Pseudolinear function: Difference between revisions

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{{Merge#REDIRECT to|[[Pseudoconvex function|discuss=Talk:Pseudoconvex#pseudolinear function#Merger proposal|date=January 2011}}]]

In mathematics, a '''pseudoconvex function''' <math>f:X\rightarrow\mathbb{R}</math> on an open convex set <math>X\subseteq\mathbb{R}^n</math> is a function that is differentiable in <math>X</math> such that for every <math>x,y\in X</math>,

:<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 of <math>-f</math>.

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A '''pseudolinear function''' is one that is both pseudoconvex and pseudoconcave.

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It can be shown (see Cambini and Martein) that <math>f</math> is pseudolinear if and only if for every <math>x,y\in X</math>,

: <math> f(x) = f(y)\text{ if and only if }\nabla f(x)^T (y - x) = 0. \, </math>

In [[mathematical optimization]], [[linear-fractional programming|linear–fractional program]]s have pseudolinear [[objective function]]s and [[linear programming|linear–inequality constraints]]: These properties allow linear-fractional problems to be solved by a variant of the [[simplex algorithm]] (of [[George B. Dantzig]]).<ref>

Chapter five: {{cite book| last=Craven|first=B. D.|title=Fractional programming|series=Sigma Series in Applied Mathematics|volume=4|publisher=Heldermann Verlag|___location=Berlin|year=1988|pages=145|isbn=3-88538-404-3 |mr=949209}}</ref><ref>{{cite article | last1=Kruk | first1=Serge|last2=Wolkowicz|first2=Henry|title=Pseudolinear programming | jstor=2653207 |journal=[[SIAM Review]]|volume=41 |year=1999 |number=4 |pages=795-805 |mr=1723002 | jstor = 2653207 | doi = 10.1137/S0036144598335259}}

</ref><ref>{{cite article | last1=Mathis|first1=Frank H.|last2=Mathis|first2=Lenora Jane|title=A nonlinear programming algorithm for hospital management |jstor=2132826|journal=[[SIAM Review]]|volume=37 |year=1995 |number=2 |pages=230-234|mr=1343214 | jstor = 2132826 | doi = 10.1137/1037046}}

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== References ==

* {{cite journal|ref=harv|first=T.|last=Rapcsak|title=On pseudolinear functions|journal=European Journal of Operational Research|volume=50|issue=3|date=1991-02-15|pages=353&ndash;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&ndash;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&ndash;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=&eta;-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&ndash;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&ndash;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|date=1993-06-11|pages=278&ndash;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&ndash;2|pages=31&ndash;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&ndash;159|format=[[Portable Document Format|PDF]]|url=http://www.gip.hk/ijotma/Internet%20IJOTMA%20V1N2/IJOTMAV1N2%20PA/IJOTMAv1n2%20pa3.pdf}}

* {{cite book|ref=harv|chapter=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=Section 3.3: Quasilinearity and Pseudolinearity|pages=50&ndash;57|doi=10.1007/978-3-540-70876-6}}

[[Category:Convex analysis]]

[[Category:Types of functions]]

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