  Solutions to network optimization problems have greatly benefited from
developments in nonlinear analysis, and, in particular, from developments in
convex optimization. A key concept that has made convex and nonconvex analysis
an important tool in science and engineering is the notion of asymptotic
function, which is often hidden in many influential studies on nonlinear
analysis and related fields. Therefore, we can also expect that asymptotic
functions are deeply connected to many results in the wireless domain, even
though they are rarely mentioned in the wireless literature. In this study, we
show connections of this type. By doing so, we explain many properties of
centralized and distributed solutions to wireless resource allocation problems
within a unified framework, and we also generalize and unify existing
approaches to feasibility analysis of network designs. In particular, we show
sufficient and necessary conditions for mappings widely used in wireless
communication problems (more precisely, the class of standard interference
mappings) to have a fixed point. Furthermore, we derive fundamental bounds on
the utility and the energy efficiency that can be achieved by solving a large
family of max-min utility optimization problems in wireless networks.
