Concave function

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.

Definition

A real-valued function $f$ on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any $x$ and $y$ in the interval and for any $t$ in [0,1],

f((1-t)x+(t)y)\geq (1-t) f(x)+(t)f(y).

A function is called strictly concave if

f((1-t)x + (t)y) > (1-t) f(x) + (t)f(y)\,

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

For a function $f : R \to R$ , this definition merely states that for every $z$ between $x$ and $y$ , the point $(z, f (z) )$ on the graph of $f$ is above the straight line joining the points $(x, f (x) )$ and $(y, f (y) )$ .

A function $f$ is quasiconcave if the upper contour sets of the function $S(a)=\{x: f(x)\geq a\}$ are convex sets.^[1]^:496

Properties

A function $f$ is concave over a convex set if and only if the function $-f$ is a convex function over the set.

A differentiable function $f$ is concave on an interval if its derivative function $f'$ is monotonically decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means non-increasing, rather than strictly decreasing, and thus allows zero slopes.)

Points where concavity changes (between concave and convex) are inflection points.

Near a local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.

If $f$ is twice-differentiable, then $f$ is concave if and only if $f''$ is non-positive (or, if the acceleration is non-positive). If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by $f (x) = - x 4$ .

If $f$ is concave and differentiable, then it is bounded above by its first-order Taylor approximation:^[1]^:489

f(y) \leq f(x) + f'(x)[y-x].

A continuous function on $C$ is concave if and only if for any $x$ and $y$ in $C$

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

If a function $f$ is concave, and $f (0) \geq 0$ , then $f$ is subadditive. Proof:

since $f$ is concave, let $y = 0$ , $f(tx) = f(tx+(1-t)\cdot 0) \ge t f(x)+(1-t)f(0) \ge t f(x)$
$f(a) + f(b) = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right) \ge \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)$

Examples

The functions $f(x)=-x^2$ and $g(x)=\sqrt{x}$ are concave on their domains, as their second derivatives $f''(x) = -2$ and $g''(x) = -\frac{1}{4 x^{1.5}}$ are always negative.
The logarithm function $f(x) = \log{x}$ is concave on its domain $(0,\infty)$ , as its derivative $\frac{1}{x}$ is a strictly decreasing function.
Any affine function $f(x)=ax+b$ is both (non-strictly) concave and convex.
The sine function is concave on the interval $[0, \pi]$ .
The function $f(B) = \log |B|$ , where $|B|$ is the determinant of a nonnegative-definite matrix B, is concave.^[2]
Practical example: rays bending in computation of radiowave attenuation in the atmosphere.

References

1 2 Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0393957357.
↑ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM Journal on Matrix Analysis and Applications 9 (3): 384–392. doi:10.1137/0609033.

Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.1375.
Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 0-470-18352-7.

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Concave function

Definition

Properties

Examples

See also

References