Convex function

Definition

Let X be a convex set in a real vector space and let f : X → R be a function.

f is called convex if:

f is called strictly convex if:

A function f is said to be (strictly) concave if −f is (strictly) convex.

Properties

1. Suppose f is a function of one real variable defined on an interval, and let

R ( x 1 , x 2 ) = f ( x 1 ) − f ( x 2 ) x 1 − x 2

(note that R(x₁, x₂) is the slope of the purple line in the above drawing; note also that the function R is symmetric in (x₁, x₂)). f is convex if and only if R(x₁, x₂) is monotonically non-decreasing in x₁, for any fixed x₂ (or vice versa). This characterization of convexity is quite useful to prove the following results.

2. A convex function f defined on some open interval C is continuous on C and Lipschitz continuous on any closed subinterval. f admits left and right derivatives, and these are monotonically non-decreasing. As a consequence, f is differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C (an example is shown in the examples' section).

3. A function is midpoint convex on an interval C if

∀ x 1 , x 2 ∈ C : f ( x 1 + x 2 2 ) ≤ f ( x 1 ) + f ( x 2 ) 2 .

This condition is only slightly weaker than convexity. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex by Sierpinski Theorem. In particular, a continuous function that is midpoint convex will be convex.

4. 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. For the basic case of a differentiable function from (a subset of) the real numbers to the real numbers, "convex" is equivalent to "increasing at an increasing rate".

5. A differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents:

f ( x ) ≥ f ( y ) + f ′ ( y ) ( x − y )

for all x and y in the interval. In particular, if f′(c) = 0, then c is a global minimum of f(x).

6. 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 f(x) = x⁴ is f ′′(x) = 12x², which is zero for x = 0, but x⁴ is strictly convex.

More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.

7. Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.

8. For a convex function f, the sublevel sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function. A function whose sublevel sets are convex is called a quasiconvex function.

9. Jensen's inequality applies to every convex function f. If X is a random variable taking values in the domain of f, then E(f(X)) ≥ f(E(X)). (Here E denotes the mathematical expectation.)

10. If a function f is convex, and f(0) ≤ 0, then f is superadditive on the positive reals.

Convex function calculus

If f and g are convex functions, then so are m ( x ) = max { f ( x ) , g ( x ) } and h ( x ) = f ( x ) + g ( x ) , i.e. the set of convex functions on a given domain form a semifield.

If f and g are convex functions and g is non-decreasing, then h ( x ) = g ( f ( x ) ) is convex. As an example, if f(x) is convex, then so is e f ( x ) , because e x is convex and monotonically increasing.

If f is concave and g is convex and non-increasing, then h ( x ) = g ( f ( x ) ) is convex.

Convexity is invariant under affine maps: that is, if f is convex with domain D f ⊆ R m , then so is g ( x ) = f ( A x + b ) , where A ∈ R m × n , b ∈ R m with domain D g ⊆ R n .

If f(x, y) is convex in x then g ( x ) = sup y ∈ C f ( x , y ) is convex in x provided g ( x ) > − ∞ for some x, even if C is not convex.

If f(x) is convex, then its perspective g ( x , t ) = t f ( x / t ) (whose domain is { ( x , t ) | x t ∈ Dom ( f ) , t > 0 } ) is convex.

The negative of a convex function is a concave function.

If f ( x ) is a convex real valued function, then f ( x ) = sup n ( a n x + b n ) for a countable collection of real numbers ( a n , b n )

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 f is called strongly convex with parameter m > 0 if the following inequality holds for all points x, y in its domain:

( ∇ f ( x ) − ∇ f ( y ) ) T ( x − y ) ≥ m ∥ x − y ∥ 2 2

or, more generally,

⟨ ∇ f ( x ) − ∇ f ( y ) , ( x − y ) ⟩ ≥ m ∥ x − y ∥ 2

where ∥ ⋅ ∥ is any norm. Some authors, such as refer to functions satisfying this inequality as elliptic functions.

An equivalent condition is the following:

f ( y ) ≥ f ( x ) + ∇ f ( x ) T ( y − x ) + m 2 ∥ 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 ∈ [ 0 , 1 ] ,

f ( t x + ( 1 − t ) y ) ≤ t f ( x ) + ( 1 − t ) f ( y ) − 1 2 m t ( 1 − t ) ∥ x − y ∥ 2 2

Notice that this definition approaches the definition for strict convexity as m → 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 f is strongly convex with parameter m if and only if ∇ 2 f ( x ) ⪰ m I for all x in the domain, where I is the identity and ∇ 2 f is the Hessian matrix, and the inequality ⪰ means that ∇ 2 f ( x ) − m I is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of ∇ 2 f ( x ) be at least m for all x. If the domain is just the real line, then ∇ 2 f ( x ) is just the second derivative f ″ ( x ) , so the condition becomes f ″ ( x ) ≥ 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 ) ≥ 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 ∇ 2 f ( x ) implies that it is strongly convex. Start by using Taylor's Theorem:

f ( y ) = f ( x ) + ∇ f ( x ) T ( y − x ) + 1 2 ( y − x ) T ∇ 2 f ( z ) ( y − x )

for some (unknown) z ∈ { t x + ( 1 − t ) y : t ∈ [ 0 , 1 ] } . Then

( y − x ) T ∇ 2 f ( z ) ( y − x ) ≥ 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 ↦ f ( x ) − m 2 ∥ x ∥ 2 is convex.

The distinction between convex, strictly convex, and strongly convex can be subtle at first glimpse. If 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 ) ≥ 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 ) ≥ m > 0 for all x.

For example, consider a function f that is strictly convex, and suppose there is a sequence of points ( x n ) such that f ″ ( x n ) = 1 n . Even though f ″ ( x n ) > 0 , the function is not strongly convex because f ″ ( x ) will become arbitrarily small.

A twice continuously differentiable function f on a compact domain X that satisfies f ″ ( x ) > 0 for all x ∈ 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.

Uniformly convex functions

A uniformly convex function, with modulus ϕ , is a function f that, for all x, y in the domain and t ∈ [0, 1], satisfies

f ( t x + ( 1 − t ) y ) ≤ t f ( x ) + ( 1 − t ) f ( y ) − t ( 1 − t ) ϕ ( ∥ x − y ∥ )

where ϕ is a function that is increasing and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking ϕ ( α ) = m 2 α 2 we recover the definition of strong convexity.

Examples

The function f ( x ) = x 2 has f ″ ( x ) = 2 > 0 at all points, so f 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 ) = 12 x 2 ≥ 0 , so f 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 function 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.

The function f ( x ) = | x | p for 1 ≤ p 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 f ( x ) is logarithmically convex if fis a convex function. The term "superconvex" is sometimes used instead.

The function 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.

The function x³ has second derivative 6x; thus it is convex on the set where x ≥ 0 and concave on the set where x ≤ 0.

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, i.e., 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 and positive homogeneity.

Examples of functions that are monotonically increasing but not convex include f ( x ) = x 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) = 1/x has f ″ ( x ) = 2 x 3 which is greater than 0 if x > 0, so f(x) is convex on the interval (0, +∞). It is concave on the interval (−∞, 0).

The function f(x) = x⁻², with f(0) = +∞, is convex on the interval (0, +∞) and convex on the interval (-∞,0), but not convex on the interval (-∞, +∞), because of the singularity at x = 0.