Power minimization under variability is formulated as a rigorous statistical robust optimization program with a guarantee of power and timing yields. Both power and timing metrics are treated probabilistically. Power reduction is performed by simultaneous sizing and dual threshold voltage assignment. An extremely fast run-time is achieved by casting the problem as a second-order conic problem and solving it using efficient interior-point optimization methods. When compared to the deterministic optimization, the new algorithm, on average, reduces static power by 31 % and total power by 17 % without the loss of parametric yield. The run time on a variety of public and industrial benchmarks is 30X faster than other known statistical power minimization algorithms.