A pseudoconvex function is a differentiable real-valued function 
on an open convex
set 
that is pseudoconvex at every point 
. Pseudoconvexity at 
means that
 |
Here, 
denotes the usual gradient of 
. More generally, the same pointwise condition can be considered
when 
is Gâteaux differentiable at 
.
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 differentiable 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.
