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Concave function

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Negative of a convex function

Inmathematics, aconcave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which thehypograph is convex. The class of concave functions is in a sense the opposite of the class ofconvex functions. A concave function is alsosynonymously calledconcave downwards,concave down,convex upwards,convex cap, orupper convex.

Definition

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A real-valuedfunction $f {\displaystyle f}$ on aninterval (or, more generally, aconvex set invector space) is said to beconcave if, for any $x {\displaystyle x}$ and $y {\displaystyle y}$ 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 calledstrictly concave if

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

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

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

A function $f {\displaystyle f}$ isquasiconcave if the upper contour sets of the function $S(a)=\{x:f(x)\geq a\}$ are convex sets.^[2]

Properties

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A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is positive

Functions of a single variable

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Adifferentiable functionf is (strictly) concave on aninterval if and only if itsderivative functionf ′ is (strictly)monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing)slope.^[3]^[4]
Points where concavity changes (between concave andconvex) areinflection points.^[5]
Iff is twice-differentiable, thenf is concaveif and only iff ′′ isnon-positive (or, informally, if the "acceleration" is non-positive). Iff ′′ isnegative thenf is strictly concave, but the converse is not true, as shown byf(x) = −x⁴.
Iff is concave and differentiable, then it is bounded above by its first-orderTaylor approximation:^[2] $f(y)\leq f(x)+f'(x)[y-x]$
ALebesgue measurable function on an intervalC is concaveif and only if it is midpoint concave, that is, for anyx andy inC $f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}$
If a functionf is concave, andf(0) ≥ 0, thenf issubadditive on $[0,\infty )$ . Proof:
- Sincef is concave and1 ≥ t ≥ 0, lettingy = 0 we have $f(tx)=f(tx+(1-t)\cdot 0)\geq tf(x)+(1-t)f(0)\geq tf(x).$
- For $a,b\in [0,\infty )$ : $f(a)+f(b)=f\left((a+b){\frac {a}{a+b}}\right)+f\left((a+b){\frac {b}{a+b}}\right)\geq {\frac {a}{a+b}}f(a+b)+{\frac {b}{a+b}}f(a+b)=f(a+b)$

Functions ofn variables

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A functionf is concave over a convex setif and only if the function−f is aconvex function over the set.
The sum of two concave functions is itself concave and so is thepointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form asemifield.
Near a strictlocal 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.
Anylocal maximum of a concave function is also aglobal maximum. Astrictly concave function will have at most one global maximum.

Examples

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The functions $f(x)=-x^{2}$ and $g(x)={\sqrt {x}}$ are concave on their domains, as their second derivatives $f''(x)=-2$ and ${\textstyle g''(x)=-{\frac {1}{4x^{3/2}}}}$ are always negative.
Thelogarithm function $f(x)=\log {x}$ is concave on its domain $(0,\infty )$ , as its derivative ${\frac {1}{x}}$ is a strictly decreasing function.
Anyaffine function $f(x)=ax+b$ is both concave and convex, but neither strictly-concave nor strictly-convex.
Thesine function is concave on the interval $[0,\pi ]$ .
The function $f(B)=\log |B|$ , where $|B|$ is thedeterminant of anonnegative-definite matrixB, is concave.^[6]

Applications

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Rays bending in thecomputation of radiowave attenuation in the atmosphere involve concave functions.
Inexpected utility theory forchoice under uncertainty,cardinal utility functions ofrisk averse decision makers are concave.
Inmicroeconomic theory,production functions are usually assumed to be concave over some or all of their domains, resulting indiminishing returns to input factors.^[7]
InThermodynamics andInformation 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 theconvex envelope of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.^[8]

References

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^Lenhart, S.; Workman, J. T. (2007).Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series. Chapman & Hall/ CRC.ISBN 978-1-58488-640-2.
^^a ^bVarian, Hal R. (1992).Microeconomic analysis (3rd ed.). New York: Norton. p. 489.ISBN 0-393-95735-7.OCLC 24847759.
^Rudin, Walter (1976).Analysis. p. 101.
^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.doi:10.1115/1.3452897.ISSN 0022-2305.
^Hass, Joel (13 March 2017).Thomas' calculus. Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006. (Fourteenth ed.). [United States]. p. 203.ISBN 978-0-13-443898-6.OCLC 965446428.{{cite book}}: CS1 maint: location missing publisher (link)
^Cover, Thomas M.; Thomas, J. A. (1988). "Determinant inequalities via information theory".SIAM Journal on Matrix Analysis and Applications.9 (3):384–392.doi:10.1137/0609033.S2CID 5491763.
^Pemberton, Malcolm; Rau, Nicholas (2015).Mathematics for Economists: An Introductory Textbook. Oxford University Press. pp. 363–364.ISBN 978-1-78499-148-7.
^Callen, Herbert B.; Callen, Herbert B. (1985). "8.1: Intrinsic Stability of Thermodynamic Systems".Thermodynamics and an introduction to thermostatistics (2nd ed.). New York: Wiley. pp. 203–206.ISBN 978-0-471-86256-7.

Further References

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Crouzeix, J.-P. (2008)."Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E (eds.).The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 815–816.doi:10.1057/9780230226203.1375.ISBN 978-0-333-78676-5.
Rao, Singiresu S. (2009).Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779.ISBN 978-0-470-18352-6.

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