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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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 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. Tags concave, convex optimization, function, ln1, numbers, positive, real, x or by Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post AEcon Economics 1 October 28th, 2013 08:57 AM Kreader12 Real Analysis 0 June 7th, 2013 09:18 PM jstarks4444 Real Analysis 1 March 30th, 2011 04:38 AM skeboy Linear Algebra 0 June 11th, 2009 10:51 PM jstarks4444 Number Theory 1 December 31st, 1969 04:00 PM

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