Convex function explained

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above 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. In simple terms, a convex function graph is shaped like a cup

\cup

(or a straight line like a linear function), while a concave function's graph is shaped like a cap

\cap

.

f(x)=cx

(where

c

is a real number), a quadratic function

cx2

(

c

as a nonnegative real number) and an exponential function

cex

(

c

as a nonnegative real number).

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

Let

X

be a convex subset of a real vector space and let

f:X\to\R

be a function.

Then

f

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

  1. For all

    0\leqt\leq1

    and all

    x1,x2\inX

    :f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)The right hand side represents the straight line between

    \left(x1,f\left(x1\right)\right)

    and

    \left(x2,f\left(x2\right)\right)

    in the graph of

    f

    as a function of

    t;

    increasing

    t

    from

    0

    to

    1

    or decreasing

    t

    from

    1

    to

    0

    sweeps this line. Similarly, the argument of the function

    f

    in the left hand side represents the straight line between

    x1

    and

    x2

    in

    X

    or the

    x

    -axis of the graph of

    f.

    So, this condition requires that the straight line between any pair of points on the curve of

    f

    to be above or just meets the graph.[2]
  2. For all

    0<t<1

    and all

    x1,x2\inX

    such that

    x1x2

    :f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)

    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,

    \left(x1,f\left(x1\right)\right)

    and

    \left(x2,f\left(x2\right)\right)

    ) between the straight line passing through a pair of points on the curve of

    f

    (the straight line is represented by the right hand side of this condition) and the curve of

    f;

    the first condition includes the intersection points as it becomes

    f\left(x1\right)\leqf\left(x1\right)

    or

    f\left(x2\right)\leqf\left(x2\right)

    at

    t=0

    or

    1,

    or

    x1=x2.

    In fact, the intersection points do not need to be considered in a condition of convex using f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right) because

    f\left(x1\right)\leqf\left(x1\right)

    and

    f\left(x2\right)\leqf\left(x2\right)

    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

\R

is also the statement used to define that are valued in the extended real number line

[-infty,infty]=\R\cup\{\pminfty\},

where such a function

f

is allowed to take

\pminfty

as a value. The first statement is not used because it permits

t

to take

0

or

1

as a value, in which case, if

f\left(x1\right)=\pminfty

or

f\left(x2\right)=\pminfty,

respectively, then

tf\left(x1\right)+(1-t)f\left(x2\right)

would be undefined (because the multiplications

0infty

and

0(-infty)

are undefined). The sum

-infty+infty

is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of

-infty

and

+infty

as a value.

The second statement can also be modified to get the definition of, where the latter is obtained by replacing

\leq

with the strict inequality

<.

Explicitly, the map

f

is called if and only if for all real

0<t<1

and all

x1,x2\inX

such that

x1x2

:f\left(t x_1 + (1-t) x_2\right) < t f\left(x_1\right) + (1-t) f\left(x_2\right)

A strictly convex function

f

is a function that the straight line between any pair of points on the curve

f

is above the curve

f

except for the intersection points between the straight line and the curve. An example of a function which is convex but not strictly convex is

f(x,y)=x2+y

. This function is not strictly convex because any two points sharing an x coordinate will have a straight line between them, while any two points NOT sharing an x coordinate will have a greater value of the function than the points between them.

The function

f

is said to be (resp. ) if

-f

(

f

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

\cup

. As an example, Jensen's inequality refers to an inequality involving a convex or convex-(down), function.[6]

Properties

Many properties of convex functions have the same simple formulation for functions of many variables 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

f

is a function of one real variable defined on an interval, and let R(x_1, x_2) = \frac (note that

R(x1,x2)

is the slope of the purple line in the first drawing; the function

R

is symmetric in

(x1,x2),

means that

R

does not change by exchanging

x1

and

x2

).

f

is convex if and only if

R(x1,x2)

is monotonically non-decreasing in

x1,

for every fixed

x2

(or vice versa). This characterization of convexity is quite useful to prove the following results.

f

of one real variable defined on some open interval

C

is continuous on

C.

