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.

1 Definition
2 Properties
3 Examples
4 See also
5 References

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],

$f(tx%2B(1-t)y)\geq t f(x)%2B(1-t)f(y).$

A function is called strictly concave if

$f(tx %2B (1-t)y) > t f(x) %2B (1-t)f(y)\,$

for any t in (0,1) and x ≠ y.

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

Properties

A function f(x) is concave over a convex set if and only if the function −f(x) 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.)

For a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave. Points where concavity changes are inflection points.

If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum.

If f(x) is twice-differentiable, then f(x) is concave if and only if f ′′(x) 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) = -x⁴.

If f is concave and differentiable then

$f(y) \leq f(x) %2B f'(x)[y-x]$ ^[2]

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

$f\left( \frac{x%2By}2 \right) \ge \frac{f(x) %2B f(y)}2$

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

since f is concave, let y = 0, $f(tx) = f(tx%2B(1-t)\cdot 0) \ge t f(x)%2B(1-t)f(0) \ge t f(x)$
$f(a) %2B f(b) = f \left((a%2Bb) \frac{a}{a%2Bb} \right) %2B f \left((a%2Bb) \frac{b}{a%2Bb} \right) \ge \frac{a}{a%2Bb} f(a%2Bb) %2B \frac{b}{a%2Bb} f(a%2Bb) = f(a%2Bb)$

Examples

The functions $f(x)=-x^2$ and $f(x)=\sqrt{x}$ are concave, as the second derivative is always negative.
Any linear function $f(x)=ax%2Bb$ is both concave and convex.
The function $f(x)=\sin(x)$ is concave on the interval $[0, \pi]$ .
The function $\log |B|$ , where $|B|$ is the determinant of a nonnegative-definite matrix B, is concave.^[3]
Practical example: rays bending in Computation of radiowave attenuation in the atmosphere.

References

^ Varian, Hal A. (1992) Microeconomic Analysis. Third Edition. W.W. Norton and Company. p. 496
^ Varian, Hal A. (1992) Microeconomic Analysis. Third Edition. W.W. Norton and Company. p. 489
^ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM journal on matrix analysis and applications 9 (3): 384–392.

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. http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008.
Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 0470183527.

Concave function

Contents

Definition

Properties

Examples

See also

References