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
(or a straight line like a linear function), while a
concave function's graph is shaped like a cap
.
(where
is a
real number), a
quadratic function
(
as a nonnegative real number) and an
exponential function
(
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
be a
convex subset of a real
vector space and let
be a function.
Then
is called
if and only if any of the following equivalent conditions hold:
- For all
and all
: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
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] - 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,
\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
(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 f\left(x1\right)\leqf\left(x1\right)
or f\left(x2\right)\leqf\left(x2\right)
at
or
or
In fact, the intersection points do not need to be considered in a condition of convex using 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
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
is allowed to take
as a value. The first statement is not used because it permits
to take
or
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
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, where the latter is obtained by replacing
with the strict inequality
Explicitly, the map
is called
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. An example of a function which is convex but not strictly convex is
. 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
is said to be
(resp.
) 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-(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
is a function of one
real variable defined on an interval, and let
(note that
is the slope of the purple line in the first drawing; the function
is
symmetric in
means that
does not change by exchanging
and
).
is convex if and only if
is
monotonically non-decreasing in
for every fixed
(or vice versa). This characterization of convexity is quite useful to prove the following results.
of one real variable defined on some open interval
is
continuous on
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,
is
differentiable at all but at most
countably many points, the set on which
is not differentiable can however still be dense. If
is closed, then
may fail to be continuous at the endpoints of
(an example is shown in the examples section).
- A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable.
- A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its tangents:[7] for all
and
in the interval.
- A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold. For example, the second derivative of
is
, which is zero for
but
is strictly convex.
- This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if
is non-negative on an interval
then
is monotonically non-decreasing on
while its converse is not true, for example,
is monotonically non-decreasing on
while its derivative
is not defined at some points on
.
is a convex function of one real variable, and
, then
is
superadditive on the
positive reals, that is
for positive real numbers
and
.
is midpoint convex on an interval
if for all
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
- A function that is marginally convex in each individual variable is not necessarily (jointly) convex. For example, the function
is
marginally linear, and thus marginally convex, in each variable, but not (jointly) convex.
valued in the
extended real numbers [-infty,infty]=\R\cup\{\pminfty\}
is convex if and only if its
epigraph is a convex set.
- A differentiable function
defined on a convex domain is convex if and only if
f(x)\geqf(y)+\nablaf(y)T ⋅ (x-y)
holds for all
in the domain.
the sublevel sets
and
with
are convex sets. A function that satisfies this property is called a
and may fail to be a convex function.
- Consequently, the set of global minimisers of a convex function
is a convex set:
- convex.
. If
is a random variable taking values in the domain of
then
\operatorname{E}(f(X))\geqf(\operatorname{E}(X)),
where
denotes the
mathematical expectation. Indeed, convex functions are exactly those that satisfies the hypothesis of
Jensen's inequality.
and
(that is, a function satisfying
for all positive real
) that is convex in one variable must be convex in the other variable.
[10] Operations that preserve convexity
is concave if and only if
is convex.
is any real number then
is convex if and only if
is convex.
- Nonnegative weighted sums:
and
are all convex, then so is
In particular, the sum of two convex functions is convex.
- this property extends to infinite sums, integrals and expected values as well (provided that they exist).
- Elementwise maximum: let
be a collection of convex functions. Then
is convex. The domain of
is the collection of points where the expression is finite. Important special cases:
are convex functions then so is
g(x)=max\left\{f1(x),\ldots,fn(x)\right\}.
If
is convex in
then
g(x)=\sup\nolimitsy\inf(x,y)
is convex in
even if
is not a convex set.
and
are convex functions and
is non-decreasing over a univariate domain, then
is convex. For example, if
is convex, then so is
because
is convex and monotonically increasing.
is concave and
is convex and non-increasing over a univariate domain, then
is convex.
- Convexity is invariant under affine maps: that is, if
is convex with domain
, then so is
, where
with domain
is convex in
then
g(x)=inf\nolimitsy\inf(x,y)
is convex in
provided that
is a convex set and that
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.
be a vector space.
is convex and satisfies
if and only if
for any
and any non-negative real numbers
that satisfy
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
is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
convex if and only if
for all
strictly convex if
for all
(note: this is sufficient, but not necessary).
strongly convex if and only if
for all
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.
