Solving nonlinear problems in communication systems using geometric programming and dualities

This dissertation uses the principles of Lagrange duality and Shannon duality, and a special type of convex optimization problems called geometric programming, to solve several nonlinear problems in communication systems.; We show that the information-theoretic limits of channel capacity for data transmission and rate distortion for data compression can be obtained through their Lagrange dual problems, in the form of geometric programs. Lagrange duality provides a rigorous characterization of Shannon duality between transmission and compression in the discrete memoryless case, and can be used to easily generate upper bounds on channel capacity and lower bounds on rate distortion. We also consider a more sophisticated model of transmission and compression with state information. By putting the known answers to eight special cases into a common form, we extend Shannon duality to cases with state information. This common form is also shown to answer the more general problems where the state information at the sender and at the receiver are different but correlated.; We then turn to a network communication system, where limited resources are allocated to competing user demands. We study a parameterization of allocation in a form that can be interpreted as the result of a relative entropy minimization. We show how geometric programs can efficiently optimize such allocations under various nonlinear Quality of Service constraints. This methodology is applied to rate allocation, admission control, and a suite of wireless cellular and ad hoc network power control problems. We continue to investigate the coupling effect between meeting user demands through transport layer mechanisms like TCP congestion control, and regulating bandwidth supply in physical layer through resource allocation like power control. Lagrange duality and convex optimization algorithms again help us understand and solve these joint problems that balance layers 1 and 4. A special case in CDMA multihop wireless networks is shown, where a provably-convergent distributed power control algorithm works with the standard TCP protocol to improve end-to-end throughput and network energy efficiency.; Through connections with large deviation principles, we also provide intuition why the nonlinearity contained in geometric programs may be useful for other problems based on stochastic models.