Pseudoconvex Function
المؤلف:
Borwein, J. and Lewis, A.
المصدر:
Convex Analysis and Nonlinear Optimization: Theory and Examples. New York: Springer Science+Business Media, 2006.
الجزء والصفحة:
...
25-7-2021
1896
Pseudoconvex Function
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.
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