×

Relationships between pre-invex concepts. (English) Zbl 1103.90398

Summary: It is well known that some properties of convex optimization problems, especially sufficient conditions for optimality, and duality conditions, are preserved when convexity is weakened to pre-invexity. However, there are several distinct pre-invex concepts-\(B\)-vex, \(B\)-pre-invex, pre-univex, \((p,r)\)-pre-invex-and these are not equivalent. The pre-invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions.

MSC:

90C30 Nonlinear programming
90C25 Convex programming
90C26 Nonconvex programming, global optimization
52A01 Axiomatic and generalized convexity
52A40 Inequalities and extremum problems involving convexity in convex geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antczak, T., \((p, r)\)-invex sets and functions, J. Math. Anal. Appl., 263, 355-379 (2001) · Zbl 1051.90018
[2] T. Antczak, \(rr\); T. Antczak, \(rr\)
[3] Bector, C. R.; Chandra, S.; Gupta, S.; Suneja, S. K., Univex sets, functions and univex nonlinear programming, (Komolosi, S.; Rapcsak; Schaible, S., Proceedings of Conference of Generalized Convexity, Pecs, Hungary (1993), Springer: Springer Berlin) · Zbl 0802.90092
[4] Bector, C. R.; Suneja, S. K.; Lalitha, C. S., Generalized \(B\)-vex functions and generalized \(B\)-vex programming, J. Optim. Theory Appl., 76, 3, 561-576 (1993) · Zbl 0802.49027
[5] Ben-Israel, A.; Mond, B., What is invexity?, J. Aust. Math. Soc. Ser. B, 28, 1-9 (1986) · Zbl 0603.90119
[6] Craven, B. D., Invex functions and constrained local minima, Bull. Aust. Math. Soc., 24, 357-366 (1981) · Zbl 0452.90066
[7] Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80, 545-550 (1981) · Zbl 0463.90080
[8] Mirtinowic, D. S., Elementarne nierówności (1972), PWN Warszawa
[9] Mohan, S. R.; Neogy, S. K., On invex sets and preinvex functions, J. Math. Anal. Appl., 189, 901-908 (1995) · Zbl 0831.90097
[10] Suneja, S. K.; Singh, C.; Bector, C. R., Generalizations of pre-invex functions and \(B\)-vex functions, J. Optim. Theory Appl., 76, 577-587 (1993) · Zbl 0802.49026
[11] Weir, T.; Jeyakumar, V., A class of nonconvex functions and mathematical programming, Bull. Aust. Math. Soc., 38, 177-189 (1988) · Zbl 0639.90082
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.