f

admits left and right derivatives, and these are monotonically non-decreasing. In addition, the left derivative is left-continuous and the right-derivative is right-continuous. As a consequence,

f

is differentiable at all but at most countably many points, the set on which

f

is not differentiable can however still be dense. If

C

is closed, then

f

may fail to be continuous at the endpoints of

C

(an example is shown in the examples section).

x

and

y

in the interval.

f(x)=x4

is

f''(x)=12x2

, which is zero for

x=0,

but

x4

is strictly convex.

f''

is non-negative on an interval

X

then

f'

is monotonically non-decreasing on

X

while its converse is not true, for example,

f'

is monotonically non-decreasing on

X

while its derivative

f''

is not defined at some points on

X

.

f

is a convex function of one real variable, and

f(0)\le0

, then

f

is superadditive on the positive reals, that is

f(a+b)\geqf(a)+f(b)

for positive real numbers

a

and

b

.

f

is midpoint convex on an interval

C

if for all

x1,x2\inC

f\!\left(\frac\right) \leq \frac. This condition is only slightly weaker than convexity. For example, a real-valued Lebesgue measurable function that is midpoint-convex is convex: this is a theorem of Sierpiński.[8] In particular, a continuous function that is midpoint convex will be convex.

Functions of several variables

f(x,y)=xy

is marginally linear, and thus marginally convex, in each variable, but not (jointly) convex.

f:X\to[-infty,infty]

valued in the extended real numbers

[-infty,infty]=\R\cup\{\pminfty\}

is convex if and only if its epigraph \ is a convex set.

f

defined on a convex domain is convex if and only if

f(x)\geqf(y)+\nablaf(y)T(x-y)

holds for all

x,y

in the domain.

f,

the sublevel sets

\{x:f(x)<a\}

and

\{x:f(x)\leqa\}

with

a\in\R

are convex sets. A function that satisfies this property is called a and may fail to be a convex function.

f

is a convex set:

{\operatorname{argmin}}f

- convex.

f

. If

X

is a random variable taking values in the domain of

f,

then

\operatorname{E}(f(X))\geqf(\operatorname{E}(X)),

where

\operatorname{E}

denotes the mathematical expectation. Indeed, convex functions are exactly those that satisfies the hypothesis of Jensen's inequality.

x

and

y,

(that is, a function satisfying

f(ax,ay)=af(x,y)

for all positive real

a,x,y>0

) that is convex in one variable must be convex in the other variable.[10]

Operations that preserve convexity

-f

is concave if and only if

f

is convex.

r

is any real number then

r+f

is convex if and only if

f

is convex.

w1,\ldots,wn\geq0

and

f1,\ldots,fn

are all convex, then so is

w1f1++wnfn.

In particular, the sum of two convex functions is convex.

\{fi\}i

be a collection of convex functions. Then

g(x)=\sup\nolimitsifi(x)

is convex. The domain of

g(x)

is the collection of points where the expression is finite. Important special cases:

f1,\ldots,fn

are convex functions then so is

g(x)=max\left\{f1(x),\ldots,fn(x)\right\}.

If

f(x,y)

is convex in

x

then

g(x)=\sup\nolimitsy\inf(x,y)

is convex in

x

even if

C

is not a convex set.

f

and

g

are convex functions and

g

is non-decreasing over a univariate domain, then

h(x)=g(f(x))

is convex. For example, if

f

is convex, then so is

ef(x)

because

ex

is convex and monotonically increasing.

f

is concave and

g

is convex and non-increasing over a univariate domain, then

h(x)=g(f(x))

is convex.

f

is convex with domain

Df\subseteqRm

, then so is

g(x)=f(Ax+b)

, where

A\inRm,b\inRm

with domain

Dg\subseteqRn.

f(x,y)

is convex in

(x,y)

then

g(x)=inf\nolimitsy\inf(x,y)

is convex in

x,

provided that

C

is a convex set and that

g(x)-infty.

f

is convex, then its perspective

g(x,t)=tf\left(\tfrac{x}{t}\right)

with domain

\left\{(x,t):\tfrac{x}{t}\in\operatorname{Dom}(f),t>0\right\}

is convex.

X

be a vector space.

f:X\toR

is convex and satisfies

f(0)\leq0

if and only if

f(ax+by)\leqaf(x)+bf(y)

for any

x,y\inX

and any non-negative real numbers

a,b

that satisfy

a+b\leq1.

