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(c) f(x,u, v) = – log(uv - x" x) on dom f = {(x,u, v) | uv > x² x, u, v > 0}.

(c) f(x,u, v) = – log(uv - x" x) on dom f = {(x,u, v) | uv > x² x, u, v > 0}.

Advanced Engineering Mathematics
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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please send handwritten solution for part c Q 3.22
3.22 Composition rules. Show that the following functions are convex. (a) f(x) = – log(– log(E, eªi =+b;)) use the fact that log(>, eyi) is convex. on dom f = {x | E, eºf =+bi < 1}. You can m vi=1 i=1 vi=1 (b) f(ӕ, и, v) fact that x'x/u is convex in (x, u) for u > 0, and that –, {(x,u, v) | uv > x"x, u, v > 0}. Use the Vx1x2 is convex on R² = -Vuv – xTx on dom f = ++· (c) f(x,u, v) = – log(uv – xTx) (d) f(x,t) = -(tº – ||x||;)"/? where p > 1 and dom f = {(x, t) | t > ||c||,}. You can use the fact that ||x||B/uP-1 is convex in (x, u) for u > 0 (see exercise 3.23), and that -x1/Pyl-1/p is convex on R? (see exercise 3.16). on dom f = {(x, u, v) | uv > x"x, u, v> 0}. (e) f(x,t) = – log(t" – ||||2) where p > 1 and dom f = {(x,t) | t > ||x||p}. You can use the fact that ||x||½/u²-! is convex in (x, u) for u > 0 (see exercise 3.23).
Transcribed Image Text:3.22 Composition rules. Show that the following functions are convex. (a) f(x) = – log(– log(E, eªi =+b;)) use the fact that log(>, eyi) is convex. on dom f = {x | E, eºf =+bi < 1}. You can m vi=1 i=1 vi=1 (b) f(ӕ, и, v) fact that x'x/u is convex in (x, u) for u > 0, and that –, {(x,u, v) | uv > x"x, u, v > 0}. Use the Vx1x2 is convex on R² = -Vuv – xTx on dom f = ++· (c) f(x,u, v) = – log(uv – xTx) (d) f(x,t) = -(tº – ||x||;)"/? where p > 1 and dom f = {(x, t) | t > ||c||,}. You can use the fact that ||x||B/uP-1 is convex in (x, u) for u > 0 (see exercise 3.23), and that -x1/Pyl-1/p is convex on R? (see exercise 3.16). on dom f = {(x, u, v) | uv > x"x, u, v> 0}. (e) f(x,t) = – log(t" – ||||2) where p > 1 and dom f = {(x,t) | t > ||x||p}. You can use the fact that ||x||½/u²-! is convex in (x, u) for u > 0 (see exercise 3.23).
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