Generalized Dual Decomposition

TL;DR

This paper introduces a generalized dual decomposition method for two-stage stochastic mixed-integer optimization that guarantees strong duality, enabling efficient parallel computation and broad applicability to complex optimization problems.

Contribution

It extends dual decomposition with nonlinear regularizers, ensuring strong duality and convergence, and applies to various complex optimization models.

Findings

GDD demonstrates strong duality in stochastic mixed-integer models.

Numerical results show GDD outperforms traditional Benders-type algorithms.

Parallel implementation significantly speeds up computation.

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.