This work explores the development of a home energy management system (HEMS) that uses weather and market forecasts to optimize the usage of home appliances and to manage battery usage and solar power production. A Moving Horizon Estimation (MHE) application is used to find the unknown home model parameters. These parameters are then updated in a Model Predictive Controller (MPC) which optimizes and balances competing comfort and economic objectives. Combining MHE and MPC applications alleviates model complexity commonly seen in HEMS by using a lumped parameter model that is adapted to fit a high-fidelity model. Heating, ventilation, and air conditioning (HVAC) on/off behaviors are simulated by using Mathematical Program with Complementarity Constraints (MPCCs) and solved in near real time with a non-linear solver. Removing HVAC on/off as a discrete variable and replacing it with an MPCC reduces solve time. The results of this work indicate that energy management optimization significantly decreases energy costs and balances energy usage more effectively throughout the day. A case study for Phoenix, Arizona shows an energy reduction of 21% and a cost reduction of 40%. This simulated home contributes less to the grid peak load and therefore improves grid stability and reduces the amplitude of load-following cycles for utilities. The case study combines renewable energy, energy storage, forecasts, cooling system, variable rate electricity plan and a multi-objective function allowing for a complete home energy optimization assessment. There remain several challenges, including improved forecast models, improved computational performance to allow the algorithms to run in real time, and mixed empirical/physics-based machine-learning methods to guide the model structure.