Answered: (#4) When is a function f(.): R" R…

(#4) When is a function f(.): R" R convex and proper? Show that f(-) is convex and proper implies epi(f) is a convex subset of R"+1

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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(#4) When is a function f(): R"→ R convex and proper? Show that f(-) is convex and proper implies epi(f) is
a convex subset of R"+1

Expert Solution

Step 1 We have to prove f(.) to be convex and proper.

Function f : Rⁿ→ R is convex if its domain is a convex set and for all x, y in its domain, and all λ ∈ [0, 1], we have
f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y).

A function f is proper if f(x) < ∞ for at least one x ∈ X.

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