Technology scaling has increased the transistor's susceptibility to process variations in nanometer very large scale integrated (VLSI) circuits. The effects of such variations are having a huge impact on performance and hence the timing yield of the integrated circuits. The circuit optimization objectives namely power, area, and delay are highly correlated and conflicting in nature. The inception of variations in process parameters have made their relationship intricate and more difficult to optimize. Traditional deterministic methods ignoring variation effects negatively impacts timing yield. A pessimistic worst case consideration of variations, on the other hand, can lead to severe over design. In this context, there is a strong need for re-invention of circuit optimization methods with a statistical perspective. In this dissertation, we model and develop novel variation aware solutions for circuit optimization methods such as gate sizing, timing based placement and buffer insertion. The uncertainty due to process variations is modeled using interval valued fuzzy numbers and a fuzzy programming based optimization is proposed to improve circuit yield without significant over design. In addition to the statistical optimization methods, we have proposed a novel technique that dynamically detects and creates the slack needed to accommodate the delay due to variations. The variation aware gate sizing technique is formulated as a fuzzy linear program and the uncertainty in delay due to process variations is modeled using fuzzy membership functions. The timing based placement technique, on the other hand, due to its quadratic dependence on wire length is modeled as nonlinear programming problem. The variations in timing based placement are modeled as fuzzy numbers in the fuzzy formulation and as chance constraints in the stochastic formulation. Further, we have proposed a piece-wise linear formulation for the variation aware buffer insertion and driver sizing (BIDS) problem. The BIDS problem is solved at the logic level, with look-up table based approximation of net lengths for early variation awareness.In the context of dynamic variation compensation, a delay detection circuit is used to identify the uncertainty in critical path delay. The delay detection circuit controls the instance of data capture in critical path memory flops to avoid a timing failure in the presence of variations. In summary, the various formulation and solution techniques developed in this dissertation achieve significantly better optimization compared to related works in the literature. The proposed methods have been rigorously tested on medium and large sized benchmarks to establish the validity and efficacy of the solution techniques.