  While globally optimal solutions to many convex programs can be computed
efficiently in polynomial time, this is, in general, not possible for nonconvex
optimization problems. Therefore, locally optimal approaches or other efficient
suboptimal heuristics are usually applied for practical implementations.
However, there is also a strong interest in computing globally optimal
solutions of nonconvex problems in offline simulations in order to benchmark
the faster suboptimal algorithms. Global solutions often rely on monotonicity
properties. A common approach is to reformulate problems into a canonical
monotonic optimization problem where the monotonicity becomes evident, but this
often comes at the cost of nested optimizations, increased numbers of
variables, and/or slow convergence. The framework of mixed monotonic
programming (MMP) proposed in this paper avoids such performance-deteriorating
reformulations by revealing hidden monotonicity properties directly in the
original problem formulation. By means of a wide range of application examples
from the area of signal processing for communications (including energy
efficiency for green communications, resource allocation in interference
networks, scheduling for fairness and quality of service, as well as beamformer
design in multiantenna systems), we demonstrate that the novel MMP approach
leads to tremendous complexity reductions compared to state-of-the-art methods
for global optimization. However, the framework is not limited to optimizing
communication systems, and we expect that similar speed-ups can be obtained for
optimization problems from other areas of research as well.
