This paper studies the class of two-stage stochastic programs (SP) with a linearly bi-parameterized recourse function defined by a convex quadratic program. A distinguishing feature of this new class of stochastic programs is that the objective function in the second stage is linearly parameterized by the first-stage decision variable, in addition to the standard linear parameterization in the constraints. Inspired by a recent result that establishes the difference-of-convexity (dc) property of such a recourse function, we analyze the almost-sure subsequential convergence of a successive sample average approximation (SAA) approach combined with the difference-of-convex algorithm (DCA) for computing a directional derivative based stationary solution of the nonconvex SP. The analysis is divided into two main cases: one, the problem admits an explicit, computationally viable dc decomposition with a differentiable concave component; and two, an implicit bivariate convex-concave property can be identified. The first case includes a strictly convex second-stage objective and a few special instances where the second-stage recourse is convex but not strictly convex. A general convex second-stage recourse function belongs to the second main case; this case requires the introduction of the notion of a generalized critical point to which the almost-sure subsequential convergence of the combined SAA, DCA and regularization is established. Overall, this research provides the first step in the investigation of this class of two-stage SPs that to the best of our knowledge, has not been the object of a focused study in the literature of two-stage SP.
The University of Southern California, December/2018
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