Multistage Distributionally Robust Mixed-Integer Programming with Decision-Dependent Moment-Based Ambiguity Sets (2002.12518v3)

Abstract: We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in (Delage and Ye, 2010) to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.

Review of Multistage Distributionally Robust Mixed-Integer Programming with Decision-Dependent Moment-Based Ambiguity Sets

The paper "Multistage Distributionally Robust Mixed-Integer Programming with Decision-Dependent Moment-Based Ambiguity Sets" by Xian Yu and Siqian Shen presents a comprehensive approach to address multistage decision problems under uncertainty. The probability distribution, which defines stage-wise uncertainties, is influenced by decisions made in previous stages, introducing an endogenous uncertainty framework. The focus is on three distinct types of decision-dependent moment-based ambiguity sets, and their reformulation into solvable mathematical programming formats.

Methodological Approach

The authors explore two core ambiguity sets where decisions impact the bounds of the first and second moments, as well as the mean and covariance matrices. For these, it is demonstrated that subproblems in each stage can be recast as mixed-integer linear programs (MILPs). A third ambiguity set extends the well-known moment-based approach of Delage and Ye (2010) to a multistage setting with decision dependencies, leading to mixed-integer semidefinite programming (MISDP) reformulations.

Furthermore, a set of computational methods are developed, leveraging the Stochastic Dual Dynamic integer Programming (SDDiP) approach and Lagrangian relaxation techniques, to efficiently solve large instances either directly or by approximation. These methods are critical in deriving both lower and upper bounds of optimal objectives in complex multistage settings.

Computational Results and Key Findings

The paper provides substantial computational results from multistage facility-location problems of various sizes and uncertainty settings. Significantly, the SDDiP method efficiently finds optimal solutions for moderate-sized instances under the two initial ambiguity sets and provides satisfactory bounds for larger MISDP instances under the third ambiguity set. Notably, the work illustrates the substantial impact of incorporating decision-dependent uncertainty in enhancing decision-making processes across stages.

Theoretical Implications

The reformulation techniques and solution algorithms proposed improve the tractability of distributionally robust optimization (DRO) models for real-world multistage decision problems. Importantly, the paper underscores the efficacy of defining ambiguity sets that closely follow decision-induced changes in uncertainty, which can lead to more accurate modeling of real-world phenomena where uncertainty is endogenous.

Practical Implications and Future Directions

Practically, the research offers advanced optimization frameworks aiding industries like logistics and supply chain management in strategic decision-making under uncertainty. With the integration of decision-dependent uncertainties offering a more realistic depiction of problems, future research can extend these models to more complex scenarios, potentially incorporating dynamic learning processes.

The paper sets the stage for further exploration of ambiguity set constructions, such as those based on divergence measures, which could offer new insights and practicality by addressing asymptotic consistency challenges. Hence, the work is both a significant contribution to multistage DRO and a foundation upon which future developments in decision-dependent uncertainty modeling can be built.