Abstract
There is a rich literature devoted to the eigenvalue analysis of variational inequalities. Of special interest is the case in which the constraint set of the variational inequality is a closed convex cone. The set of eigenvalues of a matrix A relative to a closed convex cone K is called the K-spectrum of A. Cardinality and topological results for cone spectra depend on the kind of matrices and cones that are used as ingredients. It is important to distinguish for instance between symmetric and nonsymmetric matrices and, on the other hand, between polyhedral and nonpolyhedral cones. However, more subtle subdivisions are necessary for having a better understanding of the structure of cone spectra. This work elaborates on this issue.
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References
Adly, S., Rammal, H.: A new method for solving second-order cone eigenvalue complementarity problems. J. Optim Theory Appl. 165, 563–585 (2015)
Baker, G.P.: Theory of cones. Linear Algebra Appl. 39, 263–291 (1981)
Chen, J.-S., Pan, S.: Semismooth Newton methods for the cone spectrum of linear transformations relative to Lorentz cones. Linear Nonlinear Anal. 1, 13–36 (2015)
Gajardo, P., Seeger, A.: Equilibrium problems involving the Lorentz cone. J Global Optim. 58, 321–340 (2014)
Fernandes, L., Fukushima, M., Júdice, J., Sherali, H.: The second-order cone eigenvalue complementarity problem. Optim. Methods Softw. 31, 24–52 (2016)
Fernandes, R., Júdice, J., Trevisan, V.: Complementarity eigenvalue of graphs. Linear Algebra Appl. 527, 216–231 (2017)
Hantoute, A.: Contribution à la sensibilité et à la stabilité en optimisation et en théorie métrique des points critiques. Ph.D. Thesis, Université Paul Sabatier, Toulouse (2003)
Hiriart-Urruty, J.B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52, 593–629 (2010)
Holubová, G., Nečesal, P.: A note on the relation between the Fučik spectrum and Pareto eigenvalues. J. Math. Anal. Appl. 427, 618–628 (2015)
Iusem, A., Seeger, A.: On pairs of vectors achieving the maximal angle of a convex cone. Math. Program. 104, 501–523 (2005)
Iusem, A., Seeger, A.: Searching for critical angles in a convex cone. Math Program. 120(1), 3–25 (2009)
Iusem, A., Seeger, A.: On convex cones with infinitely many critical angles. Optimization 56, 115–128 (2007)
Kalapodi, A.: Cardinality of accumulation points of infinite sets. Internat. Math. Forum 11, 539–546 (2016)
Kuc̆era, M.: A new method for the obtaining of eigenvalues of variational inequalities of the special type. Comment. Math. Univ. Carolinae 18, 205–210 (1977)
Miersemann, E.: On higher eigenvalues of variational inequalities. Comment. Math. Univ. Carolinae 24, 657–665 (1983)
Quittner, P., A note to, E.: Miersemann’s papers on higher eigenvalues of variational inequalities. Comment Math. Univ. Carolinae 26, 665–674 (1985)
Quittner, P.: Spectral analysis of variational inequalities. Comment. Math. Univ. Carolinae 27, 605–629 (1986)
Resler, J.: Stability of eigenvalues and eigenvectors of variational inequalities. Comment. Math. Univ. Carolinae 29, 541–550 (1988)
Riddell, R.C.: Eigenvalue problems for nonlinear elliptic variational inequalities on a cone. J. Funct. Anal. 26, 333–355 (1977)
Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, Berlin (1998)
Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algebra Appl. 292, 1–14 (1999)
Seeger, A.: Complementarity spectral analysis of connected graphs. Linear Algebra Appl. 543, 205–225 (2018)
Seeger, A.: Spectral classification of convex cones. Positivity 24, 1241–1261 (2020)
Seeger, A., Sossa, D.: Critical angles between two convex cones I : General theory. TOP 24, 44–65 (2016)
Seeger, A., Torki, M.: On eigenvalues induced by a cone-constraint. Linear Algebra Appl. 372, 181–206 (2003)
Seeger, A., Torki, M.: Local minima of quadratic forms on convex cones. J Global Optim. 44, 1–28 (2009)
Seeger, A., Torki, M.: On spectral maps induced by convex cones. Linear Algebra Appl. 592, 65–92 (2020)
Zhang, L.-H., Shen, C., Yang, W.H., Júdice, J.J.: A Lanczos method for large-scale extreme Lorentz eigenvalue problems. SIAM J Matrix Anal. Appl. 39, 611–631 (2018)
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Seeger, A. Cone-Constrained Eigenvalue Problems: Structure of Cone Spectra. Set-Valued Var. Anal 29, 605–619 (2021). https://doi.org/10.1007/s11228-021-00575-3
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DOI: https://doi.org/10.1007/s11228-021-00575-3
Keywords
- Complementarity eigenproblem
- Cone spectrum
- Critical values of quadratic forms on cones
- Nonpolyhedral cones