This work addresses the design of polymers with optimal levels of macroscopic properties through the use of topological indices. Specifically, two zeroth-order and two first-order connectivity indices are for the first time employed as descriptors in structure-property correlations in an optimization study. Based on these descriptors, a set of new correlations for heat capacity, cohesive energy, glass transition temperature, refractive index, and dielectric constant are proposed. These correlations are incorporated into an optimization framework. The proposed mathematical description, utilizing the concept of a basic group, accounts fully for molecular interconnectivity. When the nonconvex terms are appropriately recast in the formulation as convex inequalities, a convex mixed-integer nonlinear (MINLP) representation is obtained. Three example problems highlight the proposed molecular description, structure-property correlations, convex MINLP optimization formulation, and solution technique.