  The study of the fundamental limits of information systems is a central theme
in information theory. Both the traditional analytical approach and the
recently proposed computational approach have significant limitations, where
the former is mainly due to its reliance on human ingenuity, and the latter due
to its exponential memory and computational complexity. In this work, we
propose a new computational approach to tackle the problem with much lower
memory and computational requirements, which can naturally utilize certain
intuitions, but also can maintain the strong computational advantage of the
existing computational approach. A reformulation of the underlying optimization
problem is first proposed, which converts the large linear program to a maximin
problem. This leads to an iterative solving procedure, which uses the LP dual
to carry over learned evidence between iterations. The key in the reformulated
problem is the selection of good information inequalities, with which a relaxed
LP can be formed. A particularly powerful intuition is a potentially optimal
code construction, and we provide a method that directly utilizes it in the new
algorithm. As an application, we derive a tighter outer bound for the
storage-repair tradeoff for the $(6,5,5)$ regenerating code problem, which
involves at least 30 random variables and is impossible to compute with the
previously known computational approach.
