Concave (function)

Definition

A function g(x) is concave if
and

Or:
On one dimension:
f(x) is concave

Or:
On multiple dimension:
f is twice differentiable and f(x) is concave its hessian matrix is negative semi-definite .

f(x) is concave implies
is convex

or
Let , assume that D is a convex set.
sub f = is a convex set

: "D cross R", × is the Cartesian product.

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A function g(x) is strictly concave if
and

Or:
On one dimension:
f(x) is strictly concave

Or:
On multiple dimension:
f is twice differentiable and f(x) is strictly concave its hessian matrix is negative definite.

Property of concave function

• If f is concave, the feasible set is convex, and a is a local maximum of f, then a is a global maximum of f (convex program)
• If f is strictly concave, the feasible set is convex, and a is a local maximum of f, then a is the unique global maximum of f.
• If f is a linear function, then it is both convex and concave, but not strictly so
• Every concave function is continuous on the interior of its domain
• If f is concave, then -f is convex and vice versa
• If f and g are both concave functions, then f(x) + g(x) is also concave

Concavity of composite function

Sufficient condition to guarantee the composite function will be concave:
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 concave.

See also:

• convex (function)

Concave function and maximizer

Let f and be (continuously differentiable) concave functions mapping the open and convex set into . Define

Suppose there exists and such that the Kuhn-Tucker first-order conditions are met:

• KT-1:
• KT-2: and for

Note: is the i-th constraint
D means the derivative with respect to

Then maximizes f over if there exists an s.t. for

Note

Example