A Gauge Set Framework for Flexible Robustness Design

TL;DR

This paper introduces a unified gauge set framework for designing flexible and modular robustness in optimization under uncertainty, enabling complex ambiguity modeling and tractable reformulations.

Contribution

It develops a convex-analytic gauge set approach that decouples robustness components, provides algebraic tools for complex structures, and offers two reformulation methods for computational tractability.

Findings

Framework unifies classical robustness paradigms.

Supports complex ambiguity structures through algebraic toolkit.

Provides tractable reformulations with guarantees under uncertainty.

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.