ANALYTIC CENTER

Analytic center - Given the set, (x in X: g(x) >= 0), which we assume is non-empty and compact, such that g is concave on X, with a non-empty strict interior , its analytic center is the (unique) solution to the maximum entropy:

Note that the analytic center depends on how the set is defined -- i.e., the nature of g, rather than just the set, itself. For example, consider the analytic center of the box, [0,1]^n. One form is to have 2n functions as: (x: x >= 0 and 1-x >= 0). In this case, the analytic center is x*_j=y* for all j, where y* is the solution to: Since y* = 1/2, the analytical center is what we usually think of as the center of the box. However, the upper bounds could also be defined by 1-(x_j)^p >= 0 for all j, where p > 1 (so 1-(x_j)^p is concave). This changes the functional definition, but not the set -- it's still the unit box. The analytic center becomes skewed towards the corner because now the defining mathematical program is: The solution is y* = [1/(1+p)]^(1/p), so the analytic center approaches (1,1,...,1) as p-->inf.