In this paper, we introduce a computational framework based on an iterative twisted approximation (ITROX) and a set of associated algorithms for various sequence design problems. The proposed computational framework can be used to obtain sequences (or complementary sets of sequences) possessing good periodic or aperiodic correlation properties and, in an extended form, to construct zero (or low) correlation zone sequences. Furthermore, as constrained (e.g., finite) alphabets are of interest in many applications, we introduce a modified version of our general framework that can be useful in these cases. Several applications of ITROX are studied and numerical examples (focusing on the construction of real-valued and binary sequences) are provided to illustrate the performance of ITROX for each application.