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'convex function'
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| Title of object: |
convex function |
| Canonical Name: |
ConvexFunction |
| Type: |
Definition |
| Created on: |
2001-10-15 22:34:11 |
| Modified on: |
2006-10-25 13:06:07 |
| Classification: |
msc:26B25, msc:26A51, msc:52A41 |
| Defines: |
concave function, strictly convex function, strictly concave function |
Revision comment (for changes between this and next version):
| defines "strictly convex", "strictly concave" |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}} |
Content:
{\bf Definition} Suppose $\Omega$ is a convex set in a vector space over $\sR$
(or $\sC$), and suppose $f$ is a function $f:\Omega\to \sR$.
If for any $x,y\in \Omega$, $x\neq y$ and any $\lambda \in (0,1)$, we have
$$f\Big( \lambda x + (1-\lambda)y\Big)\leq \lambda f(x)+(1-\lambda)f(y),$$
we say that $f$ is a \emph{convex function}.
If for any $x,y\in \Omega$ and any $\lambda \in (0,1)$, we have
$$f\Big( \lambda x + (1-\lambda)y\Big)\geq \lambda f(x)+(1-\lambda)f(y),$$
we say that $f$ is a \emph{concave function}. If either of the inequalities
are strict, then we say that $f$ is a \emph{strictly convex function},
or a \emph{strictly concave function}, respectively.
\subsubsection*{Properties}
\begin{itemize}
\item A function $f$ is a (strictly) convex function if and only if $-f$ is
a (strictly) concave function. For this reason, most of the below discussion only
focuses on convex functions. Analogous result holds for concave functions.
\item On $\sR$, a continuous function is convex
if and only if for all $x,y\in \sR$, we have
$$f\left(\frac{x+y}{2}\right)\le\frac{f(x)+f(y)}{2}.$$
\item On $\sR$, a once differentiable function is convex if and only if $f'$
is monotone increasing.
\item Suppose $f$ is twice continuously differentiable function on $\sR$.
Then $f$ is convex if and only if $f'' \ge 0$.
If $f''>0$, then $f$ is strictly convex.
\item A local minimum of a convex function is a global minimum.
See \PMlinkname{this page}{ExtremalValueOfConvexconcaveFunctions}.
\end{itemize}
\subsubsection*{Examples}
\begin{itemize}
\item $e^x$,$e^{-x}$, and $x^2$ are convex functions on $\sR$. Also, $x^4$ is strictly
convex, but $12x^2$ vanishes at $x=0$.
\item A norm is a convex function.
\end{itemize} |
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