×

Invexity at a point: Generalisations and classification. (English) Zbl 0792.49014

A differentiable function \(f: X\subset R^ n\to R\) is invex if there is a vector function \(\eta: X\times X\to R^ n\) for which \(\forall x,u\in X: f(x)- f(u)\geq \eta^ T(x,u)\cdot\nabla f(u)\). When \(X\) is convex and \(\eta(x,u)= x- u\) this is usual convexity. This property has been generalized in the paper in many ways. First, for Lipschitzian and also arbitrary nonlinear functions using Clarke’s generalized derivatives. Next, the variants of the condition itself provide weak, strict, strong invexity, pseudoinvexity, quasiinvexity, etc. The relations between these properties are discussed.

MSC:

49J52 Nonsmooth analysis
90C30 Nonlinear programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Craven, J. Austral. Math. Soc. (Ser.A) 39 pp 1– (1985)
[2] DOI: 10.1080/02331938608843097 · Zbl 0591.49016 · doi:10.1080/02331938608843097
[3] Craven, Bull. Austral. Math. Soc. 24 pp 357– (1981)
[4] DOI: 10.1287/moor.1.2.165 · Zbl 0404.90100 · doi:10.1287/moor.1.2.165
[5] DOI: 10.1287/moor.8.2.231 · Zbl 0526.90077 · doi:10.1287/moor.8.2.231
[6] Preda, Stud. Cerc. Mat. 42 pp 304– (1990)
[7] Giorgi, Atti del Tredicesimo Convegno A.M.A.S.E.S., Verona pp 13– (1989)
[8] Jeyakumar, Utilitas Math. 29 pp 71– (1986)
[9] DOI: 10.1016/0022-247X(85)90099-X · Zbl 0553.90086 · doi:10.1016/0022-247X(85)90099-X
[10] Jeyakumar, Methods Oper. Res. 55 pp 109– (1985)
[11] DOI: 10.1016/0022-247X(81)90123-2 · Zbl 0463.90080 · doi:10.1016/0022-247X(81)90123-2
[12] Giorgi, Rev. Roumaine Math. Pures Appl. 38 pp 125– (1993)
[13] Mititelu, Bull. Math. Soc. Sci. Math. Roumanie 37 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.