| The goal of this thesis is to develop tractable approximation methods for geometric programming (GP) under uncertainty, including robust GP and chance-constrained GP. The optimal solution of a GP can be sensitive to variations in the problem data. Robust GP can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this thesis we propose an approximation method that seeks a compromise between accuracy and efficiency. Using the polyhedral approximations of log-sum-exp functions, a robust GP with polyhedral or ellipsoidal data uncertainty can be approximated well as a robust LP. Since robust LPs with polyhedral or ellipsoidal uncertainty can be globally solved with great efficiency, this provides a tractable approximation method for the robust GP.;The second topic concerns chance-constrained GP (CCGP). We first introduce a specific form of CCGP with one joint probability constraint. We show how to obtain feasible solutions of the joint CCGP with underlying normal variations by solving a corresponding robust GP with ellipsoid uncertainty. In addition, motivated by GP in posynomial form, we also consider a CCGP problem in which probability constraints are imposed individually. We provide a conservative approximation method based on Chebyshev inequality and give a sufficient condition under which the Chebyshev approximation reduces to a GP.;Finally, we consider specific engineering applications that reveal the effectiveness of the proposed approximation algorithms, including robust circuit sizing with minimum yield specification as well as power control in wireless shadowed fading channels. |
