Convex (function)

Definition

Convex function

On one dimension:
A function g(x) is a convex function if and only if
and

On multiple dimensions:
F is convex if and only if for every pair of of points x and y in the domain of F and every scalar between zero and one ():

or let , assume is a convex set.
epi f =
A function f is convex on if epi f is a convex set.

When the function is once differentiable:

On one dimension:
f is convex if for every x and y in the domain of f.

Note: it is not "if and only if". because convex function may not be differentiable. For example is not differentiable at .

On multiple dimensions:
F is convex if for every distinct x and y in the domain of F:

When the function is twice differentiable:

On one dimension:
f is convex if for every x in the domain of f,

On multiple dimensions:
F is convex if for every x in the domain of F,
its hessian matrix is positive semi-definite .

Strictly convex function

On one dimension:
A function g(x) is a strictly convex function over its domain if and only if
and
Note: here we require:

x and y are distinct.
λ is strictly between zero and one.

On multiple dimensions:
F is strictly convex if and only if for every pair of of points x and y in the domain of F and every scalar λ strictly between zero and one ():

When the function is once differentiable

On one dimension:
f is strictly convex if for every x and y in the domain of f.

On multiple dimensions:
F is strictly convex if for every distinct x and y in the domain of F:

When the function is twice differentiable

On one dimension:
f is strictly convex for every x in the domain of f:

On multiple dimensions:
F is strictly convex if for every x in the domain of F,
its hessian matrix is positive definite.

Properties of convex function

The properties of convexity are global in nature.

Relationship with concave function

A function is concave if the negative of the function is convex.
A function is strictly concave if the negative of the function is strictly convex.

Relationship with convex set

Consider an interval (a, b) , and x, y in (a,b).
Let f(x) be a convex function, then is convex set.

Convexity of composite function

Sufficient condition to guarantee the composite function will be convex:
If f(y) and g(x) are convex functions, and if f(y) is increasing in each of its arguments, then the composite function is convex.

Extra
If g(x) is differentiable on (a, b), then:
(1) g(x) is convex if and only if , for all a < x < y < b.
(2) g(x) is strictly convex if and only if g'(x) < g'(y), for all a < x < y < b.

If g(x) is twice differentiable on (a, b), then:
(1) g(x) is convex if and only if , for all a < x < b.
(2) g(x) is strictly convex if g''(x) > 0, for all a < x < b.