A convergent optimization method using pattern search algorithms with adaptive precision simulation

Thermal building simulation programs, such as
EnergyPlus, approximate solutions of a differential
algebraic system of equations. While the theoretical
solution is usually continuously differentiable in the building design parameters, the approximate solutions may not even be continuous, due to adaptive variations in solver iterations and the use of adaptive integration meshes. Hence, when a smooth cost function, defined on the design parameters, is evaluated using a thermal building simulation program, it becomes replaced with an approximation that fails to be even continuous. Consequently, when used in conjunction with an optimization algorithm that depends on smoothness of the cost function, the algorithm is quite likely to jam at a non-optimal point. Obviously, in such situations, the potential economic gains that optimization offers are not attained. As an illustration, we present an example, using the EnergyPlus whole building energy simulation program to evaluate our cost function, in which the Hooke-Jeeves algorithm terminates at a nonstationary point. To prevent such failures, we have developed an adaptive simulation precision control algorithm that can be used in conjunction with a family of derivative free optimization algorithms. The resulting composite algorithms are demonstrably convergent to an exact stationary point. We present the main ingredients of the composite algorithms and show by numerical experiments that using coarse approximations in the early iterations can significantly reduce the computation time.