Summary
In this paper we will establish some necessary and/or sufficient conditions for both a nonsingular and a singular matrix A (interpreted as a linear map) to be pseudomonotone. The given results are in terms of the sign of the determinants of the principal submatrices and of the cofactors of A in the nonsingular case and in terms of the structure of A in the singular case. A complete characterization of pseudomonotonicity in terms of the coefficients of a 3 × 3 matrix is given and a method for constructing a merely pseudomonotone matrix is suggested.
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
Gantmacher FR (2000) The theory of Matrices, volume one, AMS Chelsea Publishing.
Karamardian S (1976) Complementarity problems over cones with monotone and pseudomonotone maps, Journal of Optimization Theory and Applications 18: 445–454.
Karamardian S, Schaible S, Crouzeix JP (1993) Characterizations of generalized monotone maps, Journal of Optimization Theory and Applications 76: 399–413.
Pini R, Schaible S (1994) Invariance properties of generalized monotonicity, Optimization 28: 211–222.
Crouzeix JP, Schaible S (1996) Generalized monotone affine maps, SIAM Journal Matrix Anal. Appl. 17: 992–997.
Crouzeix JP (1998) Characterizations of generalized convexity and monotonicity, a survey. In Crouzeix P, Martinez-Legaz JE, Volle M (eds) Generalized convexity, generalized monotonicity, Kluwer Academic Publisher, Dordrecht.
Crouzeix JP, Hassouni A, Lahlou A, Schaible S (2000) Positive subdefinite matrices, generalized monotonicity, and linear complementarity problems, SIAM Journal Matrix Anal. Appl. 22: 66–85.
Hadjsavvas N, Schaible S (2005) Generalized monotone maps. In Hadjsavvas N, Komlosi S, Schaible S (eds) Handbook of Generalized Convexity and Generalized Monotonicity, Springer.
Marchi A, Martein L (2005) Pseudomonotonicity of an affine map: the two dimensional case, Technical report, Department of Statistics and Applied Mathematics, University of Pisa, Italy.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cambini, A., Martein, L. (2007). Pseudomonotonicity of a Linear Map on the Interior of the Positive Orthant. In: Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol 583. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-37007-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-37007-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37006-2
Online ISBN: 978-3-540-37007-9
eBook Packages: Business and EconomicsEconomics and Finance (R0)