  We provide a condition-based analysis of two interior-point methods for
unconstrained geometric programs, a class of convex programs that arise
naturally in applications including matrix scaling, matrix balancing, and
entropy maximization. Our condition numbers are natural geometric quantities
associated with the Newton polytope of the geometric program, and lead to
diameter bounds on approximate minimizers. We also provide effective bounds on
the condition numbers both in general and under combinatorial assumptions on
the Newton polytope. In this way, we generalize the iteration complexity of
recent interior-point methods for matrix scaling and matrix balancing.
Recently, there has been much work on algorithms for certain optimization
problems on Lie groups, known as capacity and scaling problems. For commutative
groups, these problems reduce to unconstrained geometric programs, which serves
as a particular source of motivation for our work.
