This work addresses constrained design optimization problems of stochastic dynamical systems involving several objective functions. In general, the objective functions are potentially conflicting performance measures, such as structural reliability and construction cost. Within this framework, solutions to multiobjective optimization problems, i.e., Pareto points, provide useful tradeoff information between alternative optimal designs. However, the characterization of the entire Pareto front is computationally prohibitive for complex structural systems under stochastic loads. Thus, the aim of this work is focused on developing an effective methodology for obtaining compromise design solutions. In particular, a compromise programming method based on the so-called aspiration levels is adopted. This formulation leads to a min-max optimization problem, which is solved directly by a stochastic optimization algorithm to account for the non-smooth nature of the compromise objective function. Specifically, a recently proposed two-phase Bayesian model updating framework is considered. An example problem is presented to illustrate the effectiveness and usefulness of the proposed approach.