Abstract A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

This paper concerns power control in wireless shadowed fading channels with independent interference. An important quality of service (QoS) requirement is that the outage probability of each transmitter/receiver pair is kept below a given level. The problem of minimizing power consumption with these outage probability specifications over shadowed fading wireless channels can be posed as a stochastic geometric program (GP), In general, it is challenging to solve this stochastic GP exactly, except for the special case of Rayleigh fading channels. In this paper, we describe a suboptimal approach based on geometric programming. We show that we can find a feasible solution to the stochastic GP by solving a (non-stochastic) geometric program obtained via the one-sided Chebyshev inequality, which can be solved globally and efficiently by interior-point methods. This GP approximation method can handle a variety of shadowed fading channel models including Nakagami/Ricean/Weibull/gamma fading with lognormal/gamma shadowing. This method gives a good compromise between computational complexity and performance, efficiently finding a power allocation that meets the outage probability specifications. Numerical examples are given to illustrate the power control method. 1

This paper describes a novel optimization-based approach to conflict resolution in air traffic control, based on geometric programming. A key feature of this approach is its ability to also take into account various metering directives issued by the traffic flow management level, in contrast to most methods that focus purely on aircraft separation issues. Moreover, the proposed methodology can account for some of the nonlinearities present in the formulations of conflict resolution problems, while incurring only a small penalty in computation time with respect to the fastest linear-programming-based approaches. Integer variables can be introduced to improve the quality of the solutions and to include combinatorial choices, for example, to optimize over aircraft sequences in merging streams. Simulation results demonstrate the efficiency of the approach on various aircraft separation problems, including miles-in-trail and minutes-in-trail restrictions through airspace fixes and boundaries.