Tractable approximate robust geometric programming

The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. Robust geometric programming 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 paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency. The method is based on approximating the robust GP as a robust linear program (LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL) convex approximation. With a polyhedral or ellipsoidal description of the uncertain data, the resulting robust LP can be formulated as a standard convex optimization problem that interior-point methods can solve. The drawback of this basic method is that the number of terms in the PWL approximations required to obtain an acceptable approximation error can be very large. To overcome the “curse of dimensionality” that arises in directly approximating the nonlinear constraint functions in the original robust GP, we form a conservative approximation of the original robust GP, which contains only bivariate constraint functions. We show how to find globally optimal PWL approximations of these bivariate constraint functions. K.-L. Hsiung ( ) · S.-J. Kim · S. Boyd Information Systems Laboratory, Electrical Engineering Department, Stanford University, Stanford, CA 94305-9510, USA e-mail: [email protected] S.-J. Kim e-mail: [email protected] S. Boyd e-mail: [email protected] 96 K.-L. Hsiung et al.