This paper is concerned with a dual object of the so-called viability kernel of a control system with mixed (state-control) constraints. This object, called the set of sustainable thresholds, describes the possible thresholds for which a given initial condition is viable. In this work, we are concerned with characterizing the weak and strong Pareto fronts of the set of sustainable thresholds and providing a practical method for computing such objects based on optimal control theory and a level-set approach. A numerical example, relying on renewable resource management, is shown to demonstrate the proposed method.