Abstract
Recently, Gowda and Sossa [Math. Program., 2019, 177: 149–171] studied the existence of solutions to weakly homogeneous variational inequalities. In particular, their main result, based on a degree-theoretic condition and a constraint on the corresponding cone complementarity problem, covers a majority of existence results on the subcategory problems of weakly homogeneous variational inequalities. In this paper, what we achieve is a new copositivity-type existence result for the weakly homogeneous variational inequality. The conditions we used are easier to check than the degree-theoretic condition and our result crosses each other with the main result established by Gowda and Sossa and the main result given by Ma, Zheng and Huang [SIAM J. Optim., 2020, 30(1): 132–148], respectively. Besides, we show the distinctiveness of our existence result by comparing it with the well-known coercivity result obtained for variational inequalities and a norm-coercivity result obtained for complementarity problems, respectively.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bai X.L., Zheng M.M., Huang Z.H., Unique solvability of weakly homogeneous generalized variational inequalities. J. Global Optim., 2021, 80(4): 921–943
Ding W.Y., Wei Y.M., Solving multi-linear systems with ℳ-tensors. J. Sci. Comput., 2016, 68(2): 689–715
Facchinei F., Pang J.S., Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I. Springer Series in Operations Research, New York: Springer-Verlag, 2003
Gowda M.S., Polynomial complementarity problems. Pac. J. Optim., 2017, 13(2): 227–241
Gowda M.S., Sossa D., Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones. Math. Program., 2019, 177: 149–171
Han L.X., A homotopy method for solving multilinear systems with ℳ-tensors. Appl. Math. Lett., 2017, 69: 49–54
Harker P.T., Pang J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program., 1990, 48(2): 161–220
He H.J., Ling C., Qi L.Q., Zhou G.L., A globally and quadratically convergent algorithm for solving multilinear systems with ℳ-tensors. J. Sci. Comput., 2018, 76(3): 1718–1741
Hieu V.T., Solution maps of polynomial variational inequalities. J. Global Optim., 2020, 77(4): 807–824
Huang Z.H., Qi L.Q., Tensor complementarity problems—part I: Basic theory. J. Optim. Theory Appl., 2019, 183(1): 1–23
Huang Z.H., Qi L.Q., Tensor complementarity problems—part III: Applications. J. Optim. Theory Appl., 2019, 183(3): 771–791
Karamardian S., An existence theorem for the complementarity problem. J. Optim. Theory Appl., 1976, 19(2): 227–232
Ling L.Y., He H.J., Ling C., On error bounds of polynomial complementarity problems with structured tensors. Optimization, 2018, 67(2): 341–358
Ling L.Y., Ling C., He H.J., Properties of the solution set of generalized polynomial complementarity problems. Pac. J. Optim., 2020, 16(1): 155–174
Ma X.X., Zheng M.M., Huang Z.H., A note on the nonemptiness and compactness of solution sets of weakly homogeneous variational inequalities. SIAM J. Optim., 2020, 30(1): 132–148
Qi L.Q., Huang Z.H., Tensor complementarity problems—part III: Solution methods. J. Optim. Theory Appl., 2019, 183(2): 365–385
Rockafellar R.T., Convex Analysis. Princeton Mathematical Series, No. 28. Princeton, NJ: Princeton University Press, 1970
Wang Y., Huang Z.H., Qi L.Q., Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl., 2018, 177(1): 137–152
Xie Z.J., Jin X.Q., Wei Y.M., Tensor methods for solving symmetric ℳ-tensor systems. J. Sci. Comput., 2018, 74(1): 412–425
Zheng M.M., Huang Z.H., Bai X.L., Nonemptiness and compactness of solution sets to weakly homogeneous generalized variational inequalities. J. Optim. Theory Appl., 2021, 189(3): 919–937
Zheng M.M., Huang Z.H., Ma X.X., Nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. J. Optim. Theory Appl., 2020, 185(1): 80–98
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (Nos. 11871051, 12371309) and the Natural Science Foundation of Hunan Province, China (No. 2022JJ40543).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Rights and permissions
About this article
Cite this article
Zheng, M., Huang, Z. A Copositivity-type Existence Result for Weakly Homogeneous Variational Inequalities. Front. Math 19, 559–576 (2024). https://doi.org/10.1007/s11464-021-0269-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-021-0269-2