Conference Agenda

A common way to model stochastic uncertainties in the constraints are chance constraints. We present a gradient-based algorithm to approximately solve the difficult class of joint chance-constrained optimization problems with continuous distributions and non-convex constraint functions. To apply our method we approximate the original problem by smoothing indicator functions and penalize a violation of the smoothed constraint in the objective function. The resulting approximation problem is solved with the Continuous Stochastic Gradient method that has recently been introduced in the literature. It is a variant of the stochastic gradient descent with improved convergence properties. Under very mild assumptions our approach is applicable to a wide range of joint chance-constrained optimization problems. Its efficiency in practice is illustrated with an application to gas networks. The experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances. Additionally, we provide a theoretical convergence theory for the smoothing and penalty approximations with sequences of increasing smoothing and penalty parameters.

In operations management, making decisions in the face of progressively revealed uncertainty presents a central challenge with vast applications across numerous sectors, including manufacturing, logistics, and finance. Against this background, we study generalized multi-stage stochastic programs that unify several commonly studied problem formulations, including stochastic, robust, and data-driven optimization. To efficiently find solutions for this new problem formulation, we generalize a class of piecewise affine policies that were previously studied for stochastic and robust settings but were not applicable to data-driven settings. While these policies yield significant improvements over affine policies in the stochastic optimization setting, no improvements were observed in the robust setting. We provide theoretical justification for this observation and overcome this shortcoming by introducing a new class of cuts that tightens the piecewise affine policy approximation. While the resulting separation problem is NP-hard in general, we identify favorable properties - fulfilled by many commonly studied uncertainty representations, including budgeted and ellipsoidal uncertainty - that render the problem tractable. Our numerical experiments reveal that our policies' new class of cuts leads to objective improvements over previous policies of up to 10% in data-driven settings and up to 20% in robust settings.

Multistage stochastic integer programs are notorious due to their rapidly growing complexity by increasing stages. Several decomposition techniques have been developed using piece-wise convex polyhedra for under-approximating cost-to-go functions. Examples of these techniques include nested decomposition and dual dynamic integer programs. Nevertheless, the performance of these approaches heavily depends on the optimality cut generated in each iteration during the search process. In this study, we develop a new stochastic nested decomposition algorithm for solving multistage integer programs with quadratic objective functions containing binary state variables. We introduce a novel type of Lagrangian cut to improve the algorithm's performance. Furthermore, we enhance the implementation performance by selectively bypassing specific algorithm steps under certain conditions. Our experiments focus on a reconfigurable matrix assembly layout design problem under uncertain demand. We compare our algorithm using the developed Lagrangian cut against classical nested decomposition and extensive mixed integer linear programming formulations solved with commercial solvers. Finally, we analyze the performance changes when the scenario tree incorporates nodes that contain shared child nodes and stage-wise independence.