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Stability is an essential indicator of proper system operation. To this end, it is often of interest to predict the Region of Asymptotic Stability (RAS) associated with an equilibrium point of interest. In the case of the presence of constraints that need to be satisfied, one is faced with the problem of estimating a Restricted RAS (RRAS). Practical methods for estimating RRASs have been proposed and demonstrated for relatively high-order systems. These methods are based on Lyapunov's second method. However, they do not address the problem of constructing an optimal Lyapunov function. In this paper, a method of finding a Lyapunov function that yields improved estimates of the restricted region of asymptotic stability is set forth. In addition, a new method for estimating RRASs based on the average of the Lyapunov derivative is also set forth. Numerical examples illustrate the utility of the proposed methods.