数理計画法特論
11
月の宿題令和
1
年12
月2
日 田地7. Show that the convexity of a function f is equivalent to the Jensen’s inequality f
( m
∑
i=1
λ i x i
)
≤ ∑ m
i=1
λ i f (x i ), where λ i ≥ 0 and ∑ m i=1 λ i = 1.
8. Verify that the functions
f 1 (x) = max(x 1 , . . . , x n ) f 2 (x) = log (e x
1+ · · · + e x
n) are convex on R n . Show also that the inequality
f 1 (x) ≤ f 2 (x) ≤ f 1 (x) + log n holds.
9. Calculate the conjugate and biconjugate functions of the function
f (x) =
x log x if x > 0 0 if x = 0 + ∞ if x < 0.
10. Let f : ℜ n → ℜ be a proper convex function. Show that its directional derivative f ′ (x; · ) : ℜ n → [ −∞ , + ∞ ] is a convex function for all direction d ∈ ℜ n with a fixed point x ∈ ℜ n .
11. Let f : ℜ 2 → ℜ be a function defined by
f(x) =
0 if x 1 = 0
2x 2 e − x
−12x 2 2 + e −2x
−12otherwise.
Show that f is differentiable at 0, but not continuous at 0.
12. Verify the Cauchy-Schwartz inequality | x T y | ≤ ∥ x ∥ 2 ∥ y ∥ 2 . 13. Let A be an n × n matrix. Show that
∥ A ∥ 1 = max
1 ≤ j ≤ n
∑ n i=1
| a ij | , ∥ A ∥ ∞ = max
1 ≤ i ≤ n
∑ n j=1
| a ij | and ∥ A ∥ 2 = √ λ, where λ is the maximum eigenvalue of A T A.
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