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Stochastic structured tensors to stochastic complementarity problems

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Abstract

This paper is concerned with the stochastic structured tensors to stochastic complementarity problems. The definitions and properties of stochastic structured tensors, such as the stochastic strong P-tensors, stochastic P-tensors, stochastic \(P_{0}\)-tensors, stochastic strictly semi-positive tensors and stochastic S-tensors are given. It is shown that the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem has a nonempty and bounded solution set. Interestingly, we partially answer the open questions proposed by Che et al. (Optim Lett 13:261–279, 2019). We also consider the expected value method of stochastic structured tensor complementarity problem with finitely many elements probability space. Finally, based on the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem, a projected gradient method is proposed for solving the stochastic structured tensor complementarity problem and the related numerical results are also given to show the efficiency of the proposed method.

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References

  1. Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem, Computer Science and Scientific Computing. Academic Press Inc., Boston (1992)

    Google Scholar 

  2. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    MATH  Google Scholar 

  3. Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)

    Article  MathSciNet  Google Scholar 

  4. Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85–96 (2016)

    Article  MathSciNet  Google Scholar 

  5. Che, M., Qi, L., Wei, Y.: Positive-definite tensor to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)

    Article  MathSciNet  Google Scholar 

  6. Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci. China Math. 61, 1695–1710 (2018)

    Article  MathSciNet  Google Scholar 

  7. Huang, Z., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bai, X., Huang, Z., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)

    Article  MathSciNet  Google Scholar 

  9. Xie, S., Li, D., Xu, H.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175, 119–136 (2017)

    Article  MathSciNet  Google Scholar 

  10. Liu, D., Li, W., Vong, S.: Tensor complementarity problems: the GVS-property and an algorithm. Linear Multilinear Alegebra 66, 1726–1749 (2018)

    Article  Google Scholar 

  11. Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to \(Z\)-tensor complementarity problems. Optim. Lett. 11, 471–482 (2017)

    Article  MathSciNet  Google Scholar 

  12. Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789–800 (2019)

    Article  MathSciNet  Google Scholar 

  13. Huang, Z., Qi, L.: Tensor complementarity problems-part I: basic theory. J. Optim. Theory Appl. 183, 1–23 (2019)

    Article  MathSciNet  Google Scholar 

  14. Qi, L., Huang, Z.: Tensor complementarity problems-part II: solution methods. J. Optim. Theory Appl. 183, 365–385 (2019)

    Article  MathSciNet  Google Scholar 

  15. Huang, Z., Qi, L.: Tensor complementarity problems-part III: applications. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01573-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, X., Che, M., Wei, Y.: Global uniqueness and solvability of tensor complementarity problems for \({\cal{H}}_+\) tensors. Numer. Algorithms (2019). https://doi.org/10.1007/s11075-019-00769-9

    Article  Google Scholar 

  17. Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 3, 1022–1038 (2005)

    Article  MathSciNet  Google Scholar 

  18. Zhang, C., Chen, X.: Smoothing projected gradient method and its application to stochastic linear complemenatrity problems. SIAM J. Optim. 20, 627–649 (2009)

    Article  MathSciNet  Google Scholar 

  19. Li, X., Liu, H., Huang, Y.: Stochastic \(P\) matrix and \(P_{0}\) matrix linear complementarity problem. J. Syst. Sci. Complex. 31, 123–128 (2011)

    MATH  Google Scholar 

  20. Lin, G., Chen, X., Fukushima, M.: Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. Math. Program. 116, 343–368 (2009)

    Article  MathSciNet  Google Scholar 

  21. Fang, H., Chen, X., Fukushima, M.: Stochastic \(R_{0}\) matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)

    Article  MathSciNet  Google Scholar 

  22. Che, M., Qi, L., Wei, Y.: Stochastic \(R_{0}\) tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019)

    Article  MathSciNet  Google Scholar 

  23. Yuan, P., You, L.: Some remark on \(P\), \(P_0\), \(B\) and \(B_0\) tensors. Linear Algebra Appl. 459, 511–521 (2014)

