Although Reliability-based Optimization (RBO) is a sound approach for design in engineering, its application has remained limited in the past due to high numerical costs associated with its solution process. In view of this fact, this contribution proposes a framework for solving most efficiently a particular class of RBO problems, namely the weight minimization of linear structures subject to dynamic Gaussian excitation. The associated reliability problem is solved by means of an efficient simulation technique while the optimization problem is addressed applying a gradient-based scheme. In order to apply this optimization scheme, an efficient procedure for estimating reliability sensitivity is applied. This procedure consists in introducing local approximations of functions related with reliability and structural performance. An example involving a two degree-of-freedom system subject to a stochastic ground acceleration illustrates the efficiency and effectiveness of the proposed framework.