Multistage stochastic convex programs: Duality and its implications

Working Paper on Augmented Lagrangian Decomposition Methods for Multistage Stochastic Programs on Augmented Lagrangian Decomposition Methods for Multistage Stochastic Programs on Augmented Lagrangian Decomposition Methods for Multistage Stochastic Programs

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two di erent ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computa...

On the Convergence of Decomposition Methods for Multistage Stochastic Convex Programs

We prove the almost-sure convergence of a class of samplingbased nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions, and uncertainty is modelled by a scenario tree. As special cases, our results imply the almost-sure convergence of SDDP, CUPPS and DOASA when applied to problems with general convex cost...

Duality gaps in nonconvex stochastic optimization

We consider multistage stochastic optimization models containing nonconvex constraints, e.g., due to logical or integrality requirements. We study three variants of Lagrangian relaxations and of the corresponding decomposition schemes, namely, scenario, nodal and geographical decomposition. Based on convex equivalents for the Lagrangian duals, we compare the duality gaps for these decomposition...

Convergence Analysis of Sampling-Based Decomposition Methods for Risk-Averse Multistage Stochastic Convex Programs

We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. We prove the al...

Hierarchical Sparsity in Multistage Convex Stochastic Programs