Difference between revisions of "Convex function"

Revision as of 21:21, 8 April 2007

A function $f: I \mapsto \mathbb{R}$ for some interval $\displaystyle I \subseteq \mathbb{R}$ is convex (sometimes written concave up) over $\displaystyle I$ if and only if the set of all points $\displaystyle (x,y)$ such that $\displaystyle y \ge f(x)$ is convex. Equivalently, $\displaystyle f$ is convex if for every $\lambda \in [0,1]$ and every $x,y \in I$ ,

$\displaystyle \lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right)$ .

Usually, when we do not specify $\displaystyle I$ , we mean $I = \mathbb{R}$ .

We say that $\displaystyle f$ is concave (or, occasionally, that it is concave down) if $\displaystyle -f$ is convex.

If $\displaystyle f$ is differentiable, then it is convex if and only if $\displaystyle f'$ is non-decreasing. Similarly, if $\displaystyle f$ is twice differentiable, we say it is convex over an interval $\displaystyle I$ if and only if $f(x) \ge 0$ for all $x \in I$ .

Note that in our previous paragraph, our requirements that $\displaystyle f$ is differentiable and twice differentiable are crucial. For a simple example, consider the function

$f(x) = \lfloor x \rfloor (x - \lfloor x \rfloor ) + {\lfloor x \rfloor \choose 2}$ ,

defined over the non-negative reals. It is piecewise differentiable, but at infinitely many points (for all natural numbers $\displaystyle x$ , to be exact) it is not differentiable. Nevertheless, it is convex. More significantly, consider the function

$f(x) = \left( |x| - 1 \right)^2$

over the interval $\displaystyle [-2, 2]$ . It is continuous, and twice differentiable at every point except $\displaystyle{} (0, 1)$ . Furthermore, its second derivative is greater than 0, wherever it is defined. But its graph is shaped like a curvy W, and it is not convex over $\displaystyle [-2,2]$ , although it is convex over $\displaystyle [-2,0]$ and over $\displaystyle [0,2]$ .

This article is a stub. Help us out by expanding it.

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-Let <math>f:\mathbb{R}\to\mathbb{R}</math> be a function, and let <math>(a,b)</math> be an [[open interval]]. (This can be done with [[closed interval]]s or [[half-open interval]]s as well.) Then <math>f</math> is said to be '''convex''' on <math>(a,b)</math> if <math>f''(x)\ge 0</math> for all <math>x\in(a,b)</math>.
+A [[function]] <math>f: I \mapsto \mathbb{R}</math> for some interval <math> \displaystyle I \subseteq \mathbb{R} </math> is ''convex'' (sometimes written ''concave up'') over <math> \displaystyle I </math> if and only if the set of all points <math> \displaystyle (x,y) </math> such that <math> \displaystyle y \ge f(x) </math> is [[convex set | convex]].  Equivalently, <math> \displaystyle f </math> is convex if for every <math> \lambda \in [0,1] </math> and every <math> x,y \in I</math>,
+<center><math> \displaystyle \lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right) </math>.</center>  Usually, when we do not specify <math> \displaystyle I </math>, we mean <math> I = \mathbb{R} </math>.
+We say that <math> \displaystyle f </math> is '''concave''' (or, occasionally, that it is ''concave down'') if <math> \displaystyle -f </math> is convex.
+If <math> \displaystyle f </math> is differentiable, then it is convex if and only if <math> \displaystyle f' </math> is non-decreasing.  Similarly, if <math> \displaystyle f </math> is twice differentiable, we say it is convex over an interval <math> \displaystyle I </math> if and only if <math> f(x) \ge 0 </math> for all <math> x \in I </math>.
+Note that in our previous paragraph, our requirements that <math> \displaystyle f </math> is differentiable and twice differentiable are crucial.  For a simple example, consider the function
+<center>
+<math>
+f(x) = \lfloor x \rfloor (x - \lfloor x \rfloor ) + {\lfloor x \rfloor \choose 2}
+</math>,
+</center>
+defined over the non-negative reals.
+It is piecewise differentiable, but at infinitely many points (for all natural numbers <math> \displaystyle x </math>, to be exact) it is not differentiable.  Nevertheless, it is convex.  More significantly, consider the function
+<center>
+<math>
+f(x) = \left( |x| - 1 \right)^2
+</math>
+</center>
+over the interval <math> \displaystyle [-2, 2] </math>.  It is continuous, and twice differentiable at every point except <math> \displaystyle{} (0, 1) </math>.  Furthermore, its second derivative is greater than 0, wherever it is defined.  But its graph is shaped like a curvy W, and it is not convex over <math> \displaystyle [-2,2] </math>, although it is convex over <math> \displaystyle [-2,0] </math> and over <math> \displaystyle [0,2] </math>.
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Difference between revisions of "Convex function"

Revision as of 21:21, 8 April 2007