Robust solution of monotone stochastic linear complementarity problems

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.

[1]  Xiaojun Chen,et al.  A penalized Fischer-Burmeister NCP-function , 2000, Math. Program..

[2]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[3]  Gül Gürkan,et al.  Sample-path solution of stochastic variational inequalities , 1999, Math. Program..

[4]  Antonio Alonso Ayuso,et al.  Introduction to Stochastic Programming , 2009 .

[5]  Xiaojun Chen,et al.  Expected Residual Minimization Method for Stochastic Linear Complementarity Problems , 2005, Math. Oper. Res..

[6]  R. Wets,et al.  Stochastic programming , 1989 .

[7]  M. Fukushima,et al.  New NCP-Functions and Their Properties , 1997 .

[8]  Michael C. Ferris,et al.  Complementarity and variational problems : state of the art , 1997 .

[9]  P. Tseng.  Growth behavior of a class of merit functions for the nonlinear complementarity problem , 1996 .

[10]  Gui-Hua Lin,et al.  New reformulations for stochastic nonlinear complementarity problems , 2006, Optim. Methods Softw..

[11]  Michael C. Ferris,et al.  Engineering and Economic Applications of Complementarity Problems , 1997, SIAM Rev..

[12]  Olvi L. Mangasarian,et al.  New improved error bounds for the linear complementarity problem , 1994, Math. Program..

[13]  K. Marti.  Stochastic Optimization Methods , 2005 .

[14]  Alexander Shapiro,et al.  The Sample Average Approximation Method for Stochastic Discrete Optimization , 2002, SIAM J. Optim..

[15]  A. Nagurney.  Migration equilibrium and variational inequalities. , 1989, Economics letters.

[16]  XIAOJUN CHEN,et al.  Stochastic R 0 Matrix Linear Complementarity , 2006 .

[17]  Defeng Sun,et al.  On NCP-Functions , 1999, Comput. Optim. Appl..

[18]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[19]  Xiaojun Chen,et al.  Stochastic R0 Matrix Linear Complementarity Problems , 2007, SIAM J. Optim..

[20]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

[21]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .