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 ≠ 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.

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 it is bounded above by its first-order Taylor approximation:

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

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) ≥ 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 are their second derivatives $f''(x) = -2$ and $g''(x) = -\frac{1}{4 x^{1.5}}$ are always negative.
• 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.
• Practical example: rays bending in computation of radiowave attenuation in the atmosphere.

See also

• Concave polygon
• Convex function
• Jensen's inequality
• Logarithmically concave function
• Quasiconcave function

References

• 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. 
• Varian, Hal R. (1992). Microeconomic Analysis (Third ed.). W.W. Norton and Company.