(514g) Rigorous Surrogate-Based Optimization Strategies That Integrate Glass Box/Black Box Process Models

Authors: 
Eason, J. P., Carnegie Mellon University
Biegler, L. T., Carnegie Mellon University
Modern nonlinear programming solvers can efficiently handle very large scale optimization problems when accurate derivative information is available. However, black box or derivative free modeling components are often unavoidable in practice when the modeled phenomena may cross length and time scales. This work is motivated by examples in chemical process optimization where most unit operations have well-known equation oriented representations, but some portion of the model (e.g. a complex reactor model) may only be available with an external function call.

Several approaches have been proposed to handle this situation. Perhaps the easiest method to implement is to consider the entire system as a black box and apply black box optimization in the space of independent variables. Inequality constraints can be handled using a penalty function method. In a flowsheeting context, this could mean repeatedly simulating the flowsheet in a sequential modular manner at various operating points. Unfortunately, converging the equality constraints (converging the flowsheet) requires many calls to the black box function for each function call used by the derivative free solver. Another approach to this problem is the use of surrogate models (i.e. reduced models). A surrogate model is an equation oriented approximation of the black box that allows traditional derivative based optimization to be applied directly. However, optimization tends to exploit approximation errors in the surrogate model leading to inaccurate solutions and repeated rebuilding of the surrogate model. However, even if the surrogate model is perfectly accurate at the solution, this only guarantees that the original problem is feasible. Because optimality conditions require gradient information, a higher degree of accuracy is required.

In this work, we consider the general problem of hybrid glass box/black box optimization, with focus on guaranteeing a sufficient accuracy level in the surrogate model to bound optimality error at a desired level. We first propose an algorithm that combines ideas from SQP filter methods and derivative free trust region methods to solve this class of problems. The black box portion of the model is replaced by a sequence of surrogate models in trust region subproblems. By carefully managing surrogate model construction, the algorithm is guaranteed to converge to optimal solutions of the original, rigorous problem. We then examine how the hybrid structure of the problem can be exploited to reduce the number of updates to the surrogate model. Moreover, with error bounds for a given surrogate model, it is sometimes possible to propagate this error through to optimality error of the surrogate model optimization problem. This information can be used to derive effective termination criteria for the trust region algorithm, or even to inform surrogate model construction by suggesting desired accuracy bounds. These concepts are demonstrated on optimization benchmarks as well as illustrative examples for process optimization.