Some characterizations of strongly preinvex functions. (English) Zbl 1093.26006

Let \(K\) be a subset of a Hilbert space \(H\) and \(\eta:K\times K\rightarrow H\) be a function. The set \(K\) is called invex with respect to \(\eta\) if for all \(u,v\in K\) and \(t\in[0,1]\), \(u+t\eta(v,u)\in K\). Given such a set, a function \(F:K\rightarrow\mathbb{R}\) is called preinvex with respect to \(\eta\) if or all \(u,v\in K\) and \(t\in[0,1]\), \(F(u+t\eta(v,u))\leq (1-t)F(u)+tF(v)\). In this paper, one considers another function \(\alpha :K\times K\rightarrow\mathbb{R}\backslash\{0\}\) and defines the set \(K\) to be \(\alpha\)-invex with respect to \(\eta\) and \(\alpha\) if for all \(u,v\in K\) and \(t\in[0,1]\), \(u+t\alpha(v,u)\eta(v,u)\in K\). Likewise, the function \(F\) is called \(\alpha\)-preinvex with respect to \(\alpha\) and \(\eta\) if for all \(u,v\in K\) and \(t\in[0,1]\), \(F(u+t\alpha(v,u)\eta(v,u))\leq (1-t)F(u)+tF(v)\). Obviously, these notions are a transcription of the previous ones, if one considers the function \(\eta_{1}(\cdot,\cdot)=\alpha(\cdot ,\cdot)\eta(\cdot,\cdot)\). Many other notions and properties introduced in this paper can be derived in the same way from the usual generalized invexity notions that can be found in other papers in the field. When this is not the case, mistakes occur frequently: for instance, in the proof of Theorem 3.2, relation (3.8) does not follow from (3.7), because \(\bar{\mu} =\mu/\alpha(v_{t},u)\) in (3,7), but \(\bar{\mu}=\mu/\alpha(v,u)\) is needed in (3.8) . In the proof of Theorem 3.5, relation (3.10) does not follow from (3.9), because (3.9) holds only for the particular choice of \(u,v\) for which the relation in line 2 of the same page holds, thus one cannot take \(v=v_{t}\).


26B25 Convexity of real functions of several variables, generalizations
90C25 Convex programming
90C46 Optimality conditions and duality in mathematical programming
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[1] Hanson, M.A., On sufficiency of the kuhn – tucker conditions, J. math. anal. appl., 80, 545-550, (1981) · Zbl 0463.90080
[2] Ben-Israel, A.; Mond, B., What is invexity?, J. aust. math. soc. ser. B, 28, 1-9, (1986) · Zbl 0603.90119
[3] Mohan, S.R.; Neogy, S.K., On invex sets and preinvex functions, J. math. anal. appl., 189, 901-908, (1995) · Zbl 0831.90097
[4] Weir, T.; Mond, B., Preinvex functions in multiobjective optimization, J. math. anal. appl., 136, 29-38, (1988) · Zbl 0663.90087
[5] Noor, M.A., Generalized convex functions, Panamer. math. J., 4, 73-89, (1994) · Zbl 0847.90124
[6] Jeyakumar, V.; Mond, B., On generalized convex mathematical programming, J. aust. math. soc. ser. B, 34, 43-53, (1992) · Zbl 0773.90061
[7] Jeyakumar, V., Strong and weak invexity in mathematical programming, Methods oper. res., 55, 109-125, (1985) · Zbl 0566.90086
[8] Noor, M.A., Invex equilibrium problems, J. math. anal. appl., 302, 463-475, (2005) · Zbl 1058.49007
[9] M.A. Noor, Properties of preinvex functions, Preprint, Etisalat College of Engineering, Sharjah, UAE, 2004
[10] Ruiz-Gaezion, G.; Osuna-Gomez, R.; Rufian-Lizan, A., Generalized invex monotonicity, European J. oper. res., 144, 501-512, (2003) · Zbl 1028.90036
[11] Yang, X.Q., Generalized convex functions and vector variational inequalities, J. optim. theory appl., 79, 563-580, (1993) · Zbl 0797.90085
[12] X.M. Yang, X.Q. Yang, K.L. Teo, Criteria for generalized invex monotonicities, European J. Oper. Res., in press · Zbl 1132.90360
[13] Yang, X.M.; Yang, X.Q.; Teo, K.L., Generalized invexity and generalized invariant monotonicity, J. optim. theory appl., 117, 607-625, (2003) · Zbl 1141.90504
[14] Schaible, S., Generalized monotonicity: concepts and uses, (), 289-299 · Zbl 0847.49013
[15] Zhu, D.L.; Marcotte, P., Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. optim., 6, 714-726, (1996) · Zbl 0855.47043
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