Radius of Robust Feasibility in Optimization

• Radius of robust feasibility is defined as the maximal uncertainty radius under which the optimization problem remains feasible, serving as a certificate for robustness.
• It leverages geometric, variational, and convex analyses to compute margins using techniques like SDP, SOCP, and dual formulations.
• Its applications span sensor networks, robust deep learning, control systems, and portfolio optimization, offering practical insights for design and stability.

The radius of robust feasibility is a fundamental concept in robust optimization and computational geometry that quantifies the largest amount of uncertainty—measured as a “radius” under a specific norm or structured set—for which robust feasibility of a given problem is retained. It serves as a certificate or quantitative margin ensuring that, under all allowable perturbations within this radius, feasibility (i.e., nonemptiness of the feasible set or satisfaction of problem constraints) is preserved. The concept is central in diverse domains including optimization under uncertainty, integer programming, robust control, network design, adversarial learning, and combinatorial optimization.

1. Core Definition and Paradigm

The radius of robust feasibility, often denoted as ρ or RRF, is the supremum of radii for which all realizations of the uncertain parameters within a corresponding norm ball or set preserve the feasibility of the optimization problem. In a canonical uncertain linear system or conic program with data uncertainty parameterized as $(A, b) \in (\bar{A}, \bar{b}) + \alpha Z$, the robust feasible set under radius $\alpha$ is

$$\mathcal{F}_R^\alpha = \{ x : A x \leq b\ \forall (A, b) \in (\bar{A}, \bar{b}) + \alpha Z \}.$$

The RRF is then

$$\rho = \sup \{ \alpha \geq 0 : \mathcal{F}_R^\alpha \neq \emptyset \}.$$

This measures the maximal admissible uncertainty (in the sense of Z's geometry and norm) for which robust feasibility is not lost (Berthold et al., 2021, 2007.07599, Datta et al., 22 Oct 2025).

The precise criteria encoded by the RRF depend on the problem class. For instance, in a robust integer program over random polytopes, it is the inscribed ball radius certifying the presence or absence of integer points, while in sensor coverage it measures maximal displacement of sensors such that coverage constraints globally persist (Chandrasekaran et al., 2011, Datta et al., 22 Oct 2025).

2. Mathematical Characterizations and Computability

The computation of the RRF is deeply tied to the problem structure. In uncertain linear and conic programming, the RRF can often be reduced to a geometric or variational distance:

• For multi-objective and conic programs under ball uncertainty, the RRF is given by the minimal Euclidean distance from a reference vector (such as $(0,1)$) to a specific hypographical or epigraphical set defined by the problem data:

$$p(V) = \inf_{(a,b) \in H(a, b)} \| (0_n, 1) - (a, b) \|$$

where $H(a, b)$ is the convex hull of data vectors augmented by a cone (Goberna et al., 2014).

• In uncertain conic programs, it is characterized as

$$C_1 \cdot \mathrm{dist}((0, 1), E(A, b, B)) \leq p(A, b) \leq C_2 \cdot \mathrm{dist}((0, 1), E(A, b, B))$$

where $E(A, b, B)$ is an epigraphical set constructed from the constraint data and a base $B$ of the dual cone (2007.07599).

• For directional sensor networks, the RRF is computed via

$$\rho = \inf_{x \in \mathcal{F}^\alpha} \min_i \frac{\bar{b}_i - \bar{a}_i^T x}{\xi_{\mathbf{Z}_i}(x, -1)}$$

where $\xi_{\mathbf{Z}_i}$ is the support function of the uncertainty set for each sensor, encoding worst-case violation margin (Datta et al., 22 Oct 2025).

In robust quadratic constraint settings (e.g., robust Markowitz portfolio), the S-lemma is used: the robust feasibility region is characterized by the existence of a multiplier $\lambda \geq 0$ such that

$A - \lambda B \succeq 0$

where $A, B$ are derived from quadratic reformulations of the robust constraints and the uncertainty set's quadratic description (Swain et al., 2023).

For integer feasibility of random polytopes, the critical thresholds are given for the inscribed ball radius $R$, with high-probability transitions between infeasibility and feasibility occurring at $c_0 \sqrt{\log(m/n)}$ and $c_1 \sqrt{\log(m/n)}$ (Chandrasekaran et al., 2011).

3. Algorithmic Strategies

The RRF is not always expressible in closed form; its computation is often reduced to amenable convex programs or semidefinite programs:

• For uncertain LPs, SDPs, SOCPs, and robust SVMs, the RRF or its bounds can be calculated by associated SOCPs or SDPs (2007.07599).
• For polytopes with random constraints, finding an integer point when the inscribed radius is above threshold leverages the Lovett-Meka algorithm (partial coloring/Edge-Walk), which controls rounding-induced discrepancy (Chandrasekaran et al., 2011).
• In robust feasibility for nonlinear systems (e.g., quadratic equations in power systems or gas networks), inner (feasibility) and outer (infeasibility) bounds on the robustness margin are computed by convex relaxations, such as sum-of-squares SDP hierarchies or lifted QCQP relaxations (Aßmann et al., 2018, Dvijotham et al., 2019).
• In settings such as robust clustering with explicit radius constraints, the feasibility region is described via quadratic constraints, often enforced by SDP or LP relaxations followed by combinatorial assignment steps (Humayun et al., 2022).

The RRF can also be operationalized via continuous optimization procedures. Notably, (Hao et al., 27 Aug 2025) shows that the family of robust solutions as the uncertainty radius varies ("robust path") is realized as a Bregman projection of a parametrized dual curve; in several important cases this can be efficiently approximated by proximal point iterates, with sharp error bounds or, in certain geometric regimes, identically zero error.

