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.

Formal definition

Consider a differentiable, 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:

Equivalently:

is the gradient

Note that the definition may also be stated in terms of the directional derivative is differentiable, this directional derivative is given by:

Properties

Relation to other types of "convexity"

is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex is quasiconvex but not pseudoconvex. This can be summarized schematically as:

1. quasiconvex

was pseudoconvex, we should have:

.

Sufficient optimality condition

For any differentiable function, we have the Fermat's theorem in an open must be a stationary point).

Pseudoconvexity is of great interest in the area of optimization, because the converse is also true for any pseudoconvex function. That is: is a stationary point. 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:

(not even a local minimum).

.

Examples

. 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:

.

Generalization to nondifferentiable functions

The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows., we can define the upper Dini derivative by:

where u is any unit vector. 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 as follows:

1. ;

denotes the line segment adjoining x and y.

Related notions

A is a function whose negative is pseudoconvex. A is a function that is both pseudoconvex and pseudoconcave. For example, linear–fractional programss 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. Dantzig).

-pseudoconvexity.

See also

• Pseudoconvexity

• Convex function

• Quasiconvex function

• Invex function

Notes

References

• .

• .