Pseudomonotone variational inequality problems: Existence of solutions. (English) Zbl 0887.90167

Summary: Necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given.


90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J40 Variational inequalities
Full Text: DOI


[1] A. Auslender, Résolution numérique d’inégalités variationnelles,RAIRO R2 (1973) 67–72.
[2] A. Auslender, Private communication.
[3] A. Auslender, R. Cominetti and J.-P. Crouzeix, Convex functions with unbounded level sets and applications to duality theory,SIAM Journal on Optimization 3 (1993) 669–687. · Zbl 0808.90102
[4] J.-P. Crouzeix and J.A. Ferland, First order criteria for generalized monotone maps,Mathematical Programming 75 (1996) 399–406. · Zbl 0892.49008
[5] J.-P. Crouzeix and S. Schaible, Generalized Monotone affine maps, to appear inSIAM Journal on Matrix Analysis and Applications 17 (1996) 992–997. · Zbl 0868.90108
[6] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations,American Journal of Mathematics 68 (1946) 214–222. · Zbl 0060.40203
[7] P.T. Harker and J.S. Pang, Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications,Mathematical Programming 48 (1990) 161–220. · Zbl 0734.90098
[8] M.S. Gowda, Affine pseudomonotone mappings and the linear complementarity problem,SIAM Journal on Matrix Analysis and Applications 11 (1990) 373–380. · Zbl 0719.90083
[9] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps,Journal of Optimization Theory and Applications 18 (1976) 445–454. · Zbl 0304.49026
[10] S. Karamardian and S. Schaible, Seven kinds of monotone maps,Journal of Optimization Theory and Applications 66 (1990) 37–46. · Zbl 0679.90055
[11] S. Karamardian, S. Schaible and J.-P. Crouzeix, Characterizations of generalized monotone maps,Journal of Optimization Theory and Applications 76 (1993) 399–413. · Zbl 0792.90070
[12] S. Komlosi, Generalized monotonicity and generalized convexity, to appear inJournal of Optimization Theory and Applications.
[13] R. Saigal, Extension of the generalized conplementarity problem,Mathematics of Operations Research 1 (3) (1976) 260–266. · Zbl 0363.90091
[14] S. Schaible, Generalized monotonicity – Concepts and uses, in: F. Giannessi and A. Maugeri, eds.,Variational Inequalities and Network Equilibrium Problems (Plenum, New York, 1995) 289–299. · Zbl 0847.49013
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.