
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
real-valued function is called convex if the
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
between any two distinct points on the
graph of the function lies above or on 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 geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
.
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
.
A twice-
differentiable function of a single variable is convex
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
its
second derivative
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
is nonnegative on its entire
domain. Well-known examples of convex functions of a single variable include a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
(where
is a
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
), a
quadratic function
In mathematics, a quadratic function of a single variable (mathematics), variable is a function (mathematics), function of the form
:f(x)=ax^2+bx+c,\quad a \ne 0,
where is its variable, and , , and are coefficients. The mathematical expression, e ...
(
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
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
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
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
. In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, a convex function applied to the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
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
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces.
The numbers an ...
.
Definition
Let
be a
convex subset of a real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 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 be above or just meeting the graph.
- 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 that are valued in the
extended real number line 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
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 , 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''. 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.
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
* Suppose
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.
* A convex function
of one real variable defined on some
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
is
continuous on
. Moreover,
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
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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:
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
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
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
.
* If
is a convex function of one real variable, and
, then
is
superadditive on the
positive reals, that is
for positive real numbers
and
.
* A function
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. 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.
* A function
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s is
positive semidefinite on the interior of the convex set.
* For a convex function
f, the
sublevel sets
\ and
\ with
a \in \R 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
f is a convex set:
\,f - convex.
* Any
local minimum of a convex function is also a
global minimum. A convex function will have at most one global minimum.
*
Jensen's inequality applies to every convex function
f. If
X is a random variable taking values in the domain of
f, then
\operatorname(f(X)) \geq f(\operatorname(X)), where
\operatorname denotes the
mathematical expectation. Indeed, convex functions are exactly those that satisfies the hypothesis of
Jensen's inequality.
* A first-order
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar (mathematics), scalar, then the function's value is multiplied by some p ...
of two positive variables
x and
y, (that is, a function satisfying
f(a x, a y) = a f(x, y) for all positive real
a, x, y > 0) that is convex in one variable must be convex in the other variable.
Operations that preserve convexity
*
-f is concave if and only if
f is convex.
* If
r is any real number then
r + f is convex if and only if
f is convex.
* Nonnegative weighted sums:
**if
w_1, \ldots, w_n \geq 0 and
f_1, \ldots, f_n are all convex, then so is
w_1 f_1 + \cdots + w_n f_n. 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
g(x) = \sup\nolimits_ f_i(x) is convex. The domain of
g(x) is the collection of points where the expression is finite. Important special cases:
**If
f_1, \ldots, f_n are convex functions then so is
g(x) = \max \left\.
**
Danskin's theorem: If
f(x,y) is convex in
x then
g(x) = \sup\nolimits_ f(x,y) is convex in
x even if
C is not a convex set.
* Composition:
**If
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
e^ because
e^x is convex and monotonically increasing.
**If
f is concave and
g is convex and non-increasing over a univariate domain, then
h(x) = g(f(x)) is convex.
**Convexity is invariant under affine maps: that is, if
f is convex with domain
D_f \subseteq \mathbf^m, then so is
g(x) = f(Ax+b), where
A \in \mathbf^, b \in \mathbf^m with domain
D_g \subseteq \mathbf^n.
* Minimization: If
f(x,y) is convex in
(x,y) then
g(x) = \inf\nolimits_ f(x,y) is convex in
x, provided that
C is a convex set and that
g(x) \neq -\infty.
* If
f is convex, then its perspective
g(x, t) = t f \left(\tfrac \right) with domain
\left\ is convex.
* Let
X be a vector space.
f : X \to \mathbf is convex and satisfies
f(0) \leq 0 if and only if
f(ax+by) \leq a f(x) + bf(y) for any
x, y \in X and any non-negative real numbers
a, b that satisfy
a + b \leq 1.
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. 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) \ge 0 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) \ge m > 0 for all
x.
