Energy systems and manufacturing processes of the 21st century are becoming increasingly dynamic and interconnected, which require new capabilities to effectively model and optimize their design and operations. Such next generation computational tools must leverage state-of-the-art techniques in optimization and be able to rapidly incorporate new advances. To address these requirements, we have developed the Institute for the Design of Advanced Energy Systems (IDAES) Integrated Platform, which builds on the strengths of both process simulators (model libraries) and algebraic modeling languages (advanced solvers). This paper specifically presents the IDAES Core Modeling Framework (IDAES-CMF), along with a case study demonstrating the application of the framework to solve process optimization problems. Capabilities provided by this framework include a flexible, modifiable, open-source platform for optimization of process flowsheets utilizing state-of-the-art solvers and solution techniques, fully open and extensible libraries of dynamic unit operations models and thermophysical property models, and integrated support for superstructure-based conceptual design and optimization under uncertainty.