Convexity and concavity

Convexity and Concavity

For some functions, every stationary point is a global maximizer, or a global minimizer. We now study classes of such functions.

We say that a twice differentiable function (i.e. a function that is differentiable and for which its derivative is differentiable) is concave if its second derivative is nonpositive and is convex if its second derivative is nonnegative.

Definition
A twice-differentiable function f of a single variable is

concave on the interval I if f ¢¢(x) £ 0 for all x in the interior of I
convex on the interval I if f ¢¢(x) ³ 0 for all x in the interior of I.

Examples are shown in the following figure.

The importance of concave and convex functions in optimization theory comes from the fact that for a concave function every stationary point is a global maximizer, and for a convex function every stationary point is a global minimizer.

Note that a function is both concave and convex if and only if it takes the form f (x) = ax + b (that is, is "affine").

Example
Is x² - 2x + 2 concave or convex on any interval? Its second derivative is 2 ³ 0, so it is convex for all values of x.

Example
Is x³ - x² concave or convex on any interval? Its second derivative is 6x - 2, so it is convex on the interval [1/3, ¥) and concave the interval (-¥, 1/3].

Example
Suppose that U and g are nondecreasing and concave functions of a single variable (that is, U¢(x) ³ 0, U²(x) £ 0, g¢(x) ³ 0, and g²(x) £ 0 for all x). Show that f (x) = g(U(x)) is nondecreasing and concave.
We have f ¢(x) = g¢(U(x))U¢(x). Since g¢(x) ³ 0 for all x and U¢(x) ³ 0 for all x, we have f ¢(x) ³ 0 for all x. That is, f is nondecreasing.
Further,
f ²(x) = g²(U(x))·U¢(x)·U¢(x) + g¢(U(x))U²(x).
Since g²(x) £ 0, g¢(x) ³ 0, and U²(x) £ 0, we have f ²(x) £ 0. That is, f is concave.
That is: a nondecreasing concave transformation of an nondecreasing concave function is nondecreasing and concave.

Exercises

Inflection points

A point at which a function changes from being convex to concave, or vice versa, is an inflection point.

Definition
c is an inflection point of a twice-differentiable function f if f ²(x) changes sign at c: for some values of a and b

either f ²(x) ³ 0 if a < x < c and f ²(x) £ 0 if c < x < b
or f ²(x) £ 0 if a < x < c and f ²(x) ³ 0 if c < x < b.

An example of an inflection point is shown in the following figure.

Proposition

If c is an inflection point of f then f ²(c) = 0.
If f ²(c) = 0 and f ² changes sign at c then c is an inflection point.

Exercise

Concavity, convexity, and global optima

If f is a differentiable concave function then f ²(x) £ 0 for all x in some interval, so f ¢(x) is decreasing in this interval. So if f ¢(c) is zero at some point c in the interior of the interval then f ¢(x) is positive for x < c and negative for x > c. So by a previous result, c is a global maximizer. That is, we have the following result.

Proposition
Suppose that the differentiable function f is defined on the interval I. Let x be in the interior of I. Then

if f is concave then x is a global maximizer of f in I if and only if it is a stationary point of f
if f is convex then x is a global minimizer of f in I if and only if it is a stationary point of f .

If we use the fact that a twice-differentiable function is concave if and only if its second derivative is nonpositive (and similarly for a convex function), we obtain the following result.

Proposition
Suppose that the twice-differentiable function f is defined on the interval I. Let x be in the interior of I. Then

if f ²(x) £ 0 for all x Î I then x is a global maximizer of f in I if and only if f ¢(x) = 0
if f ²(x) £ 0 for all x Î I then x is a global minimizer of f in I if and only if f ¢(x) = 0.

Example
Consider the problem max_x-x² subject to x Î [-1,1]. The function f is concave; its unique stationary point is 0. Thus its global maximizer is 0.

Example
A competitive firm pays w for each unit of an input and has the fixed cost F . It obtains the revenue p for each unit of output that it sells. Its output from x units of the input is Öx. For what value of x is its profit maximized?
The firm's profit is pÖx - wx. The derivative of this function is (1/2)px^-1/2 - w, and the second derivative is -(1/4)px^-3/2 £ 0, so the function is concave. So the global maximum of the function occurs at the stationary point. Hence the maximizer solves (1/2)px^-1/2 - w = 0, so that x = (p/2w)².
What happens if the production function is x²?

