We consider the problem of sequentially testing a simple null hypothesis versus a composite alternative hypothesis that consists of a finite set of densities. We study sequential tests that are based on thresholding of mixture-based likelihood ratio statistics and weighted generalized likelihood ratio statistics. It is shown that both sequential tests have several asymptotic optimality properties as error probabilities go to zero. First, for any weights, they minimize the expected sample size within a constant term under every scenario in the alternative hypothesis and at least to first order under the null hypothesis. Second, for appropriate weights that are specified up to a prior distribution, they minimize within an asymptotically negligible term a weighted expected sample size in the alternative hypothesis. Third, for a particular prior distribution, they are almost minimax with respect to the expected Kullback-Leibler divergence until stopping. Furthermore, based on high-order asymptotic expansions for the operating characteristics, we propose prior distributions that lead to a robust behavior. Finally, based on asymptotic analysis as well as on simulation experiments, we argue that both tests have the same performance when they are designed with the same weights.