Abstract
An interesting observation is that most pairs of weakly homogeneous mappings do not possess strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper focuses on studying the uniqueness and solvability of the generalized variational inequality with a pair of weakly homogeneous mappings. By using a weaker condition than the strong monotonicity and some additional conditions, we achieve several results on the unique solvability to the underlying problem, which are exported by making use of the exceptional family of elements. As an adjunct, we also obtain the nonemptiness and compactness of the solution sets to the weakly homogeneous generalized variational inequality under some appropriate conditions. The conclusions presented in this paper are new or supplements to the existing ones even when the problem comes down to its important subclasses studied in recent years.
Similar content being viewed by others
References
Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)
Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic, Boston (1992)
Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems, vol. I. Spinger, New York (2003)
Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems, vol. II. Spinger, New York (2003)
Gowda, M.S.: Applications of degree theory to linear complementarity problems. Math. Oper. Res. 18(4), 868–879 (1993)
Gowda, M.S.: Polynomial complementarity problems. Pac. J. Optim. 13(2), 227–241 (2017)
Gowda, M.S., Sossa, D.: Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones. Math. Program. 177, 149–171 (2019)
Han, J.Y., Huang, Z.H., Fang, S.C.: Solvability of variational inequality problems. J. Optim. Theory Appl. 122(3), 501–520 (2004)
Han, J.Y., Xiu, N.H., Qi, H.D.: Nonlinear Complementarity Theorey and Algorithms. Shanghai Science and Technology Press, Shanghai (2006). (in Chinese)
Hieu, V.T.: Solution maps of polynomial variational inequalities. J. Global Optim. 77, 807–824 (2020)
Huang, Z.H., Qi, L.Q.: Tensor complementarity problems part I: basic theory. J. Optim. Theory Appl. 183(1), 1–23 (2019)
Huang, Z.H., Qi, L.Q.: Tensor complementarity problems part III: applications. J. Optim. Theory Appl. 183(3), 771–791 (2019)
Isac, G., Bulavski, V., Kalashnikov, V.: Exceptional families, topological degree and complementarity problems. J. Global Optim. 10, 207–225 (1997)
Kanzow, C., Fukushima, M.: Equivalence of the generalized complementarity problem to differentiable unconstrained minimization. J. Optim. Theory Appl. 90, 581–603 (1996)
Karamardian, S.: Generalized complementarity problem. J. Optim. Theory Appl. 8(3), 161–168 (1971)
Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)
Ling, L.Y., He, H.J., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67(2), 341–358 (2018)
Ling, L.Y., Ling, C., He, H.J.: Properties of the solution set of generalized polynomial complementarity problems. Pac. J. Optim. 16(1), 155–174 (2020)
Liu, D.D., Li, W., Vong, S.W.: Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear A. 66(9), 1726–1749 (2018)
Lloyd, N.G.: Degree Theory. Cambridge University Press, Cambridge (1978)
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. 30(1), 132–148 (2020)
Megiddo, N., Kojima, M.: On the existence and uniqueness of solutions in nonlinear complementarity problems. Math. Program. 12, 110–130 (1977)
Moré, J.J.: Classes of functions and feasibility conditions in nonlinear complementarity problems. Math. Program. 6, 327–338 (1974)
Noor, M.A.: Quasi variational inequalities. Appl. Math. Lett. 1(4), 367–370 (1988)
Ortega, J.M., Rheinholdt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Pang, J.S., Yao, J.C.: On a generalization of a normal map and equation. SIAM J. Control Optim. 33, 168–184 (1995)
Wang, J., Huang, Z.H., Xu, Y.: Existence and uniqueness of solutions of the generalized polynomial variational inequality. Optim. Lett. 14, 1571–1582 (2020)
Wang, Y., Huang, Z.H., Qi, L.Q.: Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl. 177(1), 137–152 (2018)
Zhao, Y.B., Isac, G.: Quasi-\(P^*\)-maps, \(P(\tau,\alpha,\beta )\)-maps, exceptional family of elements and complementarity problem. J. Optim. Theory Appl. 105, 213–231 (2000)
Zheng, M.M., Huang, Z.H., Ma, X.X.: Nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. J. Optim. Theory Appl. 185, 80–98 (2020)
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (Grant No. 11871051).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bai, X., Zheng, M. & Huang, ZH. Unique solvability of weakly homogeneous generalized variational inequalities. J Glob Optim 80, 921–943 (2021). https://doi.org/10.1007/s10898-021-01040-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-021-01040-z