Invexity and generalized convexity. (English) Zbl 0731.26009

A differentiable function \(f:{\mathbb{R}}^ n\to {\mathbb{R}}\) is said to be invex if there exists a function \(\eta (y,x)\in {\mathbb{R}}^ n\) such that, for all \(y,x\in {\mathbb{R}}^ n\) \[ f(y)-f(x)\geq \eta (y,x)^ t\nabla f(x). \] 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 \[ f(x+t\eta (y,x))\leq tf(y)+(1-t)f(x),\quad 0\leq t\leq 1. \] 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.
Reviewer: B.Mond (Bundoora)


26B25 Convexity of real functions of several variables, generalizations
90C30 Nonlinear programming
Full Text: DOI


[1] Avriel Diewert, Generalized concavity (1988)
[2] DOI: 10.1017/S0334270000005142 · Zbl 0603.90119
[3] Bector C.R., Congressus Numerantium 52 pp 53– (1986)
[4] DOI: 10.1007/BF00932539 · Zbl 0325.26007
[5] DOI: 10.1017/S0004972700004895 · Zbl 0452.90066
[6] DOI: 10.1017/S1446788700022126
[7] DOI: 10.1016/0022-247X(81)90123-2 · Zbl 0463.90080
[8] DOI: 10.1016/0377-2217(82)90181-3 · Zbl 0501.90090
[9] Rapcsak T., Publicationes Matematicae 34 pp 35– (1987)
[10] DOI: 10.1016/0022-247X(88)90113-8 · Zbl 0663.90087
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