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Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. (English) Zbl 1168.90008
The authors consider the stochastic mathematical programs with linear complementarity constraints, which include two kinds of models: the old one, so-called lower-level wait-and-see model, and the new model called here-and-now. They study mainly the following here-and-now model:
\begin{aligned} &\underset{x,y,z}{\text{minimize}}\;E_\omega[f(x,y,\omega)+d^Tz(\omega)]\\ &\text{subject to}\quad x\in X,\quad y\geq 0,\quad F(x,y,\omega)+z(\omega)\geq 0,\\ &y^T(F(x,y,\omega)+z(\omega))=0,\quad z(\omega)\geq 0,\quad \omega\in\Omega\;\text{a.e.} \end{aligned}
Here the mapping $$F$$ is affine, $$d$$ is a vector with positive elements, $$\omega$$ is a discrete or continuous random variable; and the rest of the notations are conventional ones. The continuous problem is discretized by a quasi-Monte Carlo method.
The authors present a combined smoothing implicit programming and penalty method with appropriate convergence results. The numerical results (for a picnic vender decision problem) are also present.

##### MSC:
 90C15 Stochastic programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C30 Nonlinear programming
##### Keywords:
wait-and-see; here-and-now; quasi-Monte Carlo method
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