pseudoconvex function

Input interpretation

pseudoconvex function

Definition

Given a subset S subset R^n and a real function f which is Gâteaux differentiable at a point x element S, f is said to be pseudoconvex at x if del f(x)·(y - x)>=0, y element S⇒f(y)>=f(x). Here, del f denotes the usual gradient of f. 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 f(x) = x + x^3 is pseudoconvex and non-convex.
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