Pseudoconvex function


In convex analysis and the calculus of variations, 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.

Formal definition

Formally, a real-valued differentiable function defined on a convex open set in the finite-dimensional Euclidean space is said to be pseudoconvex if, for all such that, we have. Here is the gradient of, defined by

Properties

Every convex function is pseudoconvex, but the converse is not true. For example, the function is pseudoconvex but not convex. Any pseudoconvex function is quasiconvex, but the converse is not true since the function is quasiconvex but not pseudoconvex. Pseudoconvexity is primarily of interest because a point x* is a local minimum of a pseudoconvex function ƒ if and only if it is a stationary point of ƒ, which is to say that the gradient of ƒ vanishes at x*:

Generalization to nondifferentiable functions

The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows. Given any function we can define the upper Dini derivative of ƒ 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 ∂ƒ as follows:
A ' is a function whose negative is pseudoconvex. A ' is a function that is both pseudoconvex and pseudoconcave. 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. Given a vector-valued function η, there is a more general notion of η-pseudoconvexity and η-pseudolinearity wherein classical pseudoconvexity and pseudolinearity pertain to the case when η = y - x.