Abstract
In this paper, we establish some relationships between vector variational-like inequality and non-smooth vector optimization problems under the assumptions of α-invex non-smooth functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality, under non-smooth pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends an earlier work of Ruiz-Garzon et al. (J Oper Res 157:113–119, 2004) to a wider class of functions, namely the non-smooth pseudo-α-invex functions. Moreover, this work extends an earlier work of Mishra and Noor (J Math Anal Appl 311:78–84, 2005) to non-differentiable case.
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Chen, G.-Y., Cheng, G.M.: Vector variational inequality and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–416. Springer, Berlin (1998)
Chen G.-Y. and Craven B.D. (1990). A vector variational inequality and optimization over an efficient set. Z. Oper. Res. 34: 1–12
Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley-Interscience, New York
Dafermos S. (1990). Exchange price equilibrium and variational inequalities. Math. Program. 46: 391–402
Giannessi F. (1980). Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F. and Lions, J.-L. (eds) Variational Inequality and Complementarity Problems., pp 151–186. Wiley, New York
Giannessi, F.: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 423–432. Kluwer, London (2000)
Hanson M.A. (1981). On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80: 545–550
Mishra S.K. and Noor M.A. (2005). On vector variational-like inequality problems. J. Math. Anal. Appl. 311: 78–84
Mohan S.R. and Neogy S.K. (1995). On invex sets and preinvex functions. J. Math. Anal. Appl. 189: 901–908
Noor M.A. (1994). Variational-like inequalities. Optimization 30: 323–330
Noor M.A. (2004). Generalized mixed quasi-variational-like inequalities. Appl. Math. Comput. 156: 145–158
Noor M.A. (2004). On generalized preinvex functions and monotonicities. J. Inequal. Pure Appl. Math. 5: 1–9
Noor M.A. (2005). Invex equilibrium problems. J. Math. Anal. Appl. 302: 463–475
Osuna-Gomez R., Rufian-Lizana A. and Ruiz-Canales P. (1998). Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory Appl. 98: 651–661
Parida J., Sahoo M. and Kumar A. (1989). A variational-like inequality problems. Bull. Aust. Math. Soc. 39: 225–231
Ruiz-Garzon G., Osuna-Gomez R. and Rufian-Lizan A. (2003). Generalized invex monotonicity. Eur. J. Oper. Res. 144: 501–512
Ruiz-Garzon G., Osuna-Gomez R. and Rufian-Lizan A. (2004). Relationships between vector variational-like inequality and optimization problems. Eur. J. Oper. Res. 157: 113–119
Weir T. and Mond B. (1988). Preinvex functions in multiobjective optimization. J. Math. Anal. Appl. 136: 29–38
Yang X.Q. (1993). Generalized convex functions and vector variational inequalities. J. Optim. Theory Appl. 79: 563–580
Yang X.Q. (1997). Vector variational inequality and vector pseudolinear optimization. J. Optim. Theory Appl. 95: 729–734
Yang X.Q. and Chen G.-Y. (1992). A class of nonconvex functions and prevariational inequalities. J. Math. Anal. Appl. 169: 359–373
Yang X.Q. and Goh C.J. (1997). On vector variational inequalities: application to vector equilibria. J. Optim. Theory Appl. 95: 431–443
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Mishra, S.K., Wang, S.Y. & Lai, K.K. On non-smooth α-invex functions and vector variational-like inequality. Optimization Letters 2, 91–98 (2008). https://doi.org/10.1007/s11590-007-0045-6
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DOI: https://doi.org/10.1007/s11590-007-0045-6