convex function

Definition Suppose $\mathrm{\Omega }$ is a convex set in a vector space over $ℝ$ (or $ℂ$), and suppose $f$ is a function $f:\mathrm{\Omega }\to ℝ$. If for any $x,y\in \mathrm{\Omega }$, $x\ne y$ and any $\lambda \in (0,1)$, we have

$$f\left(\lambda x+(1-\lambda )y\right)\le \lambda f(x)+(1-\lambda )f(y),$$

we say that $f$ is a convex function. If for any $x,y\in \mathrm{\Omega }$ and any $\lambda \in (0,1)$, we have

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

we say that $f$ is a concave function. If either of the inequalities are strict, then we say that $f$ is a strictly convex function, or a strictly concave function, respectively.

Properties

• 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.

• On $ℝ$, a continuous function is convex if and only if for all $x,y\in ℝ$, we have

  $$f\left(\frac{x+y}{2}\right)\le \frac{f(x)+f(y)}{2}.$$

• On $ℝ$, a once differentiable function is convex if and only if $f^{\prime }$ is monotone increasing.

• Suppose $f$ is twice continuously differentiable function on $ℝ$. Then $f$ is convex if and only if $f^{\prime \prime }\ge 0$. If $f^{\prime \prime }>0$, then $f$ is strictly convex.

• A local minimum of a convex function is a global minimum. See this page (http://planetmath.org/ExtremalValueOfConvexconcaveFunctions).

Examples

• $e^x$,$e^{-x}$, and $x^2$ are convex functions on $ℝ$. Also, $x^4$ is strictly convex, but $12x^2$ vanishes at $x=0$.

• A norm (http://planetmath.org/NormedVectorSpace) is a convex function.

Remark.

We may generalize the above definition of a convex function to an that of an extended real-valued function whose domain is not necessarily a convex set. First, we define what an epigraph of a function is.

Let $\mathrm{\Omega }$ be a subset of a vector space over the reals, and $f$ an extended real-valued function defined on $\mathrm{\Omega }$. The epigraph of $f$, denoted by $\mathrm{epi}(f)$, is the set

$$\{(x,r)\mid x\in \mathrm{\Omega },r\ge f(x)\}.$$

An extended real-valued function $f$ defined on a subset $\mathrm{\Omega }$ of a vector space $V$ over the reals is said to be convex if its epigraph is a convex subset of $V\times ℝ$. With this definition, the domain $\mathrm{\Omega }$ of $f$ need not be convex. However, its subset , called the effective domain and denoted by $\mathrm{eff}.\mathrm{dom}(f)$, is convex. To see this, suppose $x,y\in \mathrm{eff}.\mathrm{dom}(f)$ and $z=\lambda x+(1-\lambda )y$ with $0\le \lambda \le 0$. Then $(z,\overline{z})=\lambda (x,f(x))+(1-\lambda )(y,f(y))\in \mathrm{epi}(f)$, where $\overline{z}=\lambda f(x)+(1-\lambda )f(y)$, since $\mathrm{epi}(f)$ is convex by definition. Therefore, $z\in \mathrm{dom}(f)$. In fact, , which implies that $z\in \mathrm{eff}.\mathrm{dom}(f)$.

Title	convex function
Canonical name	ConvexFunction
Date of creation	2013-03-22 11:46:26
Last modified on	2013-03-22 11:46:26
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	28
Author	matte (1858)
Entry type	Definition
Classification	msc 52A41
Classification	msc 26A51
Classification	msc 26B25
Classification	msc 55Q05
Classification	msc 18G30
Classification	msc 18B40
Classification	msc 20J05
Classification	msc 20E07
Classification	msc 18-01
Classification	msc 20L05
Related topic	JensensInequality
Related topic	LogarithmicallyConvexFunction
Defines	concave function
Defines	strictly convex function
Defines	strictly concave function
Defines	strictly convex
Defines	strictly concave
Defines	epigraph
Defines	effective domain
Defines	concave