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A Note on Piecewise Affine Decision Rules for Robust, Stochastic, and Data-Driven Optimization

Simon Thomä, Maximilian Schiffer, Wolfram Wiesemann

TL;DR

This work analyzes piecewise affine decision rules for multi-stage optimization under uncertainty by embedding uncertainties into a lifted space and solving affine policies there. It reveals that in robust settings lifting does not enhance feasibility over affine policies, while stochastic and data-driven contexts benefit from the lifted objective; the authors introduce refined liftings, conic outer approximations, and distance-based valid cuts to tighten the lifted feasibility region. They develop efficient cut-separation methods, including polynomial-time schemes under permutation invariance and broad approximation guarantees for general embeddings, and provide strong theoretical bounds. Empirically, the proposed approach yields substantial performance gains over prior piecewise affine methods across stochastic, robust, and Wasserstein-data-driven problems, notably in inventory-like multistage settings, albeit with higher compute times and sensitivity to breakpoint design. The framework thus offers a scalable, theory-backed route to improved policies with practical relevance for operations under uncertainty and data-driven decision-making.

Abstract

Multi-stage decision-making under uncertainty, where decisions are taken under sequentially revealing uncertain problem parameters, is often essential to faithfully model managerial problems. Given the significant computational challenges involved, these problems are typically solved approximately. This short note introduces an algorithmic framework that revisits a popular approximation scheme for multi-stage stochastic programs by Georghiou et al. (2015) and improves upon it to deliver superior policies in the stochastic setting, as well as extend its applicability to robust optimization and a contemporary Wasserstein-based data-driven setting. We demonstrate how the policies of our framework can be computed efficiently, and we present numerical experiments that highlight the benefits of our method.

A Note on Piecewise Affine Decision Rules for Robust, Stochastic, and Data-Driven Optimization

TL;DR

This work analyzes piecewise affine decision rules for multi-stage optimization under uncertainty by embedding uncertainties into a lifted space and solving affine policies there. It reveals that in robust settings lifting does not enhance feasibility over affine policies, while stochastic and data-driven contexts benefit from the lifted objective; the authors introduce refined liftings, conic outer approximations, and distance-based valid cuts to tighten the lifted feasibility region. They develop efficient cut-separation methods, including polynomial-time schemes under permutation invariance and broad approximation guarantees for general embeddings, and provide strong theoretical bounds. Empirically, the proposed approach yields substantial performance gains over prior piecewise affine methods across stochastic, robust, and Wasserstein-data-driven problems, notably in inventory-like multistage settings, albeit with higher compute times and sensitivity to breakpoint design. The framework thus offers a scalable, theory-backed route to improved policies with practical relevance for operations under uncertainty and data-driven decision-making.

Abstract

Multi-stage decision-making under uncertainty, where decisions are taken under sequentially revealing uncertain problem parameters, is often essential to faithfully model managerial problems. Given the significant computational challenges involved, these problems are typically solved approximately. This short note introduces an algorithmic framework that revisits a popular approximation scheme for multi-stage stochastic programs by Georghiou et al. (2015) and improves upon it to deliver superior policies in the stochastic setting, as well as extend its applicability to robust optimization and a contemporary Wasserstein-based data-driven setting. We demonstrate how the policies of our framework can be computed efficiently, and we present numerical experiments that highlight the benefits of our method.
Paper Structure (23 sections, 17 theorems, 101 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 23 sections, 17 theorems, 101 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Piecewise Affine Policies via Liftings
  3. Valid Cuts for the Lifted Support $\Xi'$
  4. Efficient Cut Separation
  5. Permutation Invariant Uncertainty Embeddings
  6. General Uncertainty Embeddings
  7. Approximation Guarantees
  8. Numerical Experiments
  9. Stochastic Programming
  10. Robust Optimization
  11. Data-Driven Optimization
  12. Proof of Lemma \ref{['lem:rectangular_hull']}
  13. Proof of Theorem \ref{['thm:negative_SP']}
  14. Proof of Observation \ref{['obs:distance_formalization']}
  15. Proof of Observation \ref{['obs:conic-reformulation']}
  16. ...and 8 more sections

Key Result

Lemma 1

The convex hull of the folded bounding box of the embedded support satisfies

Figures (9)

  • Figure 1: Visualization of the piecewise affine folding function $F_i$.
  • Figure 2: Visualization of $\mathop{\mathrm{conv}}\nolimits(F([\underline{\boldsymbol{\theta}}_g, \overline{\boldsymbol{\theta}}_g]))$. The embedded support subset $[\underline{\theta}_{gi},\overline{\theta}_{gi}]$ (green line segment on the abscissa) is folded into a two-dimensional non-convex set (two perpendicular green line segments). The convex hull of $F ([\underline{\theta}_{gi},\overline{\theta}_{gi}])$ is shown as a grey shaded set that is bounded by the inequality $\frac{\xi'_{i1} - F_{i1}(\underline{\theta}_{gi})}{F_{i1}(\overline{\theta}_{gi}) - F_{i1}(\underline{\theta}_{gi})} \geq \frac{\xi'_{i2} - F_{i2}(\underline{\theta}_{gi})}{F_{i2}(\overline{\theta}_{gi}) - F_{i2}(\underline{\theta}_{gi})}$ (dotted line).
  • Figure 3: Visualization of the breakpoint grid $Z$ (grey), a gridpoint-induced rectangle $[\boldsymbol{z}^-, \boldsymbol{z}^+]$ (blue, solid), the index sets $J^-_1 = \{1,2,3\}$, $J^+_1 = \{5\}$, $J^-_2 = \{1\}$ and $J^+_2 = \{4\}$ corresponding to the embedded uncertainty components outside the rectangle, the distance $d'(F(\boldsymbol{\theta}), [\boldsymbol{z}^-, \boldsymbol{z}^+])$ between an embedded uncertainty realization $\boldsymbol{\theta}\in\Theta$ and the rectangle, and the maximum distance $\overline{d}_g([\boldsymbol{z}^-, \boldsymbol{z}^+])$ between any point in the embedded support subset $\Theta_g$ (green, ellipsis) and the rectangle. To ease the exposition, we omit the arguments of $d'$ and $\overline{d}_g$.
  • Figure 4: Visualization of the autoregressive support $\Xi$ (green, ellipsis) and the retracted cuts $R(\{ \boldsymbol{\xi}' \in \mathop{\mathrm{conv}}\nolimits(F([\underline{\boldsymbol{\theta}}, \overline{\boldsymbol{\theta}}])) \, \colon \, d'(\boldsymbol{\xi}', [\mu\boldsymbol{e}, \mu\boldsymbol{e}]) \leq \overline{d} ([\mu\boldsymbol{e}, \mu\boldsymbol{e}]) \})$ (blue) for different autoregressive factors $\alpha$ with a single breakpoint $\mu$ in two dimensions.
  • Figure 5: Stochastic Programming. Relative objective improvements of different piecewise affine policies over affine policies for different serial demand correlations $\alpha$ and time horizons $T$. Values of AFF exceeding a $70\%$ optimality gap are not plotted at scale to ensure the visibility of relevant effects from piecewise affine policies.
  • ...and 4 more figures

Theorems & Definitions (37)

  • Example 1: Stochastic Programming
  • Example 2: Robust Optimization
  • Example 3: Data-Driven Optimization
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Proposition 1
  • Theorem 2: Polynomial-Time Separability
  • Proposition 2: Efficient Separation for Norm-Balls
  • Proposition 3: Efficient Separation for Intersections
  • ...and 27 more