Pseudoconvex function

In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex.

Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative.

The property must hold in all of the function domain, and not only for nearby points.

, defined on a (nonempty) convex open set

of the finite-dimensional Euclidean space

This function is said to be pseudoconvex if the following property holds:[1] Equivalently: Here

Note that the definition may also be stated in terms of the directional derivative of

is differentiable, this directional derivative is given by: Every convex function is pseudoconvex, but the converse is not true.

Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function

For any differentiable function, we have the Fermat's theorem necessary condition of optimality, which states that: if

Pseudoconvexity is of great interest in the area of optimization, because the converse is also true for any pseudoconvex function.

is a stationary point of a pseudoconvex function

Note also that the result guarantees a global minimum (not only local).

This last result is also true for a convex function, but it is not true for a quasiconvex function.

Consider for example the quasiconvex function: This function is not pseudoconvex, but it is quasiconvex.

Finally, note that a pseudoconvex function may not have any critical point.

An example of a function that is pseudoconvex, but not convex, is:

The figure shows this function for the case where

This example may be generalized to two variables as: The previous example may be modified to obtain a function that is not convex, nor pseudoconvex, but is quasiconvex: The figure shows this function for the case where

As can be seen, this function is not convex because of the concavity, and it is not pseudoconvex because it is not differentiable at

The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows.

, we can define the upper Dini derivative of

The function is said to be pseudoconvex if it is increasing in any direction where the upper Dini derivative is positive.

More precisely, this is characterized in terms of the subdifferential

denotes the line segment adjoining x and y.

A pseudoconcave function is a function whose negative is pseudoconvex.

A pseudolinear function is a function that is both pseudoconvex and pseudoconcave.

[4] For example, linear–fractional programs have pseudolinear objective functions and linear–inequality constraints.

These properties allow fractional-linear problems to be solved by a variant of the simplex algorithm (of George B.

-pseudolinearity; wherein classical pseudoconvexity and pseudolinearity pertain to the case when