This talk will explore optimization approaches to solving initial value problems. These differential equations describe rates of change in time-- for example, the rate of change of temperatures in an energy simulation tool like EnergyPlus, or the rate of change of pollutant concentrations in an air quality tool like CONTAM. Standard numerical approaches convert these equations into algebraic relations, rather than applying optimization. Optimization promises good error and stability control, even when taking single steps. Single-step methods are useful for simulating discontinuous systems, such as buildings, where control systems, scheduled events, and rapidly-changing driving forces make it difficult to extend algebraic methods across multiple time steps. However, optimization is also computationally expensive. Using a mix of math and geometry, I will outline the optimization approach, present a few examples, and describe the research needed to develop efficient implementations.