whose derivative is always positive: . Examples An example of a function that is pseudoconvex, but not convex, is: The figure shows this function for the case where . 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 . As can be seen, this function is not convex because of the concavity, and it is not pseudoconvex because it is not differentiable at . 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 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: where 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 , there is a more general notion of -pseudoconvexity and -pseudolinearity; wherein classical pseudoconvexity and pseudolinearity pertain to the case when . See also Pseudoconvexity Convex function Quasiconvex function Notes References . . Convex analysis Convex optimization Generalized convexity Types of functions