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Zbl 1151.90052
Fang, Haitao; Chen, Xiaojun; Fukushima, Masao
Stochastic $R_0$ matrix linear complementarity problems. (English)
[J] SIAM J. Optim. 18, No. 2, 482-506 (2007). ISSN 1052-6234; ISSN 1095-7189/e

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.
[Georg Still (Enschede)]

MSC 2000:: *90C33 Complementarity problems
90C15 Stochastic programming

Keywords: {} stochastic linear complementarity problem; $R_0$ matrix; expected residual minimization.

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