1. [10 points] (Convexity) Consider f: R R. For y,92 R, define the function...

Transcribed Image Text:

1. [10 points] (Convexity) Consider f: R → R. For y₁,92 € R", define the function g: R → R by g(x) = f(x(y₁ - y2) + y2), x≤R. (a) Prove that if f is convex on R", then g is convex on R for all y₁, y2 € Rn. Hint: You can start your proof as follows. For any y₁, y2 € R", a € [0, 1], x₁, x2 € R, g(ax₁ + (1-a)x₂) = f((ax₁ + (1 − a)x₂)(y₁ - y2) + y2) = f((ax₁ + (1 - a)x₂) (y₁ - y₂) + (a + (1 - a)) y₂) = f(a(x₁(y₁ - y2) + y2) + (1 - a) (x2(y₁ - y2) + y2)). Then use the convexity of f to continue the proof. (b) Prove that if g is convex on R for all y₁,92 € R", then f is convex on R". Hint: You can start your proof as follows. For any y₁, y2 € R", a = [0, 1], f(ayı + (1 a)y2) = f(a(y₁ - y2) + y2) = g(a) = g(a 1+ (1 - a).0). Then use the convexity of g to continue the proof. 1. [10 points] (Convexity) Consider f: R → R. For y₁,92 € R", define the function g: R → R by g(x) = f(x(y₁ - y2) + y2), x≤R. (a) Prove that if f is convex on R", then g is convex on R for all y₁, y2 € Rn. Hint: You can start your proof as follows. For any y₁, y2 € R", a € [0, 1], x₁, x2 € R, g(ax₁ + (1-a)x₂) = f((ax₁ + (1 − a)x₂)(y₁ - y2) + y2) = f((ax₁ + (1 - a)x₂) (y₁ - y₂) + (a + (1 - a)) y₂) = f(a(x₁(y₁ - y2) + y2) + (1 - a) (x2(y₁ - y2) + y2)). Then use the convexity of f to continue the proof. (b) Prove that if g is convex on R for all y₁,92 € R", then f is convex on R". Hint: You can start your proof as follows. For any y₁, y2 € R", a = [0, 1], f(ayı + (1 a)y2) = f(a(y₁ - y2) + y2) = g(a) = g(a 1+ (1 - a).0). Then use the convexity of g to continue the proof.

Answer rating: 100% (QA)

solution Since f is convex on R for all for y1 y2 Rn 01 x1x2 Rn we have fx11 x2fx11fx2 ... View the full answer