Let f:Omega sube R^n->R_(>=0) be a continuous differentiable function over Omega. Suppose that the function f is concave, and fix two points x=(x_1,…,x_n),y=(y_1,…,y_n) in Omega. If x_i≤y_i for all i=1,…,n and Omega=R^n, does it hold ∥grad_xf∥>=∥grad_yf∥?

unabuenanuevasld 2022-11-21 Answered
Let f : Ω R n R 0 be a continuous differentiable function over Ω. Suppose that the function f is concave, and fix two points x = ( x 1 , , x n ) , y = ( y 1 , , y n ) Ω, x = ( x 1 , , x n ) , y = ( y 1 , , y n ) Ω.
If x i y i for all i = 1 , , n and Ω = R n , does it hold x f ∥≥∥ y f ?
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sliceu4i
Answered 2022-11-22 Author has 16 answers
Is false. Take a concave function which is symmetric about the origin (e.g. f = x 2 ). if 0 x i and f ( x ) > f ( 0 ) , then due to symmetry we'd get x i 0 but f ( x ) < f ( 0 ) .
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