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Stochastic $R_0$ matrix linear complementarity problems. (English) Zbl 1151.90052
The authors consider the expected residual minimization method (ERM) for solving stochastic linear complementarity problems $$x \geq 0 , ~~ M(\omega) x + q(\omega) \geq 0, ~~ x^T(M(\omega) x + q(\omega)) = 0 .$$ This problem is transformed to a minimization problem $\min G(x) \text{ s.t. } x \geq 0$. The study is based on the concept of stochastic $R_0$ matrices. It is shown, that the ERM problem is solvable for any $q(\cdot)$ if and only if $M(\cdot)$ is a stochastic $R_0$ matrix. The differentiability of $G(x)$ is analysed under a certain strict complementarity condition with probability one. Necessary an sufficient optimality conditions for a solution $\overline{x}$ are given together with error bounds. Finally the authors report on experiments for solving ERM numerically. The stochastic complementarity concept is applied to a traffic equilibrium flow and a control problem.

##### MSC:
 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 90C15 Stochastic programming
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