My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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.
Efalcon is offline  
 
February 10th, 2017, 01:38 AM   #2
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 609
Thanks: 378

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by Efalcon View Post
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.
SDK is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
concave, convex optimization, function, ln1, numbers, positive, real, x or by



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
is this function concave? AEcon Economics 1 October 28th, 2013 08:57 AM
Proving a real-valued function can't be concave Kreader12 Real Analysis 0 June 7th, 2013 09:18 PM
Positive Real Numbers jstarks4444 Real Analysis 1 March 30th, 2011 04:38 AM
Positive Real Matrix Eigenvector skeboy Linear Algebra 0 June 11th, 2009 10:51 PM
Positive Real Numbers jstarks4444 Number Theory 1 December 31st, 1969 04:00 PM





Copyright © 2019 My Math Forum. All rights reserved.