Generalized Dual Decomposition

Abstract

We study two-stage stochastic optimization models with mixed-integer decision variables appearing in both stages. For these models, dual decomposition enables parallel computing implementation and can quickly provide a lower bound for the optimal value. However, the lower bound thus obtained is not exact in general due to the lack of strong duality. In this paper, we propose a generalized dual decomposition (GDD) that extends the linear regularizer used in dual decomposition to a general nonlinear one, which still admits parallelization while exhibiting strong duality. By encoding the nonlinear regularizers through parameterization and cutting planes, we establish the convergence of a GDD algorithm to achieve global optimum. In addition, we discuss strategies for solving the GDD scenario subproblems more efficiently, including pruning and valid inequalities. Furthermore, we extend GDD to a more general, constrained form that subsumes, as special cases, robust optimization, chance-constrained programs, and bilevel optimization with multiple followers. Finally, numerical experiments demonstrate the strong duality of GDD, its computational efficacy versus primal (Benders-type) decomposition algorithms, and the speedup through parallel computing.