ + 3 + ₁ ≤ 1}. -
2. Show that the following functions are convex by verifying the condition that ² f(x) 20 is satisfied for all a in the domain of f: (a) f(u₁, ₂) = n(e" +e"), (b) f(u₁, U2, U3, U4) = ln(1-₁-₂-3-4) over the domain {ue Rª|u₁ + 1₂ + 3 + ₁ ≤ 1}. -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Use hessian matrix
![2. Show that the following functions are convex by verifying the condition that
\[
\nabla^2 f(x) \succeq 0
\]
is satisfied for all \( x \) in the domain of \( f \):
(a) \( f(u_1, u_2) = \ln(e^{u_1} + e^{u_2}) \),
(b) \( f(u_1, u_2, u_3, u_4) = \ln(1 - u_1 - u_2 - u_3 - u_4) \) over the domain \(\{ u \in \mathbb{R}^4 \mid u_1 + u_2 + u_3 + u_4 \leq 1 \} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4589616-f4fd-4eb5-854c-a0440b9d838b%2Fec76fa4f-092c-4d75-9297-1283a1758b96%2F78i58c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Show that the following functions are convex by verifying the condition that
\[
\nabla^2 f(x) \succeq 0
\]
is satisfied for all \( x \) in the domain of \( f \):
(a) \( f(u_1, u_2) = \ln(e^{u_1} + e^{u_2}) \),
(b) \( f(u_1, u_2, u_3, u_4) = \ln(1 - u_1 - u_2 - u_3 - u_4) \) over the domain \(\{ u \in \mathbb{R}^4 \mid u_1 + u_2 + u_3 + u_4 \leq 1 \} \).
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