Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Distributionally Robust Multistage Convex Optimization: Data-driven Models and Performance

Shixuan Zhang, Xu Andy Sun

TL;DR

This work develops data-driven distributionally robust multistage convex optimization (DR-MCO) models with Wasserstein ambiguity sets that permit infinite uncertainty supports and embeds them in a dual dynamic programming (DDP) framework. It proves out-of-sample guarantees via measure concentration, and shows that in-sample conservatism scales linearly with the Wasserstein radius under Lipschitz conditions, enabling tunable conservatism. The authors design exact single-stage subproblem oracles (SSSO) for both concave and convex uncertain costs, enabling provable convergence of DDP under broad uncertainty structures. Through extensive numerical experiments on multi-commodity inventory and hydro-thermal power planning problems, they demonstrate DR-MCO’s advantages in small-sample regimes and provide practical guidance on radius selection and algorithmic implementation. The results offer a principled, data-driven approach to robust multistage decision making with provable guarantees and demonstrated applicability to energy and supply-chain problems.

Abstract

This paper presents a novel algorithmic study with extensive numerical experiments of distributionally robust multistage convex optimization (DR-MCO). Following the previous work on dual dynamic programming (DDP) algorithmic framework for DR-MCO, we focus on data-driven DR-MCO models with Wasserstein ambiguity sets that allow probability measures with infinite supports. These data-driven Wasserstein DR-MCO models have out-of-sample performance guarantees and adjustable in-sample conservatism. Then by exploiting additional concavity or convexity in the uncertain cost functions, we design exact single stage subproblem oracle (SSSO) implementations that ensure the convergence of DDP algorithms. We test the data-driven Wasserstein DR-MCO models against multistage robust convex optimization (MRCO), risk-neutral and risk-averse multistage stochastic convex optimization (MSCO) models on multi-commodity inventory problems and hydro-thermal power planning problems. The results show that our DR-MCO models could outperform MRCO and MSCO models when the data size is small.

On Distributionally Robust Multistage Convex Optimization: Data-driven Models and Performance

TL;DR

This work develops data-driven distributionally robust multistage convex optimization (DR-MCO) models with Wasserstein ambiguity sets that permit infinite uncertainty supports and embeds them in a dual dynamic programming (DDP) framework. It proves out-of-sample guarantees via measure concentration, and shows that in-sample conservatism scales linearly with the Wasserstein radius under Lipschitz conditions, enabling tunable conservatism. The authors design exact single-stage subproblem oracles (SSSO) for both concave and convex uncertain costs, enabling provable convergence of DDP under broad uncertainty structures. Through extensive numerical experiments on multi-commodity inventory and hydro-thermal power planning problems, they demonstrate DR-MCO’s advantages in small-sample regimes and provide practical guidance on radius selection and algorithmic implementation. The results offer a principled, data-driven approach to robust multistage decision making with provable guarantees and demonstrated applicability to energy and supply-chain problems.

Abstract

This paper presents a novel algorithmic study with extensive numerical experiments of distributionally robust multistage convex optimization (DR-MCO). Following the previous work on dual dynamic programming (DDP) algorithmic framework for DR-MCO, we focus on data-driven DR-MCO models with Wasserstein ambiguity sets that allow probability measures with infinite supports. These data-driven Wasserstein DR-MCO models have out-of-sample performance guarantees and adjustable in-sample conservatism. Then by exploiting additional concavity or convexity in the uncertain cost functions, we design exact single stage subproblem oracle (SSSO) implementations that ensure the convergence of DDP algorithms. We test the data-driven Wasserstein DR-MCO models against multistage robust convex optimization (MRCO), risk-neutral and risk-averse multistage stochastic convex optimization (MSCO) models on multi-commodity inventory problems and hydro-thermal power planning problems. The results show that our DR-MCO models could outperform MRCO and MSCO models when the data size is small.
Paper Structure (21 sections, 17 theorems, 82 equations, 12 figures, 3 algorithms)

This paper contains 21 sections, 17 theorems, 82 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Data-driven Model and Its Properties
  3. Data-driven Model Formulation
  4. Out-of-Sample Performance Guarantee
  5. Adjustable In-Sample Policy Conservatism
  6. Dual Dynamic Programming Algorithm
  7. Review of Dual Dynamic Programming Algorithms
  8. Single Stage Subproblem Oracles
  9. Subproblem Oracles: Concave Uncertain Cost Functions
  10. Subproblem Oracles: Convex Uncertain Cost Functions
  11. Numerical Experiments
  12. Baseline Models and Experiment Settings
  13. Multi-commodity Inventory Problems
  14. Inventory Problems with Uncertain Demands
  15. Inventory Problems with Uncertain Prices
  16. ...and 6 more sections

Key Result

Theorem 1

Let $d_{t,k}(\xi):=d_t(\xi,\hat{\xi}_k)$ denote the distance function to each sample $\hat{\xi}_k$ for each $k=1,\dots,n_t$. Under Assumption assum:NominalStrictFeasibility, in any stage $t\ge 2$, the expected cost-to-go function eq:MDRO-RecursiveForm can be equivalently written as

Figures (12)

  • Figure 1: Out-of-sample Cost Quantiles for Different Radii on Multi-commodity Inventory with Uncertain Demands
  • Figure 2: Comparison against Baseline Models on Multi-commodity Inventory with Uncertain Demands
  • Figure 3: In-sample and Out-of-sample Mean Costs on Multi-commodity Inventory with Uncertain Prices
  • Figure 4: Comparison against Baseline Models on Hydro-thermal Power Planning
  • Figure 5: Out-of-sample Cost Quantiles for Different Radii on Multi-commodity Inventory with Uncertain Demands, Additional Run 1
  • ...and 7 more figures

Theorems & Definitions (36)

  • Example 1: Constrained Formulation
  • Theorem 1
  • Corollary 2
  • proof
  • Theorem 3: fournier2015rate
  • Theorem 4
  • proof
  • Example 2
  • Theorem 5
  • proof
  • ...and 26 more