Abstract
Backward error analysis reveals the numerical stability of algorithms and provides elaborate stopping criteria for iterative methods. Compared with numerical linear algebra problems, the backward error analysis for optimization problems is more rarely conducted in the literature. This paper is devoted to the backward error analysis for several generalizations of tensor complementarity problems. We first present sufficient and necessary conditions for the acceptable solutions for the extended tensor complementarity problem, the vertical tensor complementarity problem, and an extended form of tensor complementarity problem. Next, the backward errors for tensor complementarity problem are also proposed, which can be used to verify the stability of the tensor complementarity problem algorithms. Finally, some numerical examples are reported to illustrate the proposed backward errors for tensor complementarity problems.
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References
Pang, J.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)
Klatte, D., Thiere, G.: Error bounds for solutions of linear equations and inequalities. Z. Oper. Res. 41, 191–214 (1995)
Gwinner, J.: Acceptable solutions of linear complementarity problems. Computing 40, 361–366 (1988)
Gwinner, J.: A note on backward error analysis for generalized linear complementarity problems. Ann. Oper. Res. 101, 391–399 (2001)
Fang, Y., Huang, N.: A characterization of an acceptable solution of the extended nonlinear complementarity problem. Z. Angew. Math. Mech. 84, 564–567 (2004)
Zheng, M., Zhang, Y., Huang, Z.: Global error bounds for the tensor complementarity problem with a \(P\)-tensor. J. Ind. Manag. Optim. 15, 933–946 (2019)
Ling, L., He, H., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67, 341–358 (2018)
Hu, S., Wang, J., Huang, Z.: Error bounds for the solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 179, 983–1000 (2018)
Mangasarian, O.: A condition number for linear inequalities and equalities. Methods Oper. Res. 43, 3–15 (1981)
Wilkinson, J.: Rounding Errors in Algebraic Processes. NJ, Prentice-Hall, Inc, Englewood Cliffs (1963)
Oettli, W., Prager, W.: Compatibility of approximate solutions of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math. 6, 405–409 (1964)
Schaback, R.: Eine rundungsgenaue formel Zur maschinellen berechnung der Prager–Oettli-Schranke. Computing 20, 177–182 (1978)
Wu, X., Ke, R.: Backward errors of the linear complementarity problem. Numer. Algorithms 83, 1249–1257 (2020)
Cottle, R., Pang, J., Stome, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)
Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)
Che, M., Wei, Y.: Theory and Computation of Complex Tensors and its Applications. Springer, Singapore (2020)
Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)
Huang, Z., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)
Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85–96 (2016)
Liu, D., Li, W., Vong, S.: Tensor complementarity problems: the GUS-property and an algorithm. Linear and Multilinear Algebra 66, 1726–1749 (2018)
Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789–800 (2019)
Huang, Z., Qi, L.: Tensor complementarity problems-Part I: Basic theory. J. Optim. Theory Appl. 183, 1–23 (2019)
Qi, L., Huang, Z.: Tensor complementarity problems-Part II: solution methods. J. Optim. Theory Appl. 183, 365–835 (2019)
Huang, Z., Qi, L.: Tensor complementarity problems-Part III: applications. J. Optim. Theory Appl. 183, 771–197 (2019)
Du, S., Che, M., Wei, Y.: Stochastic structured tensors to stochastic complementarity problems. Comput. Optim. Appl. 75, 649–668 (2020)
Hieu, V., Wei, Y., Yao, J.: Notes on the optimization problems corresponding to polynomial complementarity problems. J. Optim. Theory Appl. 184, 687–695 (2020)
Wang, X., Che, M., Qi, L., Wei, Y.: Modified gradient dynamic approach to the tensor complementarity problem. Optim. Methods Softw. 35, 394–415 (2020)
Wang, X., Che, M., Wei, Y.: Global uniqueness and solvability of tensor complementarity problems for \(\cal{H}_+\)-tensors. Numer. Algorithms 35, 394–415 (2020)
Che, M., Qi, L., Wei, Y.: The generalized order tensor complementarity problems. Numer. Math. Theory Method Appl. 13, 131–149 (2020)
Che, M., Qi, L., Wei, Y.: Stochastic \(R_0\) tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019)
Zheng, M., Huang, Z., Ma, X.: Nonemptiness and compactness of solution sets to GPCPs. J. Optim. Theory Appl. 185, 80–98 (2020)
Mathias, R., Pang, J.: Error bounds for the linear complementarity problem with a \(P\)-matrix. Linear Algebra Appl. 132, 123–136 (1990)
Luo, Z., Mangasarian, O., Ren, J., Solodov, M.: New error bounds for the linear complementarity problem. Math. Oper. Res. 19, 880–892 (1994)
Chen, X., Xiang, S.: Perturbation bounds of \(P\)-matrix linear complementarity problems. SIAM J. Optim. 18, 1250–1265 (2007)
Chen, X., Xiang, S.: Computation of error bounds for \(P\)-matrix linear complementarity problems. Math. Program. 106, 523–525 (2006)
Dai, P.: Error bounds for linear complementarity problems of DB-matrices. Linear Algebra Appl. 434, 830–840 (2011)
Higham, N.: A Survey of Componentwise Perturbation Theory in Numerical Linear Algebra, In: Gautschi, W. (ed.), Mathematics of Computation 1943–1993: A Half Century of Computational Mathematics, volume 48 of Proceedings of Symposia in Applied Mathematics, pp. 49-77, American Mathematical Society, Providence, RI, USA (1994)
Sun, M.: Monotonicity of Mangasarian’s iterative algorithm for generalized linear complementarity problems. J. Math. Anal. Appl. 144, 474–485 (1989)
Ding, W., Wei, Y.: Solving multi-linear systems with \(\cal{M}\)-tensors. J. Sci. Comput. 68, 689–715 (2016)
He, H., Ling, C., Qi, L., Zhou, G.: A globally and quadratically convergent algorithm for solving multilinear systems with \(\cal{M}\)-tensors. J. Sci. Comput. 76, 1718–1741 (2018)
Ding, W., Hou, Z., Wei, Y.: Tensor logarithmic norm and its applicatios. Numer. Linear Algebr. 23, 989–1006 (2016)
Bai, X., Huang, Z., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)
Han, L.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. 180, 949–963 (2019)
Song, Y., Qi, L.: Properties of problem and some classes of structured tensors. Ann. Appl. Math. 33, 308–323 (2017)
Acknowledgements
The authors thank the editor and the anonymous referees for numerous insightful comments, which have greatly improved this paper. S. Du is supported by the National Natural Science Foundation of China under grant 11671220. W. Ding is supported by the National Natural Science Foundation of China under grant 11801479 and the Hong Kong Research Grants Council under grant 12301619. Y. Wei is supported by the National Natural Science Foundation of China under grant 11771099 and the Innovation Program of Shanghai Municipal Education Committee.
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Communicated by Guoyin Li.
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Du, S., Ding, W. & Wei, Y. Acceptable Solutions and Backward Errors for Tensor Complementarity Problems. J Optim Theory Appl 188, 260–276 (2021). https://doi.org/10.1007/s10957-020-01774-y
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DOI: https://doi.org/10.1007/s10957-020-01774-y
Keywords
- Acceptable solutions
- Backward errors
- Tensor complementarity problem
- Extended tensor complementarity problem
- Vertical tensor complementarity problem