A more general definition of convexity and concavity

The previous definition of concavity and convexity applies only to a twice differentiable function. The following definition applies to any function, and has the additional advantage that it is more easily generalized to functions of many variables.

Definition
Let f be a function of a single variable defined on an interval. Then f is

concave if the line segment joining any two points on its graph is never above the graph
convex if the line segment joining any two points on its graph is never below the graph.

To make this definition useful we need to translate it into an algebraic condition that we can check. For a concave function the definition is illustrated in the following figure.

Every point from a to b can be written as (1 - l)a + lb, where l is a real number between 0 and 1. (When l = 0, the point is a; when l = 1 it is b.) The one feature of the figure that is not obvious is the height of the chord at the point x = (1 - l)a + lb. To find this height, first note that the equation of the chord, which passes through the points (a, f (a)) and (b, f (b)), is

y - f (a) =

f (b) - f (a)

b-a

(x-a).

Now, if we plug in x = (1-l)a + lb, we get

y =

f (b) - f (a)

b-a

((1-l)a + lb-a) + f (a) =

f (b) - f (a)

b-a

l(b - a) + f (a)

so that

y = l( f (b)- f (a)) + f (a)

= (1 - l) f (a) + l f (b).

Given this fact, the definition of concave and convex functions may be given as follows:

Definition
Let f be a function of a single variable defined on an interval. Then f is

concave on the interval I if for all a Î I, all b Î I, and all l Î (0,1) we have
f ((1-l)a + lb) ³ (1 - l) f (a) + l f (b).

convex on the interval I if for all a Î I, all b Î I, and all l Î (0,1) we have
f ((1-l)a + lb) £ (1 - l) f (a) + l f (b).

Note that this definition, unlike the original one, does not depend on f being differentiable.

Note that f is concave if and only if - f is convex.

The next example generalizes an earlier one, which applies only to differentiable functions.
Example
Let U be concave and g nondecreasing and concave. Let f (x) = g(U(x)). Show that f is concave.
We need to show that f ((1-l)a + lb) ³ (1-l) f (a) + l f (b).
By the definition of f we have
f ((1-l)a + lb) = g(U((1-l)a + lb)).
Now, since U is concave we have
U((1-l)a + lb) ³ (1 - l)U(a) + lU(b).
And since g is nondecreasing, r ³ s implies g(r) ³ g(s). Hence
g(U((1-l)a + lb)) ³ g((1-l)U(a) + lU(b)).
But now by the concavity of g we have
g((1-l)U(a) + lU(b)) ³ (1-l)g(U(a)) + lg(U(b)) = (1-l) f (a) + l f (b).
So f is concave.

Strict convexity and concavity

The inequalities in the definition of concave/convex functions are weak: a concave/convex function can have linear parts, as in the following figure.

A concave function that has no linear parts is said to be strictly concave:

Definition
The function f is

strictly concave on the interval I if for all a Î I, all b Î I with a ¹ b, and all l Î (0,1) we have
f ((1-l)a + lb) > (1 - l) f (a) + l f (b).

strictly convex on the interval I if for all a Î I, all b Î I with a ¹ b, and all l Î (0,1) we have
f ((1-l)a + lb) < (1 - l) f (a) + l f (b).

If f is twice differentiable then

f is concave on [a, b] if and only if f ²(x) £ 0 for all x Î (a, b).

If f ²(x) < 0 for all x Î (a,b) then f is strictly concave on [a, b], but the converse is not true: if f is strictly concave then its second derivative is not necessarily negative at all points. (Consider the function f (x) = -x⁴. It is concave, but its second derivative at 0 is zero.) That is,

f is strictly concave on [a, b] if f ²(x) < 0 for all x Î (a, b), but if f is strictly concave on [a, b] then f ²(x) is not necessarily negative for all x Î (a, b).

(Analogous observations apply to the case of convex and strictly convex functions, with the conditions f ²(x) ³ 0 and f ²(x) > 0 replacing the conditions f ²(x) £ 0 and f ²(x) < 0.)

Exercises