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Exercise 0.8. Write out the full details of the previous proof, and also show the converse of Proposition 1 holds. That is, if (11) holds for all o, 1 € dom(f) and 0 ≤ x ≤ 1, then f(.) is convex. Dug of in

Exercise 0.8. Write out the full details of the previous proof, and also show the converse of Proposition 1 holds. That is, if (11) holds for all o, 1 € dom(f) and 0 ≤ x ≤ 1, then f(.) is convex. Dug of in

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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Exercise 0.8. Write out the full details of the previous proof, and also show the converse of Proposition 1 holds. That is, if (11) holds for all o, ₁ € dom(f) and 0 ≤ x ≤ 1, then f(.) is convex. Proof is at the bottom Proposition 1. Suppose f(): R → R is convex. Then for every To, 1 € dom(f), To ‡ ₁ and 0<x< 1, we have See Figure 5: f(xx)-f(xo) X(x₁ - x0) f(x1) f(xo) f(xx) m1 epi(f) To f(x₁) = f(x) (x1 - x0) slope=m₁ Į XX m2 f(x₁)-f(xx) (1-X)(x₁ - xo), slope= m₂ 21 m3 slope=m3 Figure 5: The 3-point property (11) Proof. Assume f() is convex with xo x₁ € R" belonging to dom(f) and 0 < X < 1. Then f(xx) = f((1-X)x₁ + Xx₁) ≤ (1 − A)f(x) + f(x₁). (12) The right-hand side of (12) equals f(zo) + X(ƒ (x₁) − ƒ (x₁)), and then rearranging terms in (12) and dividing by A gives the first inequality in (11). The right-hand side of (12) also equals f(x₁) + (1-x)(ƒ(xo) – ƒ (x₁)), and again rearranging terms in (12) and dividing by (1 - A) gives the second inequality in (11).
Transcribed Image Text:Exercise 0.8. Write out the full details of the previous proof, and also show the converse of Proposition 1 holds. That is, if (11) holds for all o, ₁ € dom(f) and 0 ≤ x ≤ 1, then f(.) is convex. Proof is at the bottom Proposition 1. Suppose f(): R → R is convex. Then for every To, 1 € dom(f), To ‡ ₁ and 0<x< 1, we have See Figure 5: f(xx)-f(xo) X(x₁ - x0) f(x1) f(xo) f(xx) m1 epi(f) To f(x₁) = f(x) (x1 - x0) slope=m₁ Į XX m2 f(x₁)-f(xx) (1-X)(x₁ - xo), slope= m₂ 21 m3 slope=m3 Figure 5: The 3-point property (11) Proof. Assume f() is convex with xo x₁ € R" belonging to dom(f) and 0 < X < 1. Then f(xx) = f((1-X)x₁ + Xx₁) ≤ (1 − A)f(x) + f(x₁). (12) The right-hand side of (12) equals f(zo) + X(ƒ (x₁) − ƒ (x₁)), and then rearranging terms in (12) and dividing by A gives the first inequality in (11). The right-hand side of (12) also equals f(x₁) + (1-x)(ƒ(xo) – ƒ (x₁)), and again rearranging terms in (12) and dividing by (1 - A) gives the second inequality in (11).
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