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数学进展 - 2020, Vol. 49(1): 101-114
Numerical Method for Stochastic Variational Inequality Problems with Second-order Cone Constraints

史红伶1, 孙菊贺1, 李阳2,*, 李文杰1
SHI Hongling1, SUN Juhe1, LI Yang2, LI Wenjie1

1. 沈阳航空航天大学理学院, 沈阳, 辽宁, 110136;
2. 大连民族大学理学院, 大连, 辽宁, 116650
1. School of Science, Shenyang Aerospace University, Shenyang, Liaoning, 110136, P. R. China;
2. School of Science, Dalian Minzu University, Dalian, Liaoning, 116650, P. R. China

收稿日期: 2019-01-28
出版日期: 2020-03-25
2020, Vol. 49(1): 101-114
DOI: 10.11845/sxjz.2019007b

[436 KB]

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摘要 本文研究二阶锥约束随机变分不等式(SOCCSVI)问题, 运用样本均值近似(SAA)方法结合光滑Fischer-Burmeister互补函数来求解该问题.首先, 将SOCCSVI问题的Karush-Kuhn-Tucker系统转化为与之等价的方程组, 并证明了该方程组的雅可比矩阵的非奇异性.其次, 构造了光滑牛顿算法求解该方程组. 最后, 文章给出了两个数值实验证明了算法的有效性.
关键词 二阶锥约束随机变分不等式样本均值近似方法光滑FB互补函数光滑牛顿法    
Abstract:In this paper, the second order cone-constrained stochastic variational inequality problem is studied. The sample average approximation (SAA) method combined with smooth Fischer-Burmeister function is used to solve this problem. First, the Karush-Kuhn-Tucker system of the SOCCSVI problem is transformed into an equivalent equation system and after that, we prove the nonsingularity of the equation system's Jacobian. Second, the smooth Newton algorithm is constructed to solve the equation system. Finally, there are two numerical experiments to prove the effectiveness of the algorithm.
Key wordssecond order cone-constrained stochastic variational inequality    sample average approximation method    smooth FB complementary function    smooth Newton algorithm
PACS:  O221.5  
基金资助:国家自然科学基金(Nos. 11301348, 11501080)和大连民族大学自主基金(2019).
通讯作者: E-mail: *   
[1] Agdeppa R.P., Yamashita N. and Fukushima M., Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem, Pac. J. Optim., 2010, 6(1): 3-19.
[2] Belknap M.H., Chen C.-H. and Harker P.T., A gradient-based method for analyzing stochastic variational inequalities with one uncertain parameter, Working Paper 00-03-13, Philadelphia, PA: Department of Operations and Information Management, Wharton School, 2000.
[3] 陈爽, 锥约束随机变分不等式的求解及应用, 博士学位论文, 大连: 大连理工大学, 2014.
[4] Faraut, U. and Korányi, A., Analysis on Symmetric Cones, Oxford Mathematical Monographs, New York: Oxford University Press, 1994.
[5] Fukushima M., Luo Z.-Q. and Tseng P., Smoothing functions for second-order-cone complementarity problems, SIAM J. Optim., 2001, 12(2): 436-460.
[6] Gürkan G., Özge A.Y. and Robinson S.M., Sample-path solution of stochastic variational inequalities, with applications to option pricing, In: Proceedings of the 1996 Winter Simulation Conference (Charnes, J.M., Morrice, D.J., Brunner, D.T. and Swain, J.J. eds.), 1996, 337-344.
[7] Gürkan G., Özge A.Y. and Robinson S.M., Sample-path solution of stochastic variational inequalities, Math. Program.,1999, 84(2): Ser. A, 313-333.
[8] Jiang, H.Y. and Xu, H.F., Stochastic approximation approaches to the stochastic variational inequality problem, IEEE Trans. Automat. Control, 2008, 53(6): 1462-1475.
[9] Lin G.H., Luo M.J., Zhang D.L. and Zhang J., Stochastic second-order-cone complementarity problems: expected residual minimization formulation and its applications, Math. Program.,2017, 165(1): Ser. B, 197-233.
[10] Lin G.H., Luo M.J. and Zhang J., Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints, J. Global Optim., 2016, 66(3): 487-510.
[11] Ma P.-F., Chen J.-S., Huang C.-H. and Ko C.-H., Discovery of new complementarity functions for NCP and SOCCP, Comput. Appl. Math., 2018, 37(5): 5727-5749.
[12] Mak W.-K., Morton D.P. and Wood R.K., Monte Carlo bounding techniques for determining solution quality in stochastic programs, Oper. Res. Lett., 1999, 24(1/2): 47-56.
[13] Pan, S.H. and Chen, J.-S., A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions, Comput. Optim. Appl., 2010, 45(1): 59-88.
[14] Qi L.Q., Sun D.F. and Zhou G.L., A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program.,2000, 87(1): Ser. A, 1-35.
[15] Shapiro A., Dentcheva D. and Ruszczynski A., Lectures on Stochastic Programming: Modeling and Theory, Philadelphia, PA: SIAM, 2009.
[16] 孙菊贺, 锥约束变分不等式问题的数值方法的研究, 博士学位论文, 大连: 大连理工大学, 2008.
[17] Van Ryzin, G. and McGill, J.I., Revenue management without forecasting or optimization: an adaptive algorithm for determining airline seat protection levels, Management Science, 2000, 46(6): 760-775.
[18] Yang Z.-P., Zhang J., Zhu X.D. and Lin G.-H., Infeasible interior-point algorithms based on sampling average approximations for a class of stochastic complementarity problems and their applications, J. Comput. Appl. Math., 2019, 352: 382-400.
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