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, or concave up, if for any two points x and y in its domain C and any t in [0,1], we have

$f(tx+(1-t)y)\leq t f(x)+(1-t)f(y).$

Convex function on an interval.

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.

A function is called strictly convex if

$f(tx+(1-t)y) < t f(x)+(1-t)f(y)\,$

for any t in (0,1) and $x \neq y.$

A function $f$ is said to be concave if $- f$ is convex.

1 Properties
2 Convex function calculus
3 Examples
4 See also
5 References
6 External links

Properties

A function (in blue) is convex if and only if the region above its graph (in green) is a convex set.

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

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

for all x and y in C. This condition is only slightly weaker than convexity. For example, a real valued measurable function that is midpoint convex will be convex. 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) = x⁴ is f "(x) = 12 x², which is zero for x = 0, but x⁴ 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 $E f(X) \geq f(EX).$ (Here $E$ denotes the mathematical expectation.)

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 $x\in\mathbb{R}^n$ , then so is $g (y) = f (A y + b)$ , where $A\in\mathbb{R}^{n \times m},\; b\in\mathbb{R}^m.$
If $f (x, y)$ is convex in $(x, y)$ and $C$ is a convex nonempty set, then $g(x) = \inf_{y\in C} f(x,y)$ is convex in $x,$ provided $g(x) > -\infty$ for some $x .$

Examples

The function $f (x) = x 2$ has $f''(x) = 2 > 0$ 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 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 x³ has second derivative 6x; thus it is convex on the set where x ≥ 0 and concave on the set where x ≤ 0.
Every linear transformation taking values in $\mathbb{R}$ 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 $\mathbb{R}$ , 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 $f(x) = \sqrt x$ 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/x², 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.

References

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.

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.

External links

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