My Math Forum Is ln(1+x/(by+a)) concave function? (x,y,a and b are positive real numbers)

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 February 9th, 2017, 11:17 PM #1 Newbie   Joined: Feb 2017 From: lebanon Posts: 1 Thanks: 0 Is ln(1+x/(by+a)) concave function? (x,y,a and b are positive real numbers) From boyd's book: the composition of a concave function with affine function is a concave function. As we can see ln(x) is concave function and g(x,y)=x/(by+a) is linear-fractional function so ln(1+x/(by+a)) should be concave function. On the other hand, the hessian matrix of a concave function must be negative semi-definite which is not the case for this function. So in your opinion, is ln(1+x/(by+a)) concave function or not. If not can we consider it as quasi-concave function so we can use the Lagrangian method to solve such optimization problem.
February 10th, 2017, 01:38 AM   #2
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Quote:
 Originally Posted by Efalcon From boyd's book: the composition of a concave function with affine function is a concave function. As we can see ln(x) is concave function and g(x,y)=x/(by+a) is linear-fractional function so ln(1+x/(by+a)) should be concave function. On the other hand, the hessian matrix of a concave function must be negative semi-definite which is not the case for this function. So in your opinion, is ln(1+x/(by+a)) concave function or not. If not can we consider it as quasi-concave function so we can use the Lagrangian method to solve such optimization problem.
Your mistake is the claim that $\frac{x}{by+a}$ is linear. It is not. Moreover, it isn't monotone (with respct to radius) so there is no reason to expect it to be concave.

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