    Article  MathSciNet  Google Scholar 

  24. Ding, W., Luo, Z., Qi, L.: \(P\) tensors, \(P_0\) tensors, and their applications. Linear Algebra Appl. 555, 336–354 (2018)

    Article  MathSciNet  Google Scholar 

  25. Che, M., Wei, Y.: Randomized algorithms for the approximations of Tucker and the tensor train decompositions. Adv. Comput. Math. 45, 395–428 (2019)

    Article  MathSciNet  Google Scholar 

  26. Qi, L., Luo, Z.: Tensor Analysis. Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)

    Book  Google Scholar 

  27. Guo, Q., Zhang, M., Huang, Z.: Properties of \(S\)-tensors. Linear Multilinear Algebra 67, 685–696 (2019)

    Article  MathSciNet  Google Scholar 

  28. Kleywegt, A., Shapiro, A., Homen-De-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12, 479–502 (2001)

    Article  MathSciNet  Google Scholar 

  29. Calamai, P., More, J.: Projected grdient methods for linearly constrained problems. Math. Program. 39, 93–116 (1987)

    Article  Google Scholar 

  30. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  31. Chen, X., Pong, T.K., Wets, R.J.-B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program. 165, 71–111 (2017)

    Article  MathSciNet  Google Scholar 

  32. Pang, J., Sen, S., Shanbhag, V.: Two-stage non-cooperative games with risk-averse players. Math. Program. 165, 235–290 (2017)

    Article  MathSciNet  Google Scholar 

  33. Rockafellar, R.T., Wets, R.J.-B.: Stochastic variational inequalities: single-stage to multistage. Math. Program. 165, 331–360 (2017)

    Article  MathSciNet  Google Scholar 

  34. Rockafellar, R.T., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174, 453–471 (2019)

    Article  MathSciNet  Google Scholar 

  35. Chen, X., Sun, H., Xu, H.: Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems. Math. Program. 177, 255–289 (2019)

    Article  MathSciNet  Google Scholar 

  36. Chen, X., Shapiro, A., Sun, H.: Convergence analysis of sample average approximation of two-stage stochastic generalized equation. SIAM J. Optim. 29, 135–161 (2019)

    Article  MathSciNet  Google Scholar 

  37. Hieu, V.T.: A regularity condition in polynomial optimization. arXiv:1808.06100 (2018)

  38. Hieu, V.T., Wei, Y., Yao, J.: Notes on the optimization problems corresponding to polynomial complementarity problems. J. Optim. Theory Appl. (To appear) (2020)

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Acknowledgements

We would like to thank Mr. Vu Trung Hieu for his preprints [37, 38] and useful discussions. We are also grateful to the editor and three anonymous referees for valuable and helpful comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, People’s Republic of China

    Shouqiang Du

  2. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Maolin Che

  3. School of Mathematical Science and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

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  1. Shouqiang Du
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  2. Maolin Che
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  3. Yimin Wei
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Correspondence to Yimin Wei.

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Shouqiang Du is supported by the National Natural Science Foundation of China under Grant 11671220 and Shandong Provincial Nature Science Foundation under grant ZR2016AM29. Maolin Che is supported by the Fundamental Research Funds for the Central Universities under Grant JBK1801058 and the National Natural Science Foundation of China under Grant 11901471. Yimin Wei is supported by the National Natural Science Foundation of China under Grant 11771099 and Innovation Program of Shanghai Municipal Education Commission.

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Du, S., Che, M. & Wei, Y. Stochastic structured tensors to stochastic complementarity problems. Comput Optim Appl 75, 649–668 (2020). https://doi.org/10.1007/s10589-019-00144-3

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  • DOI: https://doi.org/10.1007/s10589-019-00144-3

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