Distributionally Robust SDDiP

• Distributionally Robust SDDiP is a framework that integrates decision-dependent ambiguity sets into multistage mixed-integer optimization.
• It leverages dualization and dynamic programming to reformulate worst-case expectation problems as tractable MILP or MISDP subproblems.
• Numerical studies show improved objective values and efficiency in applications like facility location under risk-averse settings.

Distributionally Robust SDDiP (Stochastic Dual Dynamic integer Programming) is a principled computational framework for solving multistage mixed-integer optimization problems under uncertainty where the probability distribution of stage-wise uncertainties is ambiguous, possibly decision-dependent, and described by moment-based ambiguity sets. This approach extends standard SDDiP to handle endogenous uncertainty via distributionally robust optimization (DRO), embedding worst-case risk measures and recourse via dynamic programming with nested ambiguity (Yu et al., 2020, Pichler et al., 2021).

1. Multistage Distributionally Robust Optimization Framework

Multistage DRO considers decisions and random parameters across a finite time horizon $T$, with recourse at each stage. At stage $t$, given prior actions and observed uncertainties $\xi_1, \ldots, \xi_t$, actions consist of a binary state expansion $x_t \in \{0,1\}^I$ (e.g., facility openings) and a mixed-integer recourse $y_t \in \mathbb{R}^{J_t}\times\mathbb{Z}^{K_t}$, subject to mixed-integer polyhedral constraints $X_t(x_{t-1},\xi_t)$. Uncertainty $\xi_t$ has a discrete support $\Xi_t = \{\xi_t^1,...,\xi_t^K\}$; in the distributionally robust context, its law is uncertain and described by an ambiguity set $\mathcal{P}_t(x_{t-1})$, potentially dependent on past states.

The objective is to minimize the expected total cost under a recursive, multi-stage, worst-case risk map: $$\min_{(x_1,y_1)\in X_1} \; g_1(x_1,y_1) + \rho_2^{(x_1)}\Big[g_2(x_2,y_2) + \cdots + \rho_T^{(x_{T-1})}[g_T(x_T,y_T)]\Big]$$ where the risk functional $t$0 (typically a mix of expectation and CVaR) is evaluated under the worst-case distribution $t$1 at each stage. The underlying dynamic program is written recursively as Bellman equations: $t$2 for $t$3, with $t$4 as the terminal stage cost (Yu et al., 2020).

2. Decision-Dependent Moment-Based Ambiguity Sets

Distributional ambiguity sets encode uncertainty about probability distributions and are constructed from finite-support distributions via the empirical $t$5 and parametric decision dependence. The main classes are:

• Type 1: Moment-Bounds Ambiguity. Dual bounds $t$6 for moments $t$7 (e.g., mean, second moment), yielding a polyhedral ambiguity set.
• Type 2: Exact Moment-Matching. Distribution matches exactly prescribed mean $t$8 and covariance $t$9, again as affine functions of the decision.
• Type 3: Delage–Ye-Type Ellipsoidal. Enforces that the mean and covariance of the measure are close (in Mahalanobis and Loewner order) to decision-dependent targets, with parameters $\xi_1, \ldots, \xi_t$0 controlling the allowable deviation.

Letting $\xi_1, \ldots, \xi_t$1 and $\xi_1, \ldots, \xi_t$2, the ellipsoidal ambiguity set is: $\xi_1, \ldots, \xi_t$3 These constructions generalize the Delage–Ye [2010] framework to the decision-dependent, multi-stage setting (Yu et al., 2020).

3. Stagewise Reformulations via Dualization

Inner maximizations (worst-case expectation over ambiguity) admit strong duals, converting the natural two-level problem into tractable MILP or MISDP forms:

• Type 1 and 2 (MILP): The maximization over polyhedral or exact moment-based ambiguity sets reduces to LP duals, yielding explicit constraints; in all cases, bilinearities (from binary decisions) are linearized via McCormick envelopes.
• Type 3 (MISDP): For Delage–Ye-type ambiguity, the dual is an SDP (with a semidefinite constraint block). The single-stage subproblem becomes a mixed-integer semi-definite program.

