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.

More …

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