Abstract
In this paper, we study quadratic complementarity problems, which form a subclass of nonlinear complementarity problems with the nonlinear functions being quadratic polynomial mappings. Quadratic complementarity problems serve as an important bridge linking linear complementarity problems and nonlinear complementarity problems. Various properties on the solution set for a quadratic complementarity problem, including existence, compactness and uniqueness, are studied. Several results are established from assumptions given in terms of the comprising matrices of the underlying tensor, henceforth easily checkable. Examples are given to demonstrate that the results improve or generalize the corresponding quadratic complementarity problem counterparts of the well-known nonlinear complementarity problem theory and broaden the boundary knowledge of nonlinear complementarity problems as well.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cottle, R.W., Pang, J.-S., Stone, R.: The Linear Complementarity Problem, Computer Science and Scientific Computing. Academic Press, Boston (1992)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1 and 2. Springer, New York (2003)
Huang, Z.H., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)
Sturmfels, B.: Solving Systems of Polynomial Equations. AMS, Providence (2002)
Mhiri, M., Prigent, J.L.: International portfolio optimization with higher moments. Int. J. Econ. Finance 2, 157–169 (2010)
Song, Y.S., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078 (2016)
Song, Y.S., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. arXiv:1412.0113 (2014)
Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)
Song, Y.S., Yu, G.H.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85–96 (2016)
Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)
Fan, J.Y., Nie, J., Zhou, A.: Tensor eigenvalue complementarity problems. Math. Program. (2017) (to appear)
Ling, C., He, H.J., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63, 143–168 (2016)
Landsberg, J.: Tensors: Geometry and Applications. Graduate Studies in Mathematics, vol. 128. AMS, Providence (2012)
Frank, M., Wolfe, P.: An algorithm for quadratic programming. Nav. Res. Logist. Q. 3, 95–110 (1956)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Boston (1970)
Andronov, V.G., Belousov, E.G., Shironin, V.M.: On solvability of the problem of polynomial programming. Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika, 4, 194–197 (1982), (in Russian). Translation Appeared in News of the Academy of Science of USSR, Dept. of Technical Sciences, Technical Cybernetics, 4, 194–197 (1982)
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11431002, 11401428, and 11771328), and Innovation Research Foundation of Tianjin University (Grant Nos. 2017XZC-0084 and 2017XRG-0015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Liqun Qi.
Rights and permissions
About this article
Cite this article
Wang, J., Hu, S. & Huang, ZH. Solution Sets of Quadratic Complementarity Problems. J Optim Theory Appl 176, 120–136 (2018). https://doi.org/10.1007/s10957-017-1205-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-017-1205-1