# The radius of stability connection

For our purposes here, the most suitable formulation of the radius of stability is this generic model: $\rho(x,\tilde{u}):= \displaystyle \max_{\rho\ge 0}\ \{\rho: C(x,u),\forall u\in N(\rho,\tilde{u})\}\ , \ x\in X$

where

• $X$ = decision space.
• $\tilde{u}$ = nominal value of the parameter of interest.
• $N(\rho,\tilde{u})$ = neighborhood of radius $\rho$ around $\tilde{u}$.
• $C(x,u)$ = list of constraints on $(x,u)$ pairs.
• $\rho(x,\tilde{u})$ = radius of stability of $x$ at $\tilde{u}$.

The neighborhoods $N(\rho,\tilde{u}),\rho\ge0,$ are subsets of some set $U$, called the uncertainty space. In words:

The radius of stability of decision $x\in X$ at $\tilde{u}$, denoted $\rho(x,\tilde{u})$, is the radius $\alpha$ of the largest neighborhood $N(\rho,\tilde{u})$ such that all the points $u$ in this neighborhood satisfy the constraints in $C(x,u)$.

Info-gap robustness model is a radius of stability model where $C(x,u)$ consists of a single constraint of the form $r^{*}\le r(x.u)$. That is, according to info-gap decision theory, the robustness of decision $x\in X$ is defined as follows: $\rho(x,\tilde{u}):= \displaystyle \max_{\rho\ge 0}\ \{\rho: r^{*}\le r(x,u),\forall u\in N(\rho,\tilde{u})\}$

where $r^{*}$ represents a critical performance level and $r(x,u)$ represents the performance level of decision $x$ given that the parameter of interest in equal to $u$.

The picture is as follows: Here the neighborhoods $N(\rho,\tilde{u}),\rho\ge0,$ are circles centered at $\tilde{u}$.

This, needless to say, is a typical worst-case analysis. To see why this is so, observe that the radius of stability of decision $x$ at $\tilde{u}$ is the distance from $\tilde{u}$ to the “nearest” point $u\in U$ that violates the performance constraint $r^{*} \le r(x,u)$.

So, insofar as robustness is concerned, the larger the info-gap robustness of a decision, the better. Hence, the best (optimal) decision is that whose robustness is the largest. Formally then, info-gap’s decision model for robustness is as follows: $\rho(\tilde{u}):= \displaystyle \max_{x\in X}\ \rho(x,\tilde{u})$

or more explicitly, $\rho(\tilde{u}):= \displaystyle \max_{\substack{x\in X\\\rho\ge 0}}\ \{\rho: r^{*}\le r(x,u),\forall u\in N(\rho,\tilde{u})\}$

Also note that, both info-gap’s robustness model and info-gap’s decision model for robustness are simple instances of Wald’s famous maximin model.

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