Abstract
Saddle point conditions of suitable functions are equivalent to particular classes of variational inequalities. Some existence theorems for variational inequalities, based on monotonicity assumptions on the operator F, allow to prove these saddle point conditions. Applications to the generalized complementarity problem and extensions to vector variational inequalities are considered.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R.W. Cottle and J.C. Yao, Pseudomonotone Complementarity Problems in Hilbert Space, Journal of Optimization Theory and Applications 75 (1992), 281–295.
R.W. Cottle, Nonlinear Programs with positively bounded Jacobians, SIAM Journal on Applied Mathematics 14 (1966) , 147–158.
K. Fan, A Minimax Inequality and Applications, in “Inequalities III” O. Shisha,(eds.), Academic Press, New York-London, 1972.
F. Giannessi, On Minty Variational Principle, to appear in “New Trends in Mathematical Programming”, Giannessi, Komlosi, Rapcsak (eds.);
F. Giannessi, Theorems of the Alternative, Quadratic Programs and Complementarity Problems, in “Variational Inequalities and Complementarity Problems”,R.W. Cottle, F. Giannessi and J.L. Lions (eds.), Wiley, New York, 1980, 151–186.
F. Giannessi and A.A. Khan, On a Separation Approach to Regularization for Generalized Variational and Quasi- Variational Inequalities, To appear.
S. Karamardian, Generalized Complementary Problem, Journal of Optimization Theory and Applications 8 (1971) , 161–167.
G. Mastroeni, Some relations between Duality theory for extremum problems and Variational Inequalities, Le Matematiche 49 (1994), 2953–304.
B. Ricceri, Existence Theorems for Nonlinear Problems, Rendiconti Accademia Nazionale delle Scienze detta dei XL 11 (1987), 77–99.
T. Tanaka,Generalized Quasiconvexities, Cone Saddle Points, and Minimax Theorem for Vector- Valued Functions, Journal of Optimization Theory and Applications 81 (1994), 355–377.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Mastroeni, G. (1998). Some Remarks on a Minimax Formulation of a Variational Inequality. In: Ricceri, B., Simons, S. (eds) Minimax Theory and Applications. Nonconvex Optimization and Its Applications, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9113-3_13
Download citation
DOI: https://doi.org/10.1007/978-94-015-9113-3_13
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5030-4
Online ISBN: 978-94-015-9113-3
eBook Packages: Springer Book Archive