For example, let
f be strictly convex, and suppose there is a sequence of points
(x_n) such that
f''(x_n) = \tfrac. Even though
f''(x_n) > 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:
(\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 \cdot, \cdot\rangle is any
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
, and
\, \cdot\, is the corresponding
norm. Some authors, such as
refer to functions satisfying this inequality as
elliptic functions.
An equivalent condition is the following:
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
[ for a strongly convex function, with parameter m, is that, for all x, y in the domain and t \in ,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 \to 0, 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 \nabla^2 f(x) \succeq mI for all x in the domain, where I is the identity and \nabla^2f is the Hessian matrix, and the inequality \succeq means that \nabla^2 f(x) - mI is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of \nabla^2 f(x) be at least m for all x. If the domain is just the real line, then \nabla^2 f(x) is just the second derivative f''(x), so the condition becomes f''(x) \ge m. If m = 0 then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that f''(x) \ge 0), 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 \nabla^2 f(x) implies that it is strongly convex. Using Taylor's Theorem there exists
z \in \
such that
f(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 function
x\mapsto f(x) -\frac m 2 \, x\, ^2
is convex.
A twice continuously differentiable function f on a compact domain X that satisfies f''(x)>0 for all x\in X 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:
* For every real number ''r'', the level set is compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
.
* The function ''f'' has a unique global minimum on ''Rn''.
Uniformly convex functions
A uniformly convex function, with modulus \phi, is a function f that, for all x, y in the domain and t \in , 1 satisfies
f(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 \alpha^2 we recover the definition of strong convexity.
It is worth noting that some authors require the modulus \phi to be an increasing function, but this condition is not required by all authors.
Examples
Functions of one variable
* The function f(x)=x^2 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.
* The function f(x)=x^4 has f''(x)=12x^2\ge 0, 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.
* The absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function f(x)=, x, is convex (as reflected in the triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
), even though it does not have a derivative at the point x = 0. It is not strictly convex.
* The function f(x)=, x, ^p for p \ge 1 is convex.
* The exponential function f(x)=e^x is convex. It is also strictly convex, since f''(x)=e^x >0 , but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function g(x) = e^ is logarithmically convex if f is a convex function. The term "superconvex" is sometimes used instead.
* The function f with domain ,1defined 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.
* The function x^3 has second derivative 6 x; thus it is convex on the set where x \geq 0 and concave on the set where x \leq 0.
* Examples of functions that are monotonically increasing but not convex include f(x)=\sqrt and g(x)=\log x.
* Examples of functions that are convex but not monotonically increasing include h(x)= x^2 and k(x)=-x.
* The function f(x) = \tfrac has f''(x)=\tfrac 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).
* The function f(x)=\tfrac 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
* LogSumExp function, also called softmax function, is a convex function.
*The function -\log\det(X) on the domain of positive-definite matrices is convex.[
* Every real-valued linear transformation is convex but not strictly convex, since if f is linear, then f(a + b) = f(a) + f(b). This statement also holds if we replace "convex" by "concave".
* Every real-valued ]affine function
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''wikt:affine, affinis'', "connected with") is a geometric transformation that preserves line (geometry), lines and parallel (geometry), parallelism, but not necessarily ...
, that is, each function of the form f(x) = a^T x + b, is simultaneously convex and concave.
* Every norm is a convex function, by the triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
and positive homogeneity.
* The spectral radius
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma ...
of a nonnegative matrix is a convex function of its diagonal elements.[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.
See also
* Concave function
* Convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theor ...
* Convex conjugate
* Convex curve
* Convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems ...
* Geodesic convexity
* Hahn–Banach theorem
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
* Hermite–Hadamard inequality
* Invex function
* Jensen's inequality
* K-convex function
* Kachurovskii's theorem, which relates convexity to monotonicity of the derivative
* Karamata's inequality
* Logarithmically convex function
* Pseudoconvex function
* Quasiconvex function
* Subderivative of a convex function
Notes
References
*
* Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
*
* Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
*
*
*
*
*
*
*
External links
*
*
{{Authority control
Convex analysis
Generalized convexity
Types of functions