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* {{cite book|ref=harv|booktitle=Generalized convexity and optimization: theory and applications|volume=616|series=Lecture Notes in Economics and Mathematical Systems|first1=Alberto|last1=Cambini|first2=Laura|last2=Martein|publisher=Springer|year=2009|isbn10=3540708758|isbn=978-3-540-70875-9|chapter=Section 3.3: Quasilinearity and Pseudolinearity|pages=50–57|doi=10.1007/978-3-540-70876-6}}
I put them here because it's not obvious to me that they should be included. I'll let someone else decide. [[User:RockMagnetist|RockMagnetist]] ([[User talk:RockMagnetist|talk]]) 21:56, 21 March 2013 (UTC)
== Generalization to nondifferentiable functions ==
The two definitions given in this section do not coincide:
Consider the function <math>f:\R\to\R</math> with <math>f(x)=-x</math> whenever <math>x\in D=\{x=\frac{n}{2^p}|\; p\in\N\cup\{0\},\, n\in\N\}</math> and <math>f(x)=|x|</math>, elsewhere.
For the upper Dini derivative it holds <math>f^+(x,1)=-1</math> for all <math>x<0</math>, <math>f^+(x,1)=+1</math> for all <math>x\geq 0</math> with <math>x\notin D</math> and <math>f^+(x,1)=+\infty</math> for all <math>x\in D</math>.
Likewise, <math>f^+(x,-1)=+1</math> for all <math>x\leq 0</math>, <math>f^+(x,-1)=-1</math> for all <math>x>0</math> with <math>x\notin D</math> and <math>f^+(x,-1)=+\infty</math> for all <math>x\in D</math>.
Now set <math>x_0=0</math>. Then <math>f^+(x_0,1)=+1</math>, but <math>f</math> is not increasing in this direction.
On the other hand, given the definition of the subdifferential as cited from https://en.wikipedia.org/wiki/Subderivative,
<math>\partial f(x)=\{x^*\in \R|\; \forall y\in \R:\; f(y)-f(x)\geq x^*(y-x) \}</math>,
thus <math>\partial f(x)=\{-1\}</math>, whenever <math>x\leq 0</math> or <math>x\in D</math> and <math>\partial f(x)=\emptyset</math>, elsewhere.
Especially, if <math>-1(y-x)\geq 0</math> for some <math>x\leq 0</math> or <math>x\in D</math>, then <math>f(y)\geq f(x)</math>, so <math>f</math> is pseudoconvex in the sense of the second definition.
From what I know, pseudoconvexity in a point can be defined with respect to a specified derivative, that is
given a derivative <math>f'(x,\cdot)</math>, then <math>f</math> is pseudoconvex in a point <math>x_0</math>, iff for all <math>x\in X</math>, <math>f(x)< f(x_0)</math> implies <math>f'(x_0,x-x_0)<0</math> and <math>f</math> is pseudoconvex with respect to this derivative, iff the same holds true everywhere.
Or it can be defined with respect to some subdifferential as is done above.
Defining
<math>\partial f(x_0)=\{x^*\in X^*|\; \forall x\in X:\, x^*(x-x_0)\leq f'(x_0,x-x_0)\}</math>
pseudoconvexity wrt to der derivatie implies pseudoconvexity wrt the corresponding subdifferential, I'm not sure about the reverse direction.
Agreed??
[[User:Catrinski|Catrinski]] ([[User talk:Catrinski|talk]]) 11:19, 27 February 2015 (UTC)
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