Xu, Huifu (2006) An implicit programming approach for a class of stochastic mathematical programs with complementarity constraints. SIAM Journal on Optimization, 16 (3), 670-696. (doi:10.1137/040608544).
Abstract
In this paper, we consider a class of stochastic mathematical programs in which the complementarity constraints are subject to random factors and the objective function is the mathematical expectation of a smooth function which depends on both upper and lower level variables and random factors. We investigate the existence, uniqueness, and differentiability of the lower level equilibrium defined by the complementarity constraints %and its dependence using a nonsmooth version of implicit function theorem. We also study the differentiability and convexity of the objective function which implicitly depends upon the lower level equilibrium. We propose numerical methods to deal with difficulties due to the continuous distribution of the random variables and intrinsic nonsmoothness of lower level equilibrium solutions due to the complementarity constraints in order that the treated programs can be readily solved by available numerical methods for deterministic mathematical programs with complementarity constraints.
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