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Designing Tractable Piecewise Affine Policies for Multi-Stage Adjustable Robust Optimization

Simon Thomä, Grit Walther, Maximilian Schiffer

Abstract

We study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are - to the best of our knowledge - the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.

Designing Tractable Piecewise Affine Policies for Multi-Stage Adjustable Robust Optimization

Abstract

We study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are - to the best of our knowledge - the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.
Paper Structure (33 sections, 19 theorems, 122 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 33 sections, 19 theorems, 122 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Related Work
  4. Our Contributions
  5. Framework For Piecewise Affine Multi-Stage Policies
  6. Construction of the Dominating Set
  7. Construction of the Domination Function
  8. Limitations
  9. Optimality Bounds for Different Uncertainty Sets
  10. Re-scaling Expensive Vertices
  11. Piecewise Affine Policies via Liftings
  12. Numerical Experiments
  13. Gaussian Instances
  14. Demand Covering Instances
  15. Discussion
  16. ...and 18 more sections

Key Result

Theorem 1

Consider an uncertainty set $\mathcal{U}$ from Problem (eq:problem_formulation) and a dominating set $\hat{\mathcal{U}}$. Let $\beta\geq1$ be such that $\forall\hat{\boldsymbol{\xi}}\in\hat{\mathcal{U}} \colon \frac{1}{\beta}\hat{\boldsymbol{\xi}}\in\mathcal{U}$. Moreover, let $Z_{AR}(\mathcal{U})$

Figures (10)

  • Figure 1: Illustration of multi-stage decision making over $T$ stages. In each stage $t$ a fraction $\boldsymbol{\xi}^t$ of the uncertainty is realized and decisions $\boldsymbol{x}^t$ are made. Here, decisions $\boldsymbol{x}^t$ may only depend on those uncertainties $\underline{\boldsymbol{\xi}}^t$ whose realization is known in stage $t$
  • Figure 2: Relations between multi-stage ARO policies compared in this paper. An arc from a policy $P$ to another policy $P'$ states $Z_P \leq Z_{P'}$, where $Z_P$ and $Z_{P'}$ are optimal objective values for the ARO Problem (\ref{['eq:problem_formulation']}) solved with policy $P, P'$ respectively. Dashed arcs only hold for hypersphere (H) or budgeted (B) uncertainty. Relations proved for the first time in this paper are highlighted (blue, bold). The compared policies are: static policies (static); piecewise affine policies via domination by Ben-Tal et al. BenTal2020 (PAPBT); our piecewise affine policies via domination (PAP), c.f., Sections \ref{['sec:framework']} and \ref{['sec:bounds']}; affine policies BenTal2004 (AFF); our piecewise affine policies with rescaling (SPAP), c.f., Section \ref{['sec:rescaling']}; near-optimal piecewise affine policies Bertsimas2015b (BG); piecewise affine policies via lifting Georghiou2015 (LIFT); our tightened piecewise affine policies via lifting (TLIFT), c.f., Section \ref{['sec:lifing_combination']}
  • Figure 3: Two dimensional hypersphere uncertainty set $\mathcal{U}$ with (dashed) dominating set $\hat{\mathcal{U}}$ (\ref{['eq:dominating_U']}) induced by the convex combination of vertices $\boldsymbol{v}_0,\boldsymbol{v}_1,\boldsymbol{v}_2$ and dominating function $\boldsymbol{h}$ (\ref{['eq:domination_function']}) that maps a point $\boldsymbol{\xi}\in\mathcal{U}$ to a point $\hat{\boldsymbol{\xi}}\in\hat{\mathcal{U}}$
  • Figure 4: Comparison of our dominating set $\hat{\mathcal{U}}$ (blue, solid frame) and the dominating set $\hat{\mathcal{U}}_\text{BT}$ proposed in Ben-Tal et al. BenTal2020 (green, dashed frame) for the hypersphere uncertainty set $\mathcal{U}$ in $m = 2$ (a) and $m=3$ (b), (c) uncertainty dimensions
  • Figure 5: Re-scaling of the expensive vertex $\boldsymbol{v}_1$ in a dominating set $\hat{\mathcal{U}}$ for uncertainty set $\mathcal{U}$. (a) shows the change of dominating set $\hat{\mathcal{U}}$ (blue, dashed) and it's vertices $\boldsymbol{v}_0, \boldsymbol{v}_1, \boldsymbol{v}_2$ to the new re-scaled dominating set $\hat{\mathcal{U}}'$ (green, solid) with vertices $\boldsymbol{v}'_0, \boldsymbol{v}'_1, \boldsymbol{v}'_2$ for $s_1=0.5, s_2=0$. (b) shows the costs for the vertex solutions $\boldsymbol{x}_i$ with maximal cost $z$ (blue, dashed) compared to the costs for the re-scaled vertex solutions $\boldsymbol{x}'_i$ with maximal cost $z'$ (green, solid)
  • ...and 5 more figures

Theorems & Definitions (36)

  • Definition 1: Domination
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4: Lemma 4 in Ben-Tal et al. BenTal2020
  • Proposition 5: Hypersphere
  • Proposition 6
  • Proposition 7: Budget
  • Proposition 8
  • Proposition 9: $p$-norm ball
  • ...and 26 more