Power-gating devices incur a small amount of voltage drop across them when they are on in active mode, degrading the maximum frequency of processors. Thus, large power-gating devices are often implemented to minimize the drop (thus the frequency degradation), requiring considerable die area. Meanwhile, adaptive voltage scaling has been used to improve yield of power-constrained processors exhibiting a large spread of maximum frequency and total power due to process variations. In this paper, first, we analyze the impact of power-gating device size on both maximum frequency and total power of processors in the presence of process variation. Second, we propose a methodology that optimizes both the size of power-gating devices and the degree of adaptive voltage scaling jointly such that we minimize the device size while maximizing performance and power efficiency of power-constrained processors. Finally, we extend our analysis and optimization for multi-core processors adopting frequency-island clocking scheme. Our experimental results using a 32nm technology model demonstrates that the joint optimization considering both die-to-die and within-die variations reduces the size of power-gating devices by more than 50% with 3% frequency improvement for power-constrained multi-core processors. Further, the optimal size of power-gating devices for multi-core processors using the frequency-island clocking scheme increases gradually while the optimal supply voltage decreases as the number of cores per die increases.