Concave function explained

In mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

Definition

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 \alpha \in [ 0 , 1 ] ,^[1] f(( 1- \alpha )x + \alpha y)\geq ( 1- \alpha ) f(x) + \alpha f(y)

A function is called strictly concave if

f(( 1- \alpha )x + \alpha y) > ( 1- \alpha) f(x) + \alpha f(y)

for any

\alpha \in ( 0 , 1 ) and x ≠ y .

For a function

f: R \to R , this second definition merely states that for every z strictly 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.^[2]

Properties

Functions of a single variable

1. A differentiable function is (strictly) concave on an interval if and only if its derivative function is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.^{[3] [4]}
2. Points where concavity changes (between concave and convex) are inflection points.^[5]
3. If is twice-differentiable, then is concave if and only if is non-positive (or, informally, if the "acceleration" is non-positive). If is negative then is strictly concave, but the converse is not true, as shown by .
4. If is concave and differentiable, then it is bounded above by its first-order Taylor approximation: $$f(y)\; \backslash leq\; f(x)\; +\; f\text{'}(x)[y-x]$$
5. A Lebesgue measurable function on an interval is concave if and only if it is midpoint concave, that is, for any and in $$f\backslash left(\backslash frac2\; \backslash right)\; \backslash ge\; \backslash frac2$$
6. If a function is concave, and, then is subadditive on

[ 0 , inf ty) . Proof:

1. • Since is concave and, letting we have $$f(tx)\; =\; f(tx+(1-t)\backslash cdot\; 0)\; \backslash ge\; t\; f(x)+(1-t)f(0)\; \backslash ge\; t\; f(x)\; .$$
   • For

a,b\in[ 0 , inf ty) : $$f(a)\; +\; f(b)\; =\; f\; \backslash left((a+b)\; \backslash frac\; \backslash right)\; +\; f\; \backslash left((a+b)\; \backslash frac\; \backslash right)\backslash ge\; \backslash frac\; f(a+b)\; +\; \backslash frac\; f(a+b)\; =\; f(a+b)$$

Functions of n variables

1. A function is concave over a convex set if and only if the function is a convex function over the set.
2. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
3. Near a strict 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.
4. Any local maximum of a concave function is also a global maximum. A strictly concave function will have at most one global maximum.

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 are always negative.

• The logarithm function

f(x) = log {x} is concave on its domain ( 0 , inf ty) , as its derivative (1)/(x) is a strictly decreasing function.f(x) = ax + b is both concave and convex, but neither strictly-concave nor strictly-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.^[6]

Applications

• Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions.
• In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave.
• In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors.^[7]
• In Thermodynamics and Information Theory, Entropy is a concave function. In the case of thermodynamic entropy, without phase transition, entropy as a function of extensive variables is strictly concave. If the system can undergo phase transition, and if it is allowed to split into two subsystems of different phase (phase separation, e.g. boiling), the entropy-maximal parameters of the subsystems will result in a combined entropy precisely on the straight line between the two phases. This means that the "Effective Entropy" of a system with phase transition is the convex envelope of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.^[8]

See also

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

Further References

• Book: Crouzeix, J.-P.. Quasi-concavity. The New Palgrave Dictionary of Economics. Steven N.. Durlauf. Lawrence E. Blume. Palgrave Macmillan. 2008. Second. 815–816. http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008. 10.1057/9780230226203.1375. 978-0-333-78676-5.
• Book: Rao, Singiresu S.. Engineering Optimization: Theory and Practice. John Wiley and Sons. 2009. 978-0-470-18352-6. 779.

Notes and References

1. Book: Lenhart . S. . Workman . J. T. . Optimal Control Applied to Biological Models . Chapman & Hall/ CRC . Mathematical and Computational Biology Series . 2007 . 978-1-58488-640-2 .
2. Book: Varian, Hal R.. Microeconomic analysis. 1992. Norton. 0-393-95735-7. 3rd. New York. 489. 24847759.
3. Book: Rudin, Walter. Analysis. 1976. 101.
4. Gradshteyn. I. S.. Ryzhik. I. M.. Hays. D. F.. 1976-07-01. Table of Integrals, Series, and Products. Journal of Lubrication Technology. 98. 3. 479. 10.1115/1.3452897. 0022-2305 . free.
5. Book: Hass, Joel . Thomas' calculus. Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006.. 13 March 2017. 978-0-13-443898-6. Fourteenth. [United States]. 203. 965446428.
6. Thomas M. Cover . Thomas M. . Cover . J. A. . Thomas . 5491763 . Determinant inequalities via information theory. SIAM Journal on Matrix Analysis and Applications. 1988. 9. 3. 384 - 392. 10.1137/0609033.
7. Book: Malcolm . Pemberton . Nicholas . Rau . Mathematics for Economists: An Introductory Textbook . Oxford University Press . 2015 . 978-1-78499-148-7 . 363–364 .
8. Book: Callen . Herbert B. . Thermodynamics and an introduction to thermostatistics . Callen . Herbert B. . 1985 . Wiley . 978-0-471-86256-7 . 2nd . New York . 203–206 . 8.1: Intrinsic Stability of Thermodynamic Systems.