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Zbl 0731.26009
Pini, Rita
Invexity and generalized convexity.
(English)
[J] Optimization 22, No.4, 513-525 (1991). ISSN 0233-1934; ISSN 1029-4945/e

A differentiable function $f:{\bbfR}\sp n\to {\bbfR}$ is said to be invex if there exists a function $\eta (y,x)\in {\bbfR}\sp n$ such that, for all $y,x\in {\bbfR}\sp n$ $$f(y)-f(x)\ge \eta (y,x)\sp t\nabla f(x).$$ Various extensions of such functions including pseudo- and quasi-invex have been defined and their relationship to each other and other generalizations of convexity have been studied. For the non- differentiable case, f is said to be pre-invex if $$f(x+t\eta (y,x))\le tf(y)+(1-t)f(x),\quad 0\le t\le 1.$$ This comprehensive paper brings together many of these scattered results which are then studied, compared, and extended. Some definitions that are introduced include $\eta$-invex subsets, pseudo and quasi-pre-invex functions. One very minor correction. Reference 2 should be to the Journal, not the Bulletin, of the Australian Mathematical Society.
[B.Mond (Bundoora)]
MSC 2000:
*26B25 Convexity and generalizations (several real variables)
90C30 Nonlinear programming

Keywords: invexity; generalizations of convexity; pre-invex; quasi-pre-invex

Cited in: Zbl 1070.26013

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