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August 8th, 2017, 03:42 AM   #1
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Convex function

Quote:
 A function $f : \mathbb{R}^n \to \mathbb{R}$ is called convex if for every pair of vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ and every $\lambda \in [0, 1]$, we have, $$f(\lambda \mathbf{x} + (1 - \lambda)\mathbf{y}) \le \lambda f(\mathbf{x}) + (1 - \lambda)f(\mathbf{y})$$
I don't understand the role of $\lambda$ here. I can see that addition is preserved but that's about it. Perhaps I'm being daft. Any insight? Thanks . August 9th, 2017, 10:02 PM   #2
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Quote:
 Originally Posted by Joppy Perhaps I'm being daft.
Yes very daft Joppy.

The argument of $f$ merely denotes all points between $\mathbf{x}$ and $\mathbf{y}$. Hence if we consider a function whose argument is all of these points, and is less than the points themselves, the function is convex. August 16th, 2017, 05:52 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 When $\displaystyle \lambda$ is 0, $\displaystyle \lambda x+ (1- \lambda)y= (0)x+ (1)y$ is just y. When $\displaystyle \lambda$ is 1, $\displaystyle \lambda x+ (1- \lambda)y= (1)x+ (0)y$ is just x. Since that is a linear formula, as $\displaystyle \lambda$ goes from 0 to 1, the point moves on the straight line from y to x. Last edited by greg1313; August 16th, 2017 at 07:02 AM. August 16th, 2017, 03:40 PM   #4
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Quote:
 Originally Posted by Country Boy When $\displaystyle \lambda$ is 0, $\displaystyle \lambda x+ (1- \lambda)y= (0)x+ (1)y$ is just y. When $\displaystyle \lambda$ is 1, $\displaystyle \lambda x+ (1- \lambda)y= (1)x+ (0)y$ is just x. Since that is a linear formula, as $\displaystyle \lambda$ goes from 0 to 1, the point moves on the straight line from y to x.
Thanks . Don't know why it took me so long to see it. Tags convex, function 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 frozenfish Calculus 1 September 10th, 2013 06:31 AM swagatopablo Algebra 0 October 10th, 2012 10:49 PM Vasily Applied Math 1 June 30th, 2012 02:57 PM coolhandluke Applied Math 2 February 16th, 2012 06:47 AM MathFreak Calculus 1 October 20th, 2011 08:04 PM

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