Given a subset  and a real function  which is Gâteaux differentiable at a point ,  is said to be pseudoconvex at  if

Here,  denotes the usual gradient of .

The term pseudoconvex is used to describe the fact that such functions share many properties of convex functions, particularly with regards to derivative properties and finding local extrema. Note, however, that pseudoconvexity is strictly weaker than convexity as every convex function is pseudoconvex though one easily checks that  is pseudoconvex and non-convex.

Similarly, every pseudoconvex function is quasi-convex, though the function  is quasi-convex and not pseudoconvex.

A function  for which  is pseudoconvex is said to be pseudoconcave.

REFERENCES:

Borwein, J. and Lewis, A. Convex Analysis and Nonlinear Optimization: Theory and Examples. New York: Springer Science+Business Media, 2006.