Convex function on an interval.
Convex function on an interval.
A function (in black) is convex if and only if the region above its graph (in green) is a convex set.
A function (in black) is convex if and only if the region above its graph (in green) is a convex set.
A graph of the bivariate convex function  x2 + xy + y2.
A graph of the bivariate convex function x2 + xy + y2.

In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function does not lie below the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.[1] Well-known examples of convex functions of a single variable include the quadratic function and the exponential function . In simple terms, a convex function refers to a function whose graph is shaped like a cup , while a concave function's graph is shaped like a cap .

Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic–geometric mean inequality and Hölder's inequality.

Definition

Visualizing a convex function and Jensen's Inequality

Let be a convex subset of a real vector space and let be a function.

Then is called convex if and only if any of the following equivalent conditions hold:

  1. For all and all :
    The right hand side represents the straight line between and in the graph of as a function of increasing from to or decreasing from to sweeps this line. Similarly, the argument of the function in the left hand side represents the straight line between and in or the -axis of the graph of So, this condition requires that the straight line between any pair of points on the curve of to be above or just meets the graph.[2]
  2. For all and all such that :
    The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (for example, and ) between the straight line passing through a pair of points on the curve of (the straight line is represented by the right hand side of this condition) and the curve of the first condition includes the intersection points as it becomes or at or or In fact, the intersection points do not need to be considered in a condition of convex using
    because and are always true (so not useful to be a part of a condition).

The second statement characterizing convex functions that are valued in the real line is also the statement used to define convex functions that are valued in the extended real number line where such a function is allowed to (but is not required to) take as a value. The first statement is not used because it permits to take or as a value, in which case, if or respectively, then would be undefined (because the multiplications and are undefined). The sum is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of and as a value.

The second statement can also be modified to get the definition of strict convexity, where the latter is obtained by replacing with the strict inequality Explicitly, the map is called strictly convex if and only if for all real and all such that :

A strictly convex function is a function that the straight line between any pair of points on the curve is above the curve except for the intersection points between the straight line and the curve.

The function is said to be concave (resp. strictly concave) if ( multiplied by -1) is convex (resp. strictly convex ).

Alternative naming

The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward.[3][4][5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph . As an example, Jensen's inequality refers to an inequality involving a convex or convex-(up), function.[6]

Properties

Many properties of convex functions have the same simple formulation for functions of many variable as for functions of one variable. See below the properties for the case of many variables, as some of them are not listed for functions of one variable.

Functions of one variable

Proof

Since is convex, by using one of the convex function definitions above and letting it follows that for all real

From this it follows that

Functions of several variables

Operations that preserve convexity

Strongly convex functions

The concept of strong convexity extends and parametrizes the notion of strict convexity. A strongly convex function is also strictly convex, but not vice versa.

A differentiable function is called strongly convex with parameter if the following inequality holds for all points in its domain:[11]

or, more generally,
where is any norm. Some authors, such as [12] refer to functions satisfying this inequality as elliptic functions.

An equivalent condition is the following:[13]

It is not necessary for a function to be differentiable in order to be strongly convex. A third definition[13] for a strongly convex function, with parameter is that, for all in the domain and

Notice that this definition approaches the definition for strict convexity as and is identical to the definition of a convex function when Despite this, functions exist that are strictly convex but are not strongly convex for any (see example below).

If the function is twice continuously differentiable, then it is strongly convex with parameter if and only if for all in the domain, where is the identity and is the Hessian matrix, and the inequality means that is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of be at least for all If the domain is just the real line, then is just the second derivative so the condition becomes . If then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that ), which implies the function is convex, and perhaps strictly convex, but not strongly convex.

Assuming still that the function is twice continuously differentiable, one can show that the lower bound of implies that it is strongly convex. Using Taylor's Theorem there exists

such that
Then
by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.

A function is strongly convex with parameter m if and only if the function

is convex.

The distinction between convex, strictly convex, and strongly convex can be subtle at first glance. If is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:

For example, let be strictly convex, and suppose there is a sequence of points such that . Even though , the function is not strongly convex because will become arbitrarily small.

A twice continuously differentiable function on a compact domain that satisfies for all is strongly convex. The proof of this statement follows from the extreme value theorem, which states that a continuous function on a compact set has a maximum and minimum.

Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets.

Uniformly convex functions

A uniformly convex function,[14][15] with modulus , is a function that, for all in the domain and satisfies

where is a function that is non-negative and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking we recover the definition of strong convexity.

It is worth noting that some authors require the modulus to be an increasing function,[15] but this condition is not required by all authors.[14]

Examples

Functions of one variable

Functions of n variables

See also

Notes

  1. ^ "Lecture Notes 2" (PDF). www.stat.cmu.edu. Retrieved 3 March 2017.
  2. ^ "Concave Upward and Downward". Archived from the original on 2013-12-18.
  3. ^ Stewart, James (2015). Calculus (8th ed.). Cengage Learning. pp. 223–224. ISBN 978-1305266643.
  4. ^ W. Hamming, Richard (2012). Methods of Mathematics Applied to Calculus, Probability, and Statistics (illustrated ed.). Courier Corporation. p. 227. ISBN 978-0-486-13887-9. Extract of page 227
  5. ^ Uvarov, Vasiliĭ Borisovich (1988). Mathematical Analysis. Mir Publishers. p. 126-127. ISBN 978-5-03-000500-3.
  6. ^ Prügel-Bennett, Adam (2020). The Probability Companion for Engineering and Computer Science (illustrated ed.). Cambridge University Press. p. 160. ISBN 978-1-108-48053-6. Extract of page 160
  7. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
  8. ^ Donoghue, William F. (1969). Distributions and Fourier Transforms. Academic Press. p. 12. ISBN 9780122206504. Retrieved August 29, 2012.
  9. ^ "If f is strictly convex in a convex set, show it has no more than 1 minimum". Math StackExchange. 21 Mar 2013. Retrieved 14 May 2016.
  10. ^ Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the National Academy of Sciences, 109(10), pp.3705-3710.
  11. ^ Dimitri Bertsekas (2003). Convex Analysis and Optimization. Contributors: Angelia Nedic and Asuman E. Ozdaglar. Athena Scientific. p. 72. ISBN 9781886529458.
  12. ^ Philippe G. Ciarlet (1989). Introduction to numerical linear algebra and optimisation. Cambridge University Press. ISBN 9780521339841.
  13. ^ a b Yurii Nesterov (2004). Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic Publishers. pp. 63–64. ISBN 9781402075537.
  14. ^ a b C. Zalinescu (2002). Convex Analysis in General Vector Spaces. World Scientific. ISBN 9812380671.
  15. ^ a b H. Bauschke and P. L. Combettes (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer. p. 144. ISBN 978-1-4419-9467-7.
  16. ^ Kingman, J. F. C. (1961). "A Convexity Property of Positive Matrices". The Quarterly Journal of Mathematics. 12: 283–284. doi:10.1093/qmath/12.1.283.
  17. ^ Cohen, J.E., 1981. Convexity of the dominant eigenvalue of an essentially nonnegative matrix. Proceedings of the American Mathematical Society, 81(4), pp.657-658.

References