# The radius of stability connection

December 28, 2011 Leave a comment

For our purposes here, the most suitable formulation of the radius of stability is this generic model:

where

- = decision space.
- = nominal value of the parameter of interest.
- = neighborhood of radius around .
- = list of constraints on pairs.
- = radius of stability of at .

The neighborhoods are subsets of some set , called the *uncertainty space*. In words:

The radius of stability of decision at , denoted , is the radius of the largest neighborhood such that all the points in this neighborhood satisfy the constraints in .

Info-gap robustness model is a radius of stability model where consists of a single constraint of the form . That is, according to info-gap decision theory, the robustness of decision is defined as follows:

where represents a *critical performance level* and represents the performance level of decision $x$ given that the parameter of interest in equal to .

The picture is as follows:

Here the neighborhoods are circles centered at .

This, needless to say, is a typical **worst-case analysis**. To see why this is so, observe that the radius of stability of decision at is the distance from to the “nearest” point that **violates** the performance constraint .

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:

or more explicitly,

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.