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 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: Properties Relation to other types of "convexity" 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: . Sufficient optimality condition For any differentiable function, we have the Fermat's theorem necessary condition of optimality, which states that: if has a local minimum at , 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. Also, the point is a critical point of , as . However, does not have a global minimum at (not even a local minimum). Finally, note that a pseudoconvex function may not have any critical point. Take for example the pseudoconvex function: ,