Chemistry - Convex function

A convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have

$f(tx+(1-t)y)\leq t f(x)+(1-t)f(y).$

I.e., a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set. A function is also said to be strictly convex if

f (t x + (1 - t) y) < t f (x) + (1 - t) f (y).

for any t in (0,1).

Contents

1 Logarithmically convex function

2 Properties of convex functions

3 Examples of convex functions

4 Reference

Logarithmically convex function

A function such that $f (x) > 0$ for all $x$ is said to be logarithmically convex function if $log f (x)$ is a convex function of $x$ .

It is easy to see that a logarithmically convex function is a convex function, but the converse is not true. For example $f (x) = x 2$ is a convex function, but $log f (x) = log x 2 = 2log x$ is not a convex function and thus $f (x) = x 2$ is not logarithmically convex. On the other hand $e^{x^2}$ is logarithmically convex since $\log e^{x^2} = x^2$ is convex. A less trivial example of a logarithmically convex function is the gamma function, if restricted to the positive reals. (See also Bohr-Mollerup_theorem.)

The definition is easily extended to functions $f\colon \R \to \R$ where still we have $f > 0$ , in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals $[a, b]$ .

Properties of convex functions

A convex function f defined on some convex open interval C is continuous on the whole C and differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the border.

A continuous function on C is convex if and only if

$f\left( \frac{x+y}2 \right) \le \frac{f(x)+f(y)}2 .$

for any x and y in C.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval.

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: f(y) ≥ f(x) + f'(x) (y - x) for all x and y 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. If its second derivative is positive then it is strictly convex, but the opposite is not true, as shown by f(x) = x⁴.

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

If two functions f and g are convex, then so is any weighted combination a f + b g with non-negative coefficients a and b.

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

For a convex function f, the level sets {x|f(x)<a} and {x|f(x)≤a} with a∈R are convex sets.

A convex function respects Jensen's inequality.

Examples of convex functions

The second derivative of x² is 2; it follows that x² is a convex function of x.
The absolute value function |x| is convex, even though it does not have a derivative at x = 0.
The function f(x) = x is convex but not strictly convex.
The function x³ has second derivative 6x; thus it is convex for x ≥ 0 and concave for x ≤ 0.

Reference

Categories: Mathematical analysis