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

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

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.

A function is called strictly convex if

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

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

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

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

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 $xinmathbb\{R\}^n$, then so is $g(y)\; =\; f(Ay+b)$, where $Ainmathbb\{R\}^\{n\; times\; m\},;\; binmathbb\{R\}^m.$
• If $f(x,y)$ is convex in $(x,y)$ and $C$ is a convex nonempty set, then $g(x)\; =\; inf\_\{yin\; C\}\; f(x,y)$ 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 $g(x,t)\; =\; tf(x/t)$ 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.

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

References

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• 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