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.
