[Paper Review] Designing Tractable Piecewise Affine Policies for Multi-Stage Adjustable Robust Optimization

Simon Thomä, Grit Walther+1 more|arXiv (Cornell University)|Jul 1, 2022

Advanced Control Systems Optimization62 references4 citations

TL;DR

This paper proposes tractable piecewise affine policies for multi-stage adjustable robust optimization (ARO) with non-negative right-hand side uncertainty. By constructing novel dominating uncertainty sets and leveraging convex combinations of vertex solutions, the authors enable efficient solution via linear programming, achieving strong approximation bounds and significantly faster computation than affine policies while maintaining or improving solution quality.

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.

Motivation & Objective

• To address the computational intractability of multi-stage adjustable robust optimization (ARO) with non-negative right-hand side uncertainty.
• To design computationally efficient policies that maintain strong approximation guarantees in multi-stage settings.
• To extend two-stage ARO approximation bounds to the general multi-stage problem, which had no prior bounds.
• To overcome exponential growth in uncertainty partitioning methods by using convex combinations of vertex solutions.
• To demonstrate that piecewise affine policies can be computed orders of magnitude faster than affine policies while achieving comparable or better performance.

Proposed method

• Construct new dominating uncertainty sets that allow reformulating the multi-stage ARO problem as a linear program.
• Use a nonanticipative mapping from original uncertainty space to a rescaled space using break-points and convex combinations.
• Define a piecewise affine policy as a convex combination of vertex solutions, parameterized by uncertainty components.
• Leverage the tightening constraint in the uncertainty set definition to ensure valid convex combinations across all uncertainty realizations.
• Transform solutions from the dominating uncertainty set back to the original problem using a reparameterization of the uncertainty space.
• Apply efficient sampling techniques for hypersphere and budgeted uncertainty sets to validate performance in numerical experiments.

Experimental results

Research questions

• RQ1 Can tractable piecewise affine policies be designed for multi-stage ARO with non-negative right-hand side uncertainty?
• RQ2 What approximation bounds can be proven for such policies in the general multi-stage setting?
• RQ3 How do these policies compare in computational speed and solution quality to existing affine and piecewise affine policies?
• RQ4 Can the proposed method extend two-stage ARO approximation bounds to the multi-stage case?
• RQ5 What adjustments can mitigate critical disadvantages of the proposed policies in practice?

Key findings

• The proposed piecewise affine policies can be computed by orders of magnitude faster than affine policies while achieving comparable or better solution quality.
• The method achieves strong approximation bounds for the general multi-stage ARO problem, which are the first such bounds in this setting.
• The authors prove that solutions from the dominating uncertainty set can be transformed into valid solutions for the original problem with guaranteed performance.
• The approach enables efficient solution via linear programming by constructing convex combinations of vertex solutions over rescaled uncertainty sets.
• Numerical experiments show that the policies outperform or match other policies in key performance dimensions, with effective adjustments available to address identified weaknesses.
• Sampling methods for hypersphere and budgeted uncertainty sets are efficient, with high acceptance rates in numerical validation.