Suppose we have a system of equations: x_1=f_1(x_1,...,x_n) x_2=f_2(x_1,...,x_n) x_n=f_n(x_1,...,x_n) where the f_i are components of the gradient triangledownf for a monotone increasing, concave function f, so f_i>0 and dfi/dxi<=0.

Eliza Gregory

Eliza Gregory

Answered question

2022-10-22

Suppose we have a system of equations:
x 1 = f 1 ( x 1 , . . . , x n ) x 2 = f 2 ( x 1 , . . . , x n ) . . . x n = f n ( x 1 , . . . , x n )
where the f i are components of the gradient f for a monotone increasing, concave function f, so f i > 0 and f i x i 0.
Does the system of equations necessarily have a solution? And is it unique if it exists?

Answer & Explanation

Milton Hampton

Milton Hampton

Beginner2022-10-23Added 16 answers

Let g = f + h where h ( x 1 , . . . , x n ) = 1 2 Σ i x i 2
Then g is a strictly concave function, because it is the sum of a concave and a strictly concave function. And stationary points of concave functions are in general global maxima, so uniqueness follows by taking the derivative of g.

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