Expected Residual Minimization Method for a Class of Stochastic Tensor Variational Inequalities

Expand
  • 1 School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning 530006, Guangxi, China;
    2 School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, Guangxi, China;
    3 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received date: 2023-08-15

  Revised date: 2024-02-08

  Online published: 2026-07-06

Supported by

The research was partially supported by Guangxi Natural Science Foundation (No. 2024GXNSFBA010345), Guangxi Science and Technology Plan Project (No. guikeAD22035021), the Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (No. 2022KY0163), the National Natural Science Foundation of China (No.12261008), the XiangsihuYoung Scholars and Innovative Research Team of GXMZU (No. 2022GXUNXSHQN02).

Abstract

This paper considers the expected residual minimization (ERM) formulation for a class of stochastic tensor variational inequalities (STVI) where the involved set contains 0. Initially, we derive some theoretical results regarding the H-eigenvalues of tensors and formulate a class of stochastic multi-person nonoperative games as an STVI. Subsequently, we transform the STVI into an ERM problem by using the regularized gap function and explore the properties of the object function. Furthermore, we use the quasi-Monte Carlo method to address the ERM problem and conduct convergence analysis. Ultimately, we conduct numerical experiments to validate our theoretical findings.

Cite this article

Jian-Xun Liu, Zhao-Feng Lan, Sheng-Jie Li . Expected Residual Minimization Method for a Class of Stochastic Tensor Variational Inequalities[J]. Journal of the Operations Research Society of China, 2026 , 14(2) : 565 -590 . DOI: 10.1007/s40305-024-00569-6

References

[1] Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84(2016)
[2] Barbagallo, A., Guarino Lo Bianco, S.: Variational inequalities on a class of structured tensors. J. Nonlinear Convex Anal. 19, 711–729(2018)
[3] Che, M.L., Qi, L.Q., Wei, Y.M.: Stochastic R0 tensors to stochastic tensor complementarity problems. Optim. Lett. 13(2), 261–279(2019)
[4] Chen, H.B., Chen, Y.N., Li, G.Y., Qi, L.Q.: A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test. Numer. Linear Algebra Appl. 25(1), e2125(2018)
[5] Chen,X.,Wets,R.J.B.,Zhang,Y.:Stochasticvariationalinequalities:residualminimizationsmoothing sample average approximations. SIAM J. Optim. 22(2), 649–673(2012)
[6] Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117(1–2), 51–80(2009)
[7] Chen, X.J., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30(4), 1022–1038(2005)
[8] Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Inc., New York (NY) (1992)
[9] Du, S.Q., Che, M.L., Wei, Y.M.: Stochastic structured tensors to stochastic complementarity problems. Comput. Optim. Appl. 75(3), 649–668(2020)
[10] Du, S.Q., Cui, L.Y., Chen, Y.Y., Wei, Y.M.: Stochastic tensor complementarity problem with discrete distribution. J. Optim. Theory Appl. 192(3), 912–929(2022)
[11] Du, S.Q., Zhang, L.P.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73(4), 789–800(2019)
[12] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003)
[13] Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53(1), 99–110(1992)
[14] Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities, with applications to option pricing. In: Proceedings of the 28th Conference on Winter Simulation, pp. 337–344(1996)
[15] Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84(2), 313–333(1999)
[16] Huang, Z.H., Qi, L.Q.: Formulating an n-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576(2017)
[17] Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884(2002)
[18] Lin, G.H., Chen, X., Fukushima, M.: New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC. Optimization 56(5–6), 641–653(2007)
[19] Liu, J.X., Li, S.J.: Unconstrained optimization reformulation for stochastic nonlinear complementarity problems. Appl. Anal. 100(6), 1158–1179(2021)
[20] Liu, J.X., Li, S.J., Xu, Y.R.: Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. J. Ind. Manag. Optim. 18(4), 2599–2610(2022)
[21] Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116(2009)
[22] Ma, H.Q., Wu, M., Huang, N.J., Xu, J.P.: Expected residual minimization method for stochastic variational inequality problems with nonlinear perturbations. Appl. Math. Comput. 219(11), 6256– 6267(2013)
[23] Manicino, O.G., Stampacchia, G.: Convex programming and variational inequalities. J. Optim. Theory Appl. 9(1), 3–23(1972)
[24] Ming, Z., Zhang, L.P., Qi, L.Q.: Expected residual minimization method for monotone stochastic tensor complementarity problem. Comput. Optim. Appl. 77(3), 871–896(2020)
[25] Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
[26] Patrick, B.: Probability and Measure. Wiley, New York (1995)
[27] Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324(2005)
[28] Qi, L.Q., Chen, H.B., Chen, Y.N.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)
[29] Shang, T.T., Tang, G.J.: Expected residual minimization method for stochastic tensor variational inequalities. J. Oper. Res. Soc. China (2022). https://doi.org/10.1007/s40305-022-00450-4
[30] Shang, T.T., Yang, J., Tang, G.J.: Generalized polynomial complementarity problems over a polyhedral cone. J. Optim. Theory Appl. 192(2), 443–483(2022)
[31] Song, Y.S., Qi, L.Q.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078(2016)
[32] Song, Y.S., Qi, L.Q.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33(3), 308–323(2017)
[33] Wang, Y., Huang, Z.H., Qi, L.Q.: Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl. 177, 137–152(2018)
[34] Xu, H.F.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia Pac. J. Oper. Res. 27(01), 103–119(2010)
[35] Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277–295(2008)
Options
Outlines

/