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. Formal definition. Consider a differentiable function formula_1, defined on a (nonempty) convex open set formula_2 of the finite-dimensional Euclidean space formula_3. This function is  said to be pseudoconvex if the following property holds: Equivalently: Here formula_4 is the gradient of formula_5, defined by: formula_6 Note that the definition may also be stated in terms of the directional derivative of formula_5, in the direction given by the vector formula_8. This is because, as formula_5 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 formula_10 is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function  formula_11 is quasiconvex but not pseudoconvex. This can be summarized schematically as: To see that formula_11 is not pseudoconvex, consider its derivative at formula_13: formula_14. Then, if formula_11 was pseudoconvex, we should have: In particular it should be true for formula_16. But it is not, as: formula_17. Sufficient optimality condition. For any differentiable function, we have the Fermat's theorem necessary condition of optimality, which states that: if formula_5 has a local minimum at formula_19, then formula_19 must be a stationary point of formula_5 (that is: formula_22). Pseudoconvexity is of great interest in the area of optimization, because the converse  is also true for any pseudoconvex function. That is: if formula_19 is a stationary point of a pseudoconvex function formula_5, then formula_5 has a global minimum at formula_19. 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 formula_27 is a critical point of formula_5, as formula_29. However, formula_5 does not have a global minimum at formula_27 (not even a local  minimum). Finally, note that a pseudoconvex function may not have any critical point. Take for example the pseudoconvex function: formula_32, whose derivative is always positive: formula_33. Examples. An example of a function that is pseudoconvex, but not convex, is: formula_34 The figure shows this function for the case where formula_35. 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 formula_36. As can be seen, this function is not convex because of the  concavity, and it is not pseudoconvex because it is not differentiable at formula_27. Generalization to nondifferentiable functions. The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows. Given any function formula_38, we can define the upper Dini derivative of formula_5 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 formula_40 as follows: where formula_41 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 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. Dantzig). Given a vector-valued function formula_42, there is a more general notion of formula_42-pseudoconvexity and formula_42-pseudolinearity; wherein classical pseudoconvexity and pseudolinearity pertain to the case when formula_45.