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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 and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C15 Stochastic programming
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