Abstract
We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.
Similar content being viewed by others
References
Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)
Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comp. Optim. Appl. 5, 97–138 (1996)
Chen, X.J., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)
Chen, X.J., Zhang, C.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627–649 (2009)
Chen, X.J., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)
Fang, H.T., Chen, X.J., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)
Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)
Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)
Li, X.L., Liu, H.W., Sun, X.J.: Feasible smooth method based on BarzilaiCBorwein method for stochastic linear complementarity problem. Numer. Algor. 57, 207–215 (2011)
Ling, C., Qi, L., Zhou, G.L., Caccetta, L.: The SC 1 property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36, 456–460 (2008)
Liu, H.W., Huang, Y.K., Li, X.L.: Partial projected Newton method for a class of stochastic linear complementarity problems. Numer. Algor. 58, 593–618 (2011)
Liu, H.W., Huang, Y.K., Li, X.L.: New reformulation and feasible semismooth Newton method for a class of stochastic linear complementarity problems. Appl. Math. Comput. 217, 9723–9740 (2011)
Liu, H.W., Li, X.L., Huang, Y.K.: Solving equations via the trust region and its application to a class of stochastic linear complementarity problems. J. Comput. Math. Appl. 61, 1646–1664 (2011)
Tang, J., Ma, C.F.: A smoothing Newton method for solving a class of stochastic linear complementarity problems. Nonlinear Anal. RWA 6, 3585–3601 (2011)
Xie, Y.J., Ma, C.F.: A smoothing Levenberg-Marquardt algorithm for solving a class of stochastic linear complementarity problem. Appl. Math. Comput. 217, 4459–4472 (2011)
Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14, 1043–1056 (2004)
Zhou, G.L., Caccetta, L.: Feasible semismooth Newton method for a class of stochastic linear complementarity problems. J. Optim. Theory Appl. 139, 379–392 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (NNSFC) under Grant No. 61072144 and No. 61179040 and the Fundamental Research Funds for the Central Universities No. K50513100007.
Rights and permissions
About this article
Cite this article
Huang, Y., Liu, H. & Zhou, S. A Barzilai–Borwein type method for stochastic linear complementarity problems. Numer Algor 67, 477–489 (2014). https://doi.org/10.1007/s11075-013-9803-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9803-y