Cloud Computing is a promising approach to handle the growing needs for computation and storage in an efficient and cost-effective manner. Towards this end, characterizing workloads in the cloud infrastructure (e.g., a data center) is essential for performing cloud optimizations such as resource provisioning and energy minimization. However, there is a huge gap between the characteristics of actual workloads (e.g., they tend to be bursty and exhibit fractal behavior) and existing cloud optimization algorithms, which tend to rely on simplistic assumptions about the workloads. To close this gap, based on fractional calculus concepts, we present a fractal model to account for the complex dynamics of cloud computing workloads (i.e., the number of request arrivals or CPU/memory usage during each time interval). More precisely, we introduce a fractal operator to account for the time-varying fractal properties of the cloud workloads. In addition, we present an efficient (online) parameter estimation algorithm, an accurate forecasting strategy, and a novel fractal-based model predictive control approach for optimizing the CPU utilization, and hence, the overall energy consumption in the system while satisfying networked architecture performance constraints like queue capacities. We demonstrate advantages of our fractal model in forecasting the complex cloud computing dynamics over conventional (non-fractal) models by using real-world cloud (Google) traces. Unlike non-fractal models, which have very poor prediction capabilities under bursty workload conditions, our fractal model can accurately predict bursty request processes, which is crucial for cloud computing workload forecasting. Finally, experimental results demonstrate that the fractal model based optimization outperforms the non-fractal based ones in terms of minimizing the resource utilization by an average of 30%.