A Gauge Set Framework for Flexible Robustness Design

TL;DR

This paper develops a unified gauge-set framework to design robustness in optimization under uncertainty, connecting SP, RO, and DRO through a convex-analytic lens. By reweighting distributions with a gauge-based distance to a nominal measure and deriving a dual upper-approximator, it decouples nominal distribution design, distance metric, and reformulation method, enabling modular robustness. It introduces a gauge-algebra toolkit for combining, decomposing, and composing gauges, and provides two tractable reformulation approaches for continuous uncertainty: functional parameterization and envelope representation; both are demonstrated in a detailed case study that shows improved out-of-sample tail performance relative to baselines. The framework generalizes common DRO constructions (moment-based, Wasserstein, and phi-divergence) and offers a principled path to tailor robustness to geometry and priorities of specific applications, with practical implications for risk-aware decision making.

Abstract

This paper proposes a unified framework for designing robustness in optimization under uncertainty using gauge sets, convex sets that generalize distance and capture how distributions may deviate from a nominal reference. Representing robustness through a gauge set reweighting formulation brings many classical robustness paradigms under a single convex-analytic perspective. The corresponding dual problem, the upper approximator regularization model, reveals a direct connection between distributional perturbations and objective regularization via polar gauge sets. This framework decouples the design of the nominal distribution, distance metric, and reformulation method, components often entangled in classical approaches, thus enabling modular and composable robustness modeling. We further provide a gauge set algebra toolkit that supports intersection, summation, convex combination, and composition, enabling complex ambiguity structures to be assembled from simpler components. For computational tractability under continuously supported uncertainty, we introduce two general finite-dimensional reformulation methods. The functional parameterization approach guarantees any prescribed gauge-based robustness through flexible selection of function bases, while the envelope representation approach yields exact reformulations under empirical nominal distributions and is asymptotically exact for arbitrary nominal choices. A detailed case study demonstrates how the framework accommodates diverse robustness requirements while admitting multiple tractable reformulations.

Figures (3)

• Figure 1: Illustration of the envelope functions under the setting $\mathcal{V}^\circ = \mathrm{Lip}_1$. Given a function $w$ with Lipschitz constant $\gamma$, each atomic envelope is defined as $\theta_{\gamma, w(\xi_i), \xi_i}(\xi) = w(\xi_i) + \gamma\|\xi - \xi_i\|$, shown as dotted lines centered at each sample $\xi_i$. Their envelope $\min_{i} \theta_{\gamma, w(\xi_i), \xi_i}(\xi)$ forms an upper approximation of $w$.
• Figure 2: Instance information. Panel (a) shows the three regions, the true distribution, and the retained $m=342$ observations. Panels (b)--(d) report Bayesian-learned distributions under different data retention levels. We fix $m=342$ and take the distribution in (c) as the nominal $\mathbb P$ for all models, except the data-driven Wasserstein CVaR baseline (WDRO), which uses the empirical measure $\bar{\mathbb P}_m=\tfrac{1}{m}\sum_{i\in[m]}\delta_{\xi_i}$ as the nominal.
• Figure 3: PAR, ENV, and WDRO solution trajectories and out-of-sample $\mathrm{CVaR}_{0.8}$ across configurations. To improve readability, the $\mathrm{CVaR}_{0.8}$ axis is split at $\rho=3$, i.e., values over $\rho\in[0,3]$ are shown on a zoomed vertical scale, while $\rho\in[3,4.8]$ is shown on the full scale.

Theorems & Definitions (114)

• Definition 1: Gauge Set
• Proposition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Definition 2: Quasi-Strong Duality
• Proposition 2: bot2009conjugate
• Theorem 1
• Remark 1
• Proposition 3