Pseudolinear function

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

$f\left(y\right)<f\left(x\right)\Rightarrow \left(y-x\right)^{T}\nabla f\left(x\right)<0.\,$

It is pseudoconcave if this is true of $-f$.

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

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

$f(x)=f(y){\text{ if and only if }}\nabla f(x)^{T}(y-x)=0.\,$

References

• Rapcsak, T. (1991-02-15). "On pseudolinear functions". European Journal of Operational Research. 50 (3): 353–360. doi:10.1016/0377-2217(91)90267-Y. ISSN 0377-2217. {{cite journal}}: Invalid |ref=harv (help)
• Mangasarian, O. L. (1965). "Pseudo-Convex Functions". Journal of the Society for Industrial and Applied Mathematics Series A. 3 (2): 281–290. doi:10.1137/0303020. ISSN 0363-0129. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)

Further reading

• Chew, Kim Lin; Choo, Eng Ung (1984). "Pseudolinearity and efficiency". Mathematical Programming. 28 (2): 226–239. doi:10.1007/BF02612363. {{cite journal}}: Invalid |ref=harv (help)
• Mishra, Shashi Kant; Giorgi, Giorgio (2008). "η-Pseudolinearity: Invexity and Generalized Monotonicity". Invexity and optimization. Nonconvex optimization and its applications. Vol. 88. Springer. ISBN 9783540785620. {{cite book}}: Invalid |ref=harv (help); Unknown parameter |isbn10= ignored (help)
• Kaul, R. N.; Lyall, Vinod; Kaur, Surjeet (1988). "Semilocal pseudolinearity and efficiency". European Journal of Operational Research. 36 (3). Elsevier Science B.V.: 402–409. doi:10.1016/0377-2217(88)90133-6. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)
• Jeyakumar, V.; Yang, X. Q. (1995). "On characterizing the solution sets of pseudolinear programs". Journal of Optimization Theory and Applications. 87 (3): 747–755. doi:10.1007/BF02192142. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)
• Komlósi, S. (1993-06-11). "First and second order characterizations of pseudolinear functions". European Journal of Operational Research. 67 (2). Elsevier Science B.V.: 278–286. doi:10.1016/0377-2217(93)90069-Y. {{cite journal}}: Invalid |ref=harv (help)
• Ansari, Qamrul Hasan; Schaible, Siegfried; Yao, Jen-Chih (1999). "η-Pseudolinearity". Decisions in Economics and Finance. 22 (1–2): 31–39. doi:10.1007/BF02912349. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)
• Giorgi, Giorgio; Rueda, Norma G. (2009). "η-Pseudolinearity and Efficiency" (PDF). International Journal of Optimization: Theory, Methods and Applications. 1 (2): 155–159. ISSN 2070-5565. {{cite journal}}: Invalid |ref=harv (help)
• Cambini, Alberto; Martein, Laura (2009). "Section 3.3: Quasilinearity and Pseudolinearity". Lecture Notes in Economics and Mathematical Systems. Vol. 616. Springer. pp. 50–57. doi:10.1007/978-3-540-70876-6. ISBN 9783540708759. {{cite book}}: Invalid |ref=harv (help); Missing or empty |title= (help); Unknown parameter |isbn10= ignored (help)