(621d) Surrogate-Based Optimization for Mixed-Integer Nonlinear Problems

The design and synthesis of chemical engineering system depends on high-fidelity computer simulations. While these simulations provide more accurate data to represent a physical phenomenon, these simulations often are not in the form of a tractable system of algebraic equations, making it difficult to obtain near optimal solutions. One way to overcome this challenge is surrogate-based optimization. A surrogate-based optimization, also known as black-box optimization, involves constructing an approximation model to replace a computationally expensive computer simulation. The surrogate models are then optimized using deterministic solvers. This framework has been studied extensively for nonlinear problems (NLPs) and several algorithms currently exist [1-3] for continuous variables.

However, many chemical engineering problems contain both continuous and discrete variables. For example, process synthesis problems contain binary variables for the selection of process design and continuous variables for operating conditions; for the optimization of distillation columns, the number of stages