The resulting single-level reformulation for each $\xi_1, \ldots, \xi_t$4 can be summarized as: | Ambiguity Set | Stagewise Reformulation | |---------------|------------------------| | Type 1 | Mixed-integer linear program (MILP) | | Type 2 | MILP with quadratic constraints (exact moment) | | Type 3 | Mixed-integer semidefinite program (MISDP) |

These dualizations are essential for embedding robust multistage recourse within SDDiP (Yu et al., 2020).

4. Distributionally Robust SDDiP Algorithm

Distributionally Robust SDDiP (DR-SDDiP) generalizes the SDDiP framework to handle the inner DRO maximizations and their dual-based reformulations:

• Forward Pass: Simulate sample paths of uncertainties, solve the (approximated) MILP/MISDP along each scenario trajectory, and collect trial decisions and costs.
• Backward Pass: For each visited state and scenario, solve the dual of the stagewise subproblem to obtain valid subgradients (“Lagrangian cuts”), add these as affine minorants (cuts) of the recourse function $\xi_1, \ldots, \xi_t$5.
• Termination: Iterate until the bound gap is within a desired tolerance.

This cutting-plane approximation maintains global lower bounds (via dual cuts) and can be enhanced by upper-bounding schemes in the presence of MISDPs (see below).

A key feature is that each backward pass operates via the dual of the stage’s robust subproblem, yielding cuts that account for the current ambiguity set and its dependence on past decisions. The approach enables scalable optimization even under endogenous and non-rectangular ambiguity (Yu et al., 2020, Pichler et al., 2021).

5. Bounding Schemes and MISDP Approximation

Because generic MISDPs are computationally demanding, Yu and Shen (Yu et al., 2020) introduce bounding schemes:

• Lower Bounds: Relax the state-linking constraint in the backward pass (dual variable-based cuts), producing valid under-estimators of the true dynamic value function.
• Upper Bounds: Inner-approximate the semidefinite (PSD) cone (e.g., using diagonally-dominant polyhedral cones), converting the MISDP into an MILP; a sequence of increasingly tight MILP relaxations yields upper bounds convergent to the MISDP optimum.

This approach guarantees monotonic tightening of the upper bound and convergence as the inner-approximation is refined. Empirical results demonstrate that the MILP-based upper bounds are 3–5× faster than direct MISDP lower-bound passes on moderate instance sizes.

6. Numerical Performance and Applications

Numerical studies focus on multistage facility location with up to 50 potential sites, 100 customer zones, and 100 scenario support points. All three ambiguity models were tested:

• Type 1 & 2: SDDiP converged to global optima (for $\xi_1, \ldots, \xi_t$6, $\xi_1, \ldots, \xi_t$7) in minutes, with decision-dependent ambiguity improving objective values by 5–10% over decision-independent benchmarks.
• Type 3: The lower–upper bound gap under relaxed-cut and inner-approximation schemes was typically 2–4% for moderate instances ($\xi_1, \ldots, \xi_t$8); the inner-approximation MILP yielded faster runtimes.
• Scalability: Stagewise MILP solve time is linear in scenario count and time horizon but exponential in the number of binary variables, due to McCormick expansions.

Practical implications include the efficacy of incorporating endogenous (decision-dependent) ambiguity, tractability of large-scale instances for MILP cases, and the computational manageability of inner-approximated MISDPs.

7. Theoretical Connections and Extensions

Distributionally robust SDDiP relies foundationally on the mathematical underpinnings of conditional risk measures and their dual representations. The constructions in (Pichler et al., 2021) formalize the equivalence between static and dynamically nested risk measures, conditional risk functionals, and rectangularity conditions for ambiguity sets. Validity of value function lower bounds in the SDDiP recursion is ensured by the convex-analytic structure of the coherent risk functionals and their strong duals. Extensions to Wasserstein and AVaR ambiguity sets follow directly, supporting the broad applicability of DR-SDDiP in multistage, risk-averse optimization.

• Yu & Shen. "Multistage Distributionally Robust Mixed-Integer Programming with Decision-Dependent Moment-Based Ambiguity Sets" (Yu et al., 2020).
• Pflug, Shapiro, et al., "Mathematical Foundations of Distributionally Robust Multistage Optimization" (Pichler et al., 2021).