Strongly convex functions

The concept of strong convexity extends and parametrizes the notion of strict convexity. Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function.[11] A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function

f

is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:

f

convex if and only if

f''(x)\ge0

for all

x.

f

strictly convex if

f''(x)>0

for all

x

(note: this is sufficient, but not necessary).

f

strongly convex if and only if

f''(x)\gem>0

for all

x.

For example, let

f

be strictly convex, and suppose there is a sequence of points

(xn)

such that

f''(xn)=\tfrac{1}{n}

. Even though

f''(xn)>0

, the function is not strongly convex because

f''(x)

will become arbitrarily small.

More generally, a differentiable function

f

is called strongly convex with parameter

m>0

if the following inequality holds for all points

x,y

in its domain:[12] (\nabla f(x) - \nabla f(y))^T (x-y) \ge m \|x-y\|_2^2 or, more generally,\langle \nabla f(x) - \nabla f(y), x-y \rangle \ge m \|x-y\|^2 where

\langle,\rangle

is any inner product, and

\|\|

is the corresponding norm. Some authors, such as [13] refer to functions satisfying this inequality as elliptic functions.

An equivalent condition is the following:[14] f(y) \ge f(x) + \nabla f(x)^T (y-x) + \frac \|y-x\|_2^2

It is not necessary for a function to be differentiable in order to be strongly convex. A third definition[14] for a strongly convex function, with parameter

m,

is that, for all

x,y

in the domain and

t\in[0,1],

f(tx+(1-t)y) \le t f(x)+(1-t)f(y) - \frac m t(1-t) \|x-y\|_2^2

Notice that this definition approaches the definition for strict convexity as

m\to0,

and is identical to the definition of a convex function when

m=0.

Despite this, functions exist that are strictly convex but are not strongly convex for any

m>0

(see example below).

If the function

f

is twice continuously differentiable, then it is strongly convex with parameter

m

if and only if

\nabla2f(x)\succeqmI

for all

x

in the domain, where

I

is the identity and

\nabla2f

is the Hessian matrix, and the inequality

\succeq

means that

\nabla2f(x)-mI

is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of

\nabla2f(x)

be at least

m

for all

x.

If the domain is just the real line, then

\nabla2f(x)

is just the second derivative

f''(x),

so the condition becomes

f''(x)\gem

. If

m=0

then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that

f''(x)\ge0

), 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

\nabla2f(x)

implies that it is strongly convex. Using Taylor's Theorem there existsz \in \such thatf(y) = f(x) + \nabla f(x)^T (y-x) + \frac (y-x)^T \nabla^2f(z) (y-x)Then(y-x)^T \nabla^2f(z) (y-x) \ge m (y-x)^T(y-x) by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.

A function

f

is strongly convex with parameter m if and only if the functionx\mapsto f(x) -\frac m 2 \|x\|^2is convex.

A twice continuously differentiable function

f

on a compact domain

X

that satisfies

f''(x)>0

for all

x\inX

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.

Properties of strongly-convex functions

If f is a strongly-convex function with parameter m, then:[15]

Uniformly convex functions

A uniformly convex function,[16] [17] with modulus

\phi

, is a function

f

that, for all

x,y

in the domain and

t\in[0,1],

satisfiesf(tx+(1-t)y) \le t f(x)+(1-t)f(y) - t(1-t) \phi(\|x-y\|)where

\phi

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

\phi(\alpha)=\tfrac{m}{2}\alpha2

we recover the definition of strong convexity.

It is worth noting that some authors require the modulus

\phi

to be an increasing function,[18] but this condition is not required by all authors.[19]

Examples

Functions of one variable

f(x)=x2

has

f''(x)=2>0

, so is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.

f(x)=x4

has

f''(x)=12x2\ge0

, so is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex.

f(x)=|x|

is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point

x=0.

