concave function

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 $\backslash alpha\; \backslash in\; [0,1]$,{{cite book |last1=Lenhart |first1=S. |last2=Workman |first2=J. T. |title=Optimal Control Applied to Biological Models |publisher=Chapman & Hall/ CRC |series=Mathematical and Computational Biology Series |year=2007 |isbn=978-1-58488-640-2 }}

:$f((1-\backslash alpha\; )x+\backslash alpha\; y)\backslash geq\; (1-\backslash alpha\; )\; f(x)+\backslash alpha\; f(y)$

A function is called strictly concave if

:$f((1-\backslash alpha\; )x+\backslash alpha\; y)\; >\; (1-\backslash alpha\; )\; f(x)+\backslash alpha\; f(y)$

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

For a function $f:\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}$, 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 $(y,\; f(y))$.

Image:ConcaveDef.png

A function $f$ is quasiconcave if the upper contour sets of the function $S(a)=\backslash \{x:\; f(x)\backslash geq\; a\backslash \}$ are convex sets.{{Cite book|last=Varian, Hal R.|url=https://www.worldcat.org/oclc/24847759|title=Microeconomic analysis|date=1992|publisher=Norton|isbn=0-393-95735-7|edition=3rd|location=New York|pages=489|oclc=24847759}}

File:cubic_graph_special_points_repeated.svg

1. A differentiable function {{mvar|f}} is (strictly) concave on an interval if and only if its derivative function {{mvar|f ′}} is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.{{Cite book| last=Rudin| first=Walter| title=Analysis| year=1976| pages= 101}}{{Cite journal |last1=Gradshteyn|first1=I. S.| last2=Ryzhik|first2=I. M.| last3=Hays|first3=D. F.| date=1976-07-01| title=Table of Integrals, Series, and Products| journal=Journal of Lubrication Technology| volume=98|issue=3|pages=479| doi=10.1115/1.3452897|issn=0022-2305 |doi-access=free}}
2. Points where concavity changes (between concave and convex) are inflection points.{{Cite book|last=Hass, Joel |url=https://www.worldcat.org/oclc/965446428| title=Thomas' calculus| others=Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006.|date=13 March 2017| isbn=978-0-13-443898-6| edition=Fourteenth| location=[United States]| pages=203| oclc=965446428}}
3. If {{mvar|f}} is twice-differentiable, then {{mvar|f}} is concave if and only if {{mvar|f ′′}} is non-positive (or, informally, if the "acceleration" is non-positive). If {{mvar|f ′′}} is negative then {{mvar|f}} is strictly concave, but the converse is not true, as shown by {{math|1=f(x) = −x⁴}}.
4. If {{mvar|f}} 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 {{math|C}} is concave if and only if it is midpoint concave, that is, for any {{mvar|x}} and {{mvar|y}} in {{math|C}} $$f\backslash left(\; \backslash frac\{x+y\}2\; \backslash right)\; \backslash ge\; \backslash frac\{f(x)\; +\; f(y)\}2$$
6. If a function {{mvar|f}} is concave, and {{math|f(0) ≥ 0}}, then {{mvar|f}} is subadditive on $[0,\backslash infty)$. Proof:
7. * Since {{mvar|f}} is concave and {{math|1 ≥ t ≥ 0}}, letting {{math|1=y = 0}} we have $$f(tx)\; =\; f(tx+(1-t)\backslash cdot\; 0)\; \backslash ge\; t\; f(x)+(1-t)f(0)\; \backslash ge\; t\; f(x)\; .$$
8. * For $a,b\backslash in[0,\backslash infty)$: $$f(a)\; +\; f(b)\; =\; f\; \backslash left((a+b)\; \backslash frac\{a\}\{a+b\}\; \backslash right)\; +\; f\; \backslash left((a+b)\; \backslash frac\{b\}\{a+b\}\; \backslash right)$$

\ge \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)

1. A function {{mvar|f}} is concave over a convex set if and only if the function {{mvar|−f}} 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.

• The functions $f(x)=-x^2$ and $g(x)=\backslash sqrt\{x\}$ are concave on their domains, as their second derivatives $f$(x) = -2 and $g$(x) =-\frac{1}{4 x^{3/2}} are always negative.
• The logarithm function $f(x)\; =\; \backslash log\{x\}$ is concave on its domain $(0,\backslash infty)$, as its derivative $\backslash frac\{1\}\{x\}$ is a strictly decreasing function.
• Any affine 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 $[0,\; \backslash pi]$.
• The function $f(B)\; =\; \backslash log\; |B|$, where $|B|$ is the determinant of a nonnegative-definite matrix B, is concave.{{cite journal |author-link=Thomas M. Cover |first1=Thomas M. |last1=Cover |first2=J. A. |last2=Thomas |s2cid=5491763 |title=Determinant inequalities via information theory| journal=SIAM Journal on Matrix Analysis and Applications| year=1988| volume=9|number=3| pages=384–392| doi=10.1137/0609033}}

• 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.{{cite book |first1=Malcolm |last1=Pemberton |first2=Nicholas |last2=Rau |title=Mathematics for Economists: An Introductory Textbook |publisher=Oxford University Press |year=2015 |isbn=978-1-78499-148-7 |pages=363–364 |url=https://books.google.com/books?id=9j5_DQAAQBAJ&pg=PA363 }}
• 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.{{Cite book |last1=Callen |first1=Herbert B. |title=Thermodynamics and an introduction to thermostatistics |last2=Callen |first2=Herbert B. |date=1985 |publisher=Wiley |isbn=978-0-471-86256-7 |edition=2nd |location=New York |pages=203–206 |chapter=8.1: Intrinsic Stability of Thermodynamic Systems}}

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

{{reflist}}

• {{cite book|last=Crouzeix|first=J.-P.|chapter=Quasi-concavity|title=The New Palgrave Dictionary of Economics|editor-first=Steven N.|editor-last=Durlauf|editor2-first=Lawrence E|editor2-last=Blume|publisher=Palgrave Macmillan|year=2008|edition= Second|pages=815–816|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008|doi=10.1057/9780230226203.1375|isbn=978-0-333-78676-5}}
• {{cite book |title=Engineering Optimization: Theory and Practice|first=Singiresu S.|last=Rao|

publisher=John Wiley and Sons|year=2009|isbn=978-0-470-18352-6|page=779}}

{{Calculus topics}}

{{Convex analysis and variational analysis}}

{{Authority control}}

Category:Convex analysis

Category:Types of functions