More generally, a differentiable function
is called strongly convex with parameter
if the following inequality holds for all points
in its domain:
[12] or, more generally,
where
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]
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
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.
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.
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
, 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
\phi(\alpha)=\tfrac{m}{2}\alpha2
we recover the definition of strong convexity.
It is worth noting that some authors require the modulus
to be an increasing function,
[18] but this condition is not required by all authors.
[19] Examples
Functions of one variable
has
, so is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
has
, 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.
is convex (as reflected in the
triangle inequality), even though it does not have a derivative at the point
It is not strictly convex.
for
is convex.
is convex. It is also strictly convex, since
, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function
is
logarithmically convex if
is a convex function. The term "superconvex" is sometimes used instead.
[20]
with domain [0,1] defined by
for
is convex; it is continuous on the open interval
but not continuous at 0 and 1.
has second derivative
; thus it is convex on the set where
and
concave on the set where
and
.
and
.
has
which is greater than 0 if
so
is convex on the interval
. It is concave on the interval
.
with
, is convex on the interval
and convex on the interval
, but not convex on the interval
, because of the singularity at
Functions of n variables
- LogSumExp function, also called softmax function, is a convex function.
- The function
on the domain of
positive-definite matrices is convex.
is linear, then
. This statement also holds if we replace "convex" by "concave".
is simultaneously convex and concave.
See also
References
- Book: Bertsekas
, Dimitri
. Dimitri Bertsekas . Convex Analysis and Optimization . Athena Scientific . 2003.
- Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
- Book: Donoghue
, William F.
. Distributions and Fourier Transforms . Academic Press . 1969.
- Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
- Book: Krasnosel'skii M.A., Rutickii Ya.B. . Convex Functions and Orlicz Spaces . P.Noordhoff Ltd . Groningen . 1961.
- Book: Lauritzen
, Niels
. Undergraduate Convexity . World Scientific Publishing . 2013.
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, David
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, David
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, R. T.
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, Brian
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- Book: Zălinescu, C.. Convex analysis in general vector spaces. World Scientific Publishing Co., Inc. River Edge, NJ. 2002. xx+367. 981-238-067-1. 1921556.
Notes and References
- Web site: Lecture Notes 2. www.stat.cmu.edu. 3 March 2017.
- 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 .
- Book: Stewart, James. Calculus. Cengage Learning. 2015. 978-1305266643. 8th. 223–224.
- 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
- Book: Uvarov. Vasiliĭ Borisovich. Mathematical Analysis. Mir Publishers. 1988. 978-5-03-000500-3. 126-127.
- 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
- Book: Convex Optimization. Stephen P.. Boyd . Lieven. Vandenberghe . 2004 . Cambridge University Press. 978-0-521-83378-3. pdf . October 15, 2011.
- Book: Donoghue, William F.. Distributions and Fourier Transforms. 1969. Academic Press . 9780122206504 . August 29, 2012. 12.
- 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.
- Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the National Academy of Sciences, 109(10), pp.3705-3710.
- Web site: Strong convexity · Xingyu Zhou's blog . 2023-09-27 . xingyuzhou.org.
- Book: 72. Convex Analysis and Optimization. limited. Dimitri Bertsekas. Contributors: Angelia Nedic and Asuman E. Ozdaglar. Athena Scientific. 2003. 9781886529458.
- Book: Introduction to numerical linear algebra and optimisation. Philippe G. Ciarlet. Cambridge University Press . 1989 . 9780521339841.
- Book: 63–64. Introductory Lectures on Convex Optimization: A Basic Course. limited. Yurii Nesterov. Kluwer Academic Publishers. 2004. 9781402075537.
- Web site: Nemirovsky and Ben-Tal . 2023 . Optimization III: Convex Optimization .
- Book: Convex Analysis in General Vector Spaces. C. Zalinescu. World Scientific. 2002. 9812380671.
- 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.
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- 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.