[Paper Review] Beyond Worst-case: A Probabilistic Analysis of Affine Policies in Dynamic Optimization

Omar El Housni, Vineet Goyal|arXiv (Cornell University)|Jun 18, 2017

Risk and Portfolio Optimization12 references3 citations

TL;DR

This paper provides a probabilistic analysis of affine policies in two-stage dynamic robust optimization, showing that under random uncertainty distributions, affine policies achieve near-optimal performance with significantly better approximation bounds than the worst-case $O(igsqrt{m})$ bound. The key contribution is a theoretical justification for the empirically observed strong performance of affine policies on random instances, validated through experiments on uniform and folded normal distributions.

ABSTRACT

Affine policies (or control) are widely used as a solution approach in dynamic optimization where computing an optimal adjustable solution is usually intractable. While the worst case performance of affine policies can be significantly bad, the empirical performance is observed to be near-optimal for a large class of problem instances. For instance, in the two-stage dynamic robust optimization problem with linear covering constraints and uncertain right hand side, the worst-case approximation bound for affine policies is $O(\sqrt m)$ that is also tight (see Bertsimas and Goyal (2012)), whereas observed empirical performance is near-optimal. In this paper, we aim to address this stark-contrast between the worst-case and the empirical performance of affine policies. In particular, we show that with high probability affine policies give a good approximation for two-stage dynamic robust optimization problems on random instances generated from a large class of distributions; thereby, providing a theoretical justification of the observed empirical performance. The approximation bound depends on the distribution, but it is significantly better than the worst-case bound for a large class of distributions.

Motivation & Objective

• To address the discrepancy between the poor worst-case performance bounds and strong empirical performance of affine policies in dynamic robust optimization.
• To analyze the approximation quality of affine policies under random uncertainty distributions, rather than worst-case adversarial settings.
• To provide theoretical justification for the near-optimal empirical performance of affine policies observed in computational experiments.
• To derive distribution-dependent approximation bounds that are significantly tighter than the worst-case $O(igsqrt{m})$ bound for a broad class of distributions.
• To validate the theoretical findings through computational experiments on uniform and folded normal distributions.

Proposed method

• Formulates a two-stage adjustable robust optimization problem with linear covering constraints and uncertain right-hand sides in a compact convex uncertainty set $\mathcal{U}$.
• Analyzes affine policies where second-stage decisions are linear functions of the uncertain parameters, enabling efficient computation.
• Derives probabilistic approximation bounds by modeling the uncertainty vector $\boldsymbol{h}$ as being drawn from a large class of distributions, including uniform and folded normal.
• Uses duality and linear programming reformulations to analyze the performance of affine policies relative to the optimal adjustable solution.
• Employs a mixed-integer programming (MIP) formulation to compute the optimal adjustable solution for small instances, enabling comparison with affine policies.
• Applies digitized linearization techniques to handle bilinear terms in the MIP formulation, enabling exact solution via standard solvers.

Experimental results

Research questions

• RQ1 Can affine policies achieve better-than-worst-case approximation performance when uncertainty is drawn from random distributions rather than adversarial ones?
• RQ2 How do the approximation bounds of affine policies depend on the underlying distribution of the uncertain parameters?
• RQ3 To what extent does the empirical performance of affine policies on random instances match or exceed theoretical worst-case bounds?
• RQ4 What is the computational trade-off between computing the optimal adjustable solution and using affine policies on random instances?
• RQ5 Are there specific distributional assumptions under which affine policies are provably near-optimal with high probability?

Key findings

• For i.i.d. uniform random instances, the theoretical worst-case bound for affine policies is a 2-approximation (Corollary 2.2), but empirical results show an average optimality gap of only 1%, corresponding to an approximation ratio of at most 1.04.
• For Gaussian and folded normal distributions, the theoretical bound is $O(\sqrt{\log(mn)})$, yet empirical performance remains strong, with average gaps below 4%.
• Affine policies scale efficiently, with average computation times under 15 seconds even for $m = 50$, while the optimal adjustable solution could not be computed within a 3-hour time limit for $m \geq 40$.
• The optimal adjustable solution was computable only for small instances ($m \leq 30$) using MIP, but the affine policy consistently outperformed the worst-case theoretical bounds in practice.
• In all tested distributions, the maximum ratio $z_{\text{Aff}}/z_{\text{AR}}$ remained below 1.04, indicating robust near-optimality across diverse random instances.
• The results demonstrate that affine policies are not only computationally tractable but also provably near-optimal with high probability under a wide class of random distributions, explaining their empirical success.