Concave function


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.

Contents

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

A function is called strictly concave if

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

for any t in (0,1) and xy.

For a function f:RR, 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) ). ConcaveDef.png

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) + 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+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) = − x2 and f(x)=\sqrt{x} are concave, as the second derivative is always negative.
  • Any linear function f(x) = ax + b is both concave and convex.
  • The function f(x) = sin(x) is concave on the interval [0,π].
  • The function log  | B | , where | B | is the determinant of a nonnegative-definite matrix B, is concave.[3]
  • Practical application: rays bending in Computation of radiowave attenuation in the atmosphere.

See also

References

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

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