4. Applications Across Domains

The RRF has pervasive utility in both modeling and computational aspects of robust and stochastic systems:

• Sensor networks: The RRF quantifies the maximal deployment error permissible for each sensor such that coverage over its Voronoi region remains intact under orientation and placement uncertainties (Datta et al., 22 Oct 2025).
• Power and gas networks: The RRF corresponds to the maximal uncertainty in physical parameters (e.g., generation, pressure loss factors) so that Kirchhoff or Weymouth constraints are solvable for all process conditions (Aßmann et al., 2018, Dvijotham et al., 2019).
• Robust classification and deep learning: In the context of adversarial robustness, the certified radius (a form of RRF) is the largest admissible input perturbation (in ℓ₂ or other norms) that provably leaves model predictions invariant (Zhai et al., 2020, Zhen et al., 2021, Seferis et al., 26 Apr 2024, Wang et al., 29 Jan 2025).
• Portfolio optimization: RRF criteria become eigenvalue conditions guaranteeing that worst-case expected return constraints hold for all admissible asset return scenarios, as encoded via positive-semidefinite matrix inequalities (Swain et al., 2023).
• Control theory: The structured real stability radius generalizes RRF to the stability margin against structured system perturbations; in large-scale control, quadratic-convergent subspace algorithms enable practical estimation of this critical value (Aliyev, 2021).
• Multi-objective programming: RRF ensures feasibility of possibly infinite systems of constraints under simultaneous (possibly dependent) objective and constraint uncertainties (Goberna et al., 2014, Berthold et al., 2021).

5. Theoretical and Geometric Properties

Theoretical underpinnings of the RRF are drawn from convex analysis, variational geometry, and combinatorics:

• In linear and conic settings, feasibility under data uncertainty is tightly linked to the separation of the reference vector from the convex hull or epigraph of possible constraint data (Goberna et al., 2014, 2007.07599).
• The S-lemma and its variants serve as the principal certifying mechanism for robust quadratic arrays and convex QCQPs (Swain et al., 2023).
• Connections to discrepancy theory, seen in integer feasibility over random polytopes, reveal sharp phase transitions for the RRF and establish that, for random constraints, much smaller inscribed ball radii than in worst-case deterministically constructed polytopes suffice (Chandrasekaran et al., 2011).
• When the feasible region and the uncertainty set possess a dual (polar) relationship or monotonicity properties (e.g., simplex under ellipsoidal uncertainty), the robust solution path as the radius varies can be exactly recovered from proximal point sequences, yielding computational savings and elementary geometric intuition (Hao et al., 27 Aug 2025).

Non-uniform or adaptive notions of the robustness radius (e.g., "local RRF" in classification) achieve maximally robust predictions without sacrificing Bayes-optimal accuracy by tailoring the radius to the local data geometry rather than enforcing a uniform criterion (Bhattacharjee et al., 2021).

6. Implications and Limitations

The RRF provides a quantitative guideline for system design and operational policies, directly measuring the tolerance of systems to uncertainty. It guides:

• Model validation (problem is not robustly feasible if RRF is zero).
• Trade-off analysis between robustness and optimality—larger RRFs may imply more conservative or less optimal solutions.
• Algorithm tailoring by using the RRF as a stopping or calibration criterion during model or hyperparameter selection (Hao et al., 27 Aug 2025, Seferis et al., 26 Apr 2024).

However, several challenges and limitations persist:

• Computing exact RRFs for complex nonconvex systems typically requires relaxations or approximations, and the resulting certificates may be conservative (Aßmann et al., 2018, Dvijotham et al., 2019).
• Strong duality or convexity is typically required for the most tractable RRF formulations; in highly nonconvex or combinatorial settings, only probabilistic or approximate thresholds are currently available.
• The RRF is inherently “worst-case”—in high dimensions, or with adversarial structure in the uncertainty, even moderate radii can lead to infeasibility, necessitating further statistical or distributional modeling if realistic average-case performance is desired.

7. Summary Table: RRF Instances Across Domains

Problem Type	Uncertainty Model	RRF/Certificate Formula
Linear/Conic Prog.	Ball/Normed Set	$\inf_{(a,b)\in H(a,b)} \|(0_n,1)-(a,b)\|$
Integer Feasibility (Random Polytopes)	Random Gaussian Constraints	$R \gtrless c_0, c_1 \sqrt{\log(m/n)}$ threshold
Sensors in Uncertain Terrain	Ball in $\mathbb{R}^2$	$$\rho = \inf_x \min_i \frac{\bar{b}_i - \bar{a}_i^T x}{\xi_{\mathbf{Z}_i}(x, -1)}$$
Quadratic Optimization/Portfolio	Ellipsoidal/Box/Polyhedral	$A - \lambda B \succeq 0$ for some $\lambda \geq 0$
Power/Gas Networks	Hypercube/Interval radius	Decided by polynomial optimization or LP/SDP relaxations
Deep Learning / Classification	ℓ₂ ball (certified radius)	$$r = \frac{\sigma}{2}[ \Phi^{-1}(p_y) - \Phi^{-1}(p_{y'}^{\max}) ]$$
Robust Control (Stability Margin)	Structured norm ball	$$r_\mathbb{R}(A; B, C) = \inf \{ \| \Delta \|_2 : A+B\Delta C \text{ unstable} \}$$

This consolidation demonstrates the broad applicability and significance of the radius of robust feasibility as both a theoretical construct and a practical computational tool in certifying, quantifying, and operationalizing robustness in modern optimization, data science, and control domains.