×

zbMATH — the first resource for mathematics

Some remarks on the Minty vector variational inequality. (English) Zbl 1140.90492
Summary: We establish some relations between a Minty vector variational inequality and a vector optimization problem under pseudoconvexity or pseudomonotonicity, respectively. Our results generalize those of F. Giannessi [Appl. Optim. 13, 93–99 (1998; Zbl 0909.90253)].

MSC:
90C29 Multi-objective and goal programming
49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] GIANNESSI, F., On Minty Variational Principle, New Trends in Mathematical Programming, Edited by F. Giannessi, T. Rapcsák, and S. Komlósi, Kluwer Academic Publishers, Dordrecht, Netherlands, 1997. · Zbl 0909.90253
[2] GIANNESSI, F., Theorems of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley, New York, NY, 151–186, 1980.
[3] CHEN, G. Y., and YANG, X. Q., The Vector Complementarity Problem and Its Equivalence with the Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol.153, 136–158, 1990. · Zbl 0719.90078
[4] YANG, X. Q., Vector Variational Inequality and Vector Pseudolinear Optimization, Journal of Optimization Theory and Applications, Vol.95, 729–734, 1997. · Zbl 0901.90162
[5] LEE, G. M., KIM, D. S., LEE, B. S., and YEN, N. D., Vector Variational Inequality as a Tool for Studying Vector Optimization Problems, Nonlinear Analysis, Vol.34, 745–765, 1998. · Zbl 0956.49007
[6] LEE, G. M., On Relations between Vector Variational Inequality and Vector Optimization Problem, Progress in Optimization, X. Q. Yang, A. I. Mees, M. Fisher, and L. Jennings, Kluwer Academic Publishers, Dordrecht, Netherlands, 2000. · Zbl 0969.49003
[7] WARD, D. E., and LEE, G. M., On Relations between Vector Optimization Problems and Vector Variational Inequalities, Journal of Optimization Theory and Applications, Vol.113, 583–596, 2002. · Zbl 1022.90024
[8] BAZARAA, M. S., SHERALI, H. D., and SHETTY, C. M., Nonlinear Programming: Theory and Applications, John Wiley, New York, NY, 1993.
[9] KARAMARDIAN, S., and SCHAIBLE, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol.66, 37–46, 1990. · Zbl 0679.90055
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.