Concave function

Summary

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 a convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

Definition edit

A real-valued function   on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any   and   in the interval and for any  ,[1]

 

A function is called strictly concave if

 

for any   and  .

For a function  , this second definition merely states that for every   strictly between   and  , the point   on the graph of   is above the straight line joining the points   and  .

 

A function   is quasiconcave if the upper contour sets of the function   are convex sets.[2]

Properties edit

Functions of a single variable edit

  1. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ 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 f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the "acceleration" is non-positive). If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by f(x) = −x4.
  4. If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:[2]
     
  5. A Lebesgue measurable function on an interval C is concave if and only if it is midpoint concave, that is, for any x and y in C
     
  6. If a function f is concave, and f(0) ≥ 0, then f is subadditive on  . Proof:
    • Since f is concave and 1 ≥ t ≥ 0, letting y = 0 we have
       
    • For  :
       

Functions of n variables edit

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

Examples edit

  • The functions   and   are concave on their domains, as their second derivatives   and   are always negative.
  • The logarithm function   is concave on its domain  , as its derivative   is a strictly decreasing function.
  • Any affine function   is both concave and convex, but neither strictly-concave nor strictly-convex.
  • The sine function is concave on the interval  .
  • The function  , where   is the determinant of a nonnegative-definite matrix B, is concave.[6]

Applications edit

See also edit

References edit

  1. ^ 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.
  2. ^ a b Varian, Hal R. (1992). Microeconomic analysis (3rd ed.). New York: Norton. p. 489. ISBN 0-393-95735-7. OCLC 24847759.
  3. ^ Rudin, Walter (1976). Analysis. p. 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. doi:10.1115/1.3452897. ISSN 0022-2305.
  5. ^ 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)
  6. ^ 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.
  7. ^ Pemberton, Malcolm; Rau, Nicholas (2015). Mathematics for Economists: An Introductory Textbook. Oxford University Press. pp. 363–364. ISBN 978-1-78499-148-7.

Further References edit

  • 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.