# Convex function

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In mathematics, a real-valued function *f* defined on an interval (or on any convex subset of some vector space) is called **convex**, **concave upwards**, **concave up** or **convex cup**, if for any two points *x* and *y* in its domain *C* and any *t* in [0,1], we have

In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set.

Pictorially, a function is called 'convex' if the function lies below the straight line segment connecting two points, for any two points in the interval.^{[1]}

A function is called **strictly convex** if

for any *t* in (0,1) and

A function *f* is said to be concave if − *f* is convex.

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## [edit] Properties

A convex function *f* defined on some open interval *C* is continuous on *C* and differentiable at all but at most countably many points. If *C* is closed, then *f* may fail to be continuous at the endpoints of *C*.

A function is midpoint convex on an interval *C* if

for all *x* and *y* in *C*. This condition is only slightly weaker than convexity. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex ^{[2]}. In particular, a continuous function that is midpoint convex will be convex.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval.

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: *f*(*y*) ≥ *f*(*x*) + *f* '(*x*) (*y* − *x*) for all *x* and *y* in the interval. In particular, if *f* '(*c*) = *0*, then *c* is a global minimum of *f*(*x*).

A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the converse does not hold. For example, the second derivative of *f*(*x*) = *x*^{4} is *f* "(*x*) = 12 *x*^{2}, which is zero for *x* = 0, but *x*^{4} is strictly convex.

More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.

Any local minimum of a convex function is also a global minimum. A *strictly* convex function will have at most one global minimum.

For a convex function *f*, the sublevel sets {*x* | *f*(*x*) < *a*} and {*x* | *f*(*x*) ≤ *a*} with *a* ∈ **R** are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function; such a function is called a *quasiconvex function*.

Jensen's inequality applies to every convex function *f*. If *X* is a random variable taking values in the domain of *f*, then (Here *E* denotes the mathematical expectation.)

## [edit] Convex function calculus

- If
*f*and*g*are convex functions, then so are*m*(*x*) = max{*f*(*x*),*g*(*x*)} and*h*(*x*) =*f*(*x*) +*g*(*x*). - If
*f*and*g*are convex functions and if*g*is increasing, then*h*(*x*) =*g*(*f*(*x*)) is convex. - Convexity is invariant under affine maps: that is, if
*f*(*x*) is convex with , then so is*g*(*y*) =*f*(*A**y*+*b*), where - If
*f*(*x*,*y*) is convex in (*x*,*y*) and*C*is a convex nonempty set, then is convex in*x*, provided for some*x*.

## [edit] Examples

- The function
*f*(*x*) =*x*^{2}has at all points, so*f*is a (strictly) convex function. - The absolute value function
*f*(*x*) = |*x*| is convex, even though it does not have a derivative at the point*x*= 0. - The function
*f*(*x*) = |*x*|^{p}for 1 ≤*p*is convex. - The exponential function
*f*(*x*) =*e*^{x}is convex. More generally, the function*g*(*x*) =*e*^{f(x)}is logarithmically convex if f is a convex function. - The function
*f*with domain [0,1] defined by*f*(0)=*f*(1)=1,*f*(*x*)=0 for 0<*x*<1 is convex; it is continuous on the open interval (0,1), but not continuous at 0 and 1. - The function
*x*^{3}has second derivative 6*x*; thus it is convex on the set where*x*≥ 0 and concave on the set where*x*≤ 0. - Every linear transformation taking values in is convex but not strictly convex, since if
*f*is linear, then*f*(*a*+*b*) =*f*(*a*) +*f*(*b*). This statement also holds if we replace "convex" by "concave". - Every affine function taking values in , i.e., each function of the form
*f*(*x*) =*a*^{T}*x*+*b*, is simultaneously convex and concave. - Every norm is a convex function, by the triangle inequality.
- If
*f*is convex, the*perspective function**g*(*x*,*t*) =*t**f*(*x*/*t*) is convex for*t*> 0. - Examples of functions that are monotonically increasing but not convex include and
*g*(*x*) = log(*x*). - Examples of functions that are convex but not monotonically increasing include
*h*(*x*) =*x*^{2}and*k*(*x*) = −*x*. - The function
*f*(*x*) = 1/*x*^{2}, with*f*(0)=+∞, is convex on the interval (0,+∞) and convex on the interval (-∞,0), but not convex on the interval (-∞,+∞), because of the singularity at*x*= 0.

## [edit] See also

- Convex optimization
- Geodesic convexity
- Kachurovskii's theorem, which relates convexity to monotonicity of the derivative
- Logarithmically convex function
- Quasiconvex function
- Subderivative of a convex function
- Jensen's inequality
- Hermite–Hadamard inequality

## [edit] References

- Moon, Todd. "Tutorial: Convexity and Jensen's inequality". http://www.neng.usu.edu/classes/ece/7680/lecture2/node5.html. Retrieved on 2008-09-04.
- Rockafellar, R. T. (1970).
*Convex analysis*. Princeton: Princeton University Press. - Luenberger, David (1984).
*Linear and Nonlinear Programming*. Addison-Wesley. - Luenberger, David (1969).
*Optimization by Vector Space Methods*. Wiley & Sons. - Bertsekas, Dimitri (2003).
*Convex Analysis and Optimization*. Athena Scientific. - Thomson, Brian (1994).
*Symmetric Properties of Real Functions*. CRC Press. - Donoghue, William F. (1969).
*Distributions and Fourier Transforms*. Academic Press.

- Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
- Krasnosel'skii M.A., Rutickii Ya.B. (1961).
*Convex Functions and Orlicz Spaces*. Groningen: P.Noordhoff Ltd. - Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.

## [edit] External links

- Stephen Boyd and Lieven Vandenberghe, Convex Optimization (PDF)
- Jon Dattorro, Convex Optimization & Euclidean Distance Geometry (book pdf)