It is not strictly convex.

f(x)=|x|p

for

p\ge1

is convex.

f(x)=ex

is convex. It is also strictly convex, since

f''(x)=ex>0

, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function

g(x)=ef(x)

is logarithmically convex if

f

is a convex function. The term "superconvex" is sometimes used instead.[20]

f

with domain [0,1] defined by

f(0)=f(1)=1,f(x)=0

for

0<x<1

is convex; it is continuous on the open interval

(0,1),

but not continuous at 0 and 1.

x3

has second derivative

6x

; thus it is convex on the set where

x\geq0

and concave on the set where

x\leq0.

f(x)=\sqrt{x}

and

g(x)=logx

.

h(x)=x2

and

k(x)=-x

.

f(x)=\tfrac{1}{x}

has

f''(x)=\tfrac{2}{x3}

which is greater than 0 if

x>0

so

f(x)

is convex on the interval

(0,infty)

. It is concave on the interval

(-infty,0)

.

f(x)=\tfrac{1}{x2}

with

f(0)=infty

, is convex on the interval

(0,infty)

and convex on the interval

(-infty,0)

, but not convex on the interval

(-infty,infty)

, because of the singularity at

x=0.

Functions of n variables

-log\det(X)

on the domain of positive-definite matrices is convex.

f

is linear, then

f(a+b)=f(a)+f(b)

. This statement also holds if we replace "convex" by "concave".

f(x)=aTx+b,

is simultaneously convex and concave.

See also

References

Notes and References

  1. Web site: Lecture Notes 2. www.stat.cmu.edu. 3 March 2017.
  2. Web site: Concave Upward and Downward. live. https://web.archive.org/web/20131218034748/http://www.mathsisfun.com:80/calculus/concave-up-down-convex.html . 2013-12-18 .
  3. Book: Stewart, James. Calculus. Cengage Learning. 2015. 978-1305266643. 8th. 223–224.
  4. Book: W. Hamming. Richard. Methods of Mathematics Applied to Calculus, Probability, and Statistics. Courier Corporation. 2012. 978-0-486-13887-9. illustrated. 227. Extract of page 227
  5. Book: Uvarov. Vasiliĭ Borisovich. Mathematical Analysis. Mir Publishers. 1988. 978-5-03-000500-3. 126-127.
  6. Book: The Probability Companion for Engineering and Computer Science . illustrated . Adam . Prügel-Bennett . Cambridge University Press . 2020 . 978-1-108-48053-6 . 160 . Extract of page 160
  7. Book: Convex Optimization. Stephen P.. Boyd . Lieven. Vandenberghe . 2004 . Cambridge University Press. 978-0-521-83378-3. pdf . October 15, 2011.
  8. Book: Donoghue, William F.. Distributions and Fourier Transforms. 1969. Academic Press . 9780122206504 . August 29, 2012. 12.
  9. Web site: If f is strictly convex in a convex set, show it has no more than 1 minimum . Math StackExchange . 21 Mar 2013 . 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. Web site: Strong convexity · Xingyu Zhou's blog . 2023-09-27 . xingyuzhou.org.
  12. Book: 72. Convex Analysis and Optimization. limited. Dimitri Bertsekas. Contributors: Angelia Nedic and Asuman E. Ozdaglar. Athena Scientific. 2003. 9781886529458.
  13. Book: Introduction to numerical linear algebra and optimisation. Philippe G. Ciarlet. Cambridge University Press . 1989 . 9780521339841.
  14. Book: 63–64. Introductory Lectures on Convex Optimization: A Basic Course. limited. Yurii Nesterov. Kluwer Academic Publishers. 2004. 9781402075537.
  15. Web site: Nemirovsky and Ben-Tal . 2023 . Optimization III: Convex Optimization .
  16. Book: Convex Analysis in General Vector Spaces. C. Zalinescu. World Scientific. 2002. 9812380671.
  17. Book: 144. Convex Analysis and Monotone Operator Theory in Hilbert Spaces . limited. H. Bauschke and P. L. Combettes . Springer . 2011 . 978-1-4419-9467-7.
  18. Book: 144. Convex Analysis and Monotone Operator Theory in Hilbert Spaces . limited. H. Bauschke and P. L. Combettes . Springer . 2011 . 978-1-4419-9467-7.
  19. Book: Convex Analysis in General Vector Spaces. C. Zalinescu. World Scientific. 2002. 9812380671.
  20. Kingman . J. F. C. . 10.1093/qmath/12.1.283 . A Convexity Property of Positive Matrices . The Quarterly Journal of Mathematics . 12 . 283–284 . 1961 .
  21. 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.