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. Consider a differentiable function , 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: Here is the gradient of , defined by: Note that the definition may also be stated in terms of the directional derivative of , in the direction given by the vector . This is because, as is differentiable, this directional derivative is given by: Every convex function is pseudoconvex, but the converse is not true. For example, the function is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function is quasiconvex but not pseudoconvex. This can be summarized schematically as: To see that is not pseudoconvex, consider its derivative at : . Then, if was pseudoconvex, we should have: In particular it should be true for . But it is not, as: . For any differentiable function, we have the Fermat's theorem necessary condition of optimality, which states that: if has a local minimum at in an open domain, then must be a stationary point of (that is: ). Pseudoconvexity is of great interest in the area of optimization, because the converse is also true for any pseudoconvex function. That is: if is a stationary point of a pseudoconvex function , then has a global minimum at . 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.