Pseudoconvex function

Pseudoconvex functions play a crucial role in nonlinear optimization . The strong requirement of convexity in objective functions or constraints is not met in many cases. With attenuating convexity concepts such as quasi-convexity or pseudoconvexity, one then tries to save certain properties in order to use them in algorithms. In the following, a real-valued function is differentiable on an open subset . If the function fulfills the following property, it is called pseudoconvex: The following applies to all : ${\displaystyle f}$ ${\displaystyle \Gamma \subseteq \mathbb {R} ^{n}}$ ${\displaystyle \forall x_{1},x_{2}\in \Gamma }$

From follows . ${\displaystyle \nabla f\left(x_{1}\right)^{T}\left(x_{2}-x_{1}\right)\geq 0}$ ${\displaystyle f\left(x_{2}\right)\geq f\left(x_{1}\right)}$

Even applies

Out and follows . ${\displaystyle \nabla f\left(x_{1}\right)^{T}\left(x_{2}-x_{1}\right)\geq 0}$ ${\displaystyle x_{1}\not =x_{2}}$ ${\displaystyle f\left(x_{2}\right)>f\left(x_{1}\right)}$

so one calls the function strictly pseudoconvex. It denotes the gradient of at the point . ${\displaystyle \nabla f(x)}$ ${\displaystyle f}$ ${\displaystyle x\in \Gamma }$

If (so ) then the condition for pseudoconvexity is simply: ${\displaystyle \Gamma \subset \mathbb {R} }$ ${\displaystyle n=1}$

From follows . ${\displaystyle f'(x_{1})(x_{2}-x_{1})\geq 0}$ ${\displaystyle f(x_{2})\geq f(x_{1})}$

A function is called pseudoconcave if the negative of the function is pseudoconvex.

Examples and characteristics

blue:, red: ${\displaystyle f(x):=xe^{x}}$ ${\displaystyle \textstyle g(x):={\frac {-1}{1+x^{2}}}}$

Differentiable convex functions are pseudoconvex. The functions

${\displaystyle f(x):=xe^{x}}$ and

${\displaystyle g(x):={\frac {-1}{1+x^{2}}}}$

are examples of pseudoconvex functions that are not convex. ${\displaystyle \mathbb {R} \rightarrow \mathbb {R} }$

Pseudoconvex functions on convex areas are strictly quasi-convex .

Importance for optimization

If the derivative of a pseudoconvex function disappears in the point , then there is a minimum. This follows immediately from the definition, because in this case the premise is independent of fulfilled and it follows . The definition of pseudoconvexity is designed in such a way that this is true. ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle f(x_{2})\geq f(x_{1})}$

Individual evidence

1. ^ Karl-Heinz Borgwardt, Optimization, Operations Research, Game Theory , Birkhäuser, Basel 2001, ISBN 3-7643-6519-6 , definition 12.14
2. ^ L. Collatz, W. Wetterling: Optimization Tasks , Springer-Verlag (1966), ISBN 0-387-05616-5 , paragraph 6.4
3. ↑ L. Collatz, W. Wetterling: Optimization Tasks , Springer-Verlag (1966), ISBN 0-387-05616-5 , §6, sentence 10
4. ↑ D. Jungnickel: Optimization Methods , Springer-Verlag (2008), ISBN 3-540-76789-4 , Corollary 3.4.14
5. ↑ D. Jungnickel: Optimization Methods , Springer-Verlag (2008), ISBN 3-540-76789-4 , sentence 3.4.15