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#REDIRECT [[Pseudoconvex function#pseudolinear function]]
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A function <math>f:X\rightarrow\mathbb{R}</math> is said to be '''pseudoconvex''' in <math>X\subseteq\mathbb{R}</math> if it is differentiable in <math>X</math> and for any <math>x,y\in X</math>,
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:<math>f\left(y\right)<f\left(x\right)\Rightarrow\left(y-x\right)^{T}\nabla f\left(x\right)<0</math>.
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It is '''pseudoconcave''' if this is true for <math>-f</math>.
A '''pseudolinear function''' is one that is both pseudoconvex and pseudoconcave.
== References ==
* {{cite journal|ref=harv|first=T.|last=Rapcsak|title=On pseudolinear functions|journal=European Journal of Operational Research|volume=50|issue=3|day=15|month=February|year=1991|pages=353–360|issn=0377-2217|doi=10.1016/0377-2217(91)90267-Y}}
* {{cite journal|ref=harv|title=Pseudo-Convex Functions|journal=Journal of the Society for Industrial and Applied Mathematics Series A|volume=3|issue=2|pages=281–290 |month=January|year=1965|doi=10.1137/0303020|first=O. L.|last=Mangasarian|issn=0363-0129}}
== Further reading ==
* {{cite journal|ref=harv|journal=Mathematical Programming|volume=28|issue=2|pages=226–239|doi=10.1007/BF02612363|title=Pseudolinearity and efficiency|first1=Kim Lin|last1=Chew|first2=Eng Ung|last2=Choo|year=1984}}
* {{cite book|ref=harv|title=Invexity and optimization|volume=88|series=Nonconvex optimization and its applications|first1=Shashi Kant|last1=Mishra|first2=Giorgio|last2=Giorgi|publisher=Springer|year=2008|isbn10=3540785620|isbn=9783540785620|chapter=η-Pseudolinearity: Invexity and Generalized Monotonicity}}
* {{cite journal|ref=harv|journal=European Journal of Operational Research|volume=36|issue=3|month=September|year=1988|pages=402–409|doi=10.1016/0377-2217(88)90133-6|publisher=Elsevier Science B.V.|title=Semilocal pseudolinearity and efficiency|first1=R. N.|last1=Kaul|first2=Vinod|last2=Lyall|first3=Surjeet|last3=Kaur}}
* {{cite journal|ref=harv|journal=Journal of Optimization Theory and Applications|volume=87|issue=3|pages=747–755|doi=10.1007/BF02192142|title=On characterizing the solution sets of pseudolinear programs|first1=V.|last1=Jeyakumar|first2=X. Q.|last2=Yang|month=December|year=1995}}
* {{cite journal|ref=harv|journal=European Journal of Operational Research|volume=67|issue=2|day=11|month=June|year=1993|pages=278–286|doi=10.1016/0377-2217(93)90069-Y|publisher=Elsevier Science B.V.|title=First and second order characterizations of pseudolinear functions|first=S.|last=Komlósi}}
* {{cite journal|ref=harv|journal=Decisions in Economics and Finance|volume=22|issue=1–2|pages=31–39|doi=10.1007/BF02912349|title=η-Pseudolinearity|first1=Qamrul Hasan|last1=Ansari|first2=Siegfried|last2=Schaible|first3=Jen-Chih|last3=Yao|month=March|year=1999}}
* {{cite journal|ref=harv|first1=Giorgio|last1=Giorgi|first2=Norma G.|last2=Rueda|title=η-Pseudolinearity and Efficiency|journal=International Journal of Optimization: Theory, Methods and Applications|issn=2070-5565|year=2009|volume=1|issue=2|pages=155–159|format=[[Portable Document Format|PDF]]|url=http://www.gip.hk/ijotma/Internet%20IJOTMA%20V1N2/IJOTMAV1N2%20PA/IJOTMAv1n2%20pa3.pdf}}
* {{cite book|ref=harv|title=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=9783540708759|chapter=Qualilinearity and Pseudolinearity|pages=50–57|doi=10.1007/978-3-540-70876-6}}
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