TWO-STAGE STOCHASTIC PROGRAMMING WITH LINEARLY BI-PARAMETERIZED QUADRATIC RECOURSE

This paper studies the class of two-stage stochastic programs with a linearly bi-parameterized recourse function defined by a convex quadratic program. A distinguishing feature of this new class of nonconvex 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. While a recent result has established that the resulting recourse function is of the difference-of-convex (dc) kind, the associated dc decomposition of the recourse function does not provide an easy way to compute a directional stationary solution of the two-stage stochastic program. Based on an implicit convex-concave property of the bi-parameterized recourse function, we introduce the concept of a generalized critical point of such a recourse function and provide a sufficient condition for such a point to be a directional stationary point of the stochastic program. We describe an iterative algorithm that combines regularization, convexification, and sampling and establish the subsequential convergence of the algorithm to a generalized critical point, with probability