IssaRice: Created page with "a function $f : \mathbf R \to \mathbf R$ is often defined to be convex if $f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$ for all $x,y \in \mathbf R$ ..."

2020-01-14T00:05:12Z

Created page with "a function <math>f : \mathbf R \to \mathbf R</math> is often defined to be convex if <math>f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)</math> for all <math>x,y \in \mathbf R</math>..."

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a function <math>f : \mathbf R \to \mathbf R</math> is often defined to be convex if <math>f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)</math> for all <math>x,y \in \mathbf R</math> and all <math>t \in [0,1]</math>.

two ways to think about this:

* if you want to use the idea that the set of [[wikipedia:convex combination]]s of two points is the line segment between them: take the points (x, f(x)) and (y, f(y)). then each convex combination has the form <math>(tx + (1-t)y, tf(x) + (1-t)f(y))</math> where t is between 0 and 1. so now we are saying that if we fix some x value tx + (1-t)y along this line segment, then the y value of the y segment, tf(x) + (1-t)f(y), is at least the y value of the function, f(tx + (1-t)y). in other words, the line segment between any two points lies above the function.
* if you want to reason just from middle-school algebra: the line between (x, f(x)) and (y, f(y)) is given by <math>l(z) = f(x) + \frac{f(y)-f(x)}{y-x}(z-x)</math>. rewrite this as <math>l(z) = f(x) + (f(y)-f(x))\frac{z-x}{y-x}</math>. if we consider just the line segment between the points, then <math>z \in [x,y]</math>. thus <math>\frac{z-x}{y-x} \in [0, 1]</math>. so if we make the substitution <math>t = \frac{z-x}{y-x}</math> we get <math>f(x) + (f(y)-f(x))t = (1-t)f(x) + tf(y)</math>. now you could say x and y are symmetric here, we could swap them around to get the thing in the definition of convex function, or you could substitute again, say s=1-t, to get it in the right form.
:t measures what fraction of the distance from x to y we have traversed (just inspect its definition). the x-value for a given value of t is (just solve for z): <math>z = x + (y-x)t = (1-t)x + ty</math>. the y-value of the line segment at this point is (1-t)f(x) + tf(y) as we saw above. thus we are saying that the function value at this x-value, f((1-t)x + ty), is below the line segment's y-value, (1-t)f(x) + tf(y).

User:IssaRice/Convex function - Revision history

IssaRice: Created page with "a function f : \mathbf R \to \mathbf R is often defined to be convex if f(tx + (1-t)y) \leq tf(x) + (1-t)f(y) for all x,y \in \mathbf R..."

IssaRice: Created page with "a function $f : \mathbf R \to \mathbf R$ is often defined to be convex if $f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$ for all $x,y \in \mathbf R$ ..."