Abstract
We consider the stochastic linear complementarity problem (SLCP) involving a random matrix whose expectation matrix is positive semi-definite. We show that the expected residual minimization (ERM) formulation of this problem has a nonempty and bounded solution set if the expected value (EV) formulation, which reduces to the LCP with the positive semi-definite expectation matrix, has a nonempty and bounded solution set. We give a new error bound for the monotone LCP and use it to show that solutions of the ERM formulation are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in SLCP. Numerical examples including a stochastic traffic equilibrium problem are given to illustrate the characteristics of the solutions.
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References
Birge J.R. and Louveaux F. (1997). Introduction to Stochastic Programming. Springer, New York
Chen B., Chen X. and Kanzow C. (2000). A penalized Fischer–Burmeister NCP-function. Math. Program. 88: 211–216
Chen X. and Fukushima M. (2005). Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30: 1022–1038
Cottle R.W., Pang J.-S. and Stone R.E. (1992). The Linear Complementarity Problem. Academic, New York
Daffermos S. (1980). Traffic equilibrium and variational inequalities. Transportation Sci. 14: 42–54
Facchinei F. and Pang J.-S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, I and II. Springer, New York
Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)
Ferris, M.C.: ftp://ftp.cs.wisc.edu/math-prog/matlab/lemke.m, Department of Computer Science, University of Wisconsin (1998)
Ferris M.C. and Pang J.-S. (1997). Engineering and economic applications of complementarity problems. SIAM Rev. 39: 669–713
Frank M. and Wolfe P. (1956). An algorithm for quadratic programming. Nav. Res. Logist. 3: 95–110
Gürkan G., Özge A.Y. and Robinson S.M. (1999). Sample-path solution of stochastic variational inequalities. Math. Program. 84: 313–333
Kall P. and Wallace S.W. (1994). Stochastic Programming. Wiley, Chichester
Kanzow C., Yamashita N. and Fukushima M. (1997). New NCP-functions and their properties. J. Optim. Theory Appl. 94: 115–135
Kleywegt A.J., Shapiro A. and Homen-De-Mello T. (2001). The sample average approximation method for stochastic disete optimization. SIAM J. Optim. 12: 479–502
Lin G.H. and Fukushima M. (2006). New reformulations for stochastic nonlinear complementarity problems. Optim. Methods Softw. 21: 551–564
Luo Z.-Q. and Tseng P. (1997). A new class of merit functions for the nonlinear complementarity problem. In: Ferris, M.C. and Pang, J.-S. (eds) Complementarity and Variational Problems: State of the Art., pp 204–225. SIAM, Philadelphia
Mangasarian O.L. and Ren J. (1994). New improved error bounds for the linear complementarity problem. Math. Program. 66: 241–255
Marti K. (2005). Stochastic Optimization Methods. Springer, Berlin
Tseng P. (1996). Growth behavior of a class of merit functions for the nonlinear complementarity problem. J. Optim. Theory Appl. 89: 17–37
Sun D. and Qi L. (1999). On NCP-functions. Comp. Optim. Appl. 13: 201–220
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It is our great pleasure and honor to dedicate this paper to Professor Steve Robinson on the occasion of his 65th birthday. His original ideas and deep insight have always been so inspiring and beneficial to our work. This paper is just one of such instances.
This work was supported in part by a Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.
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Chen, X., Zhang, C. & Fukushima, M. Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009). https://doi.org/10.1007/s10107-007-0163-z
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DOI: https://doi.org/10.1007/s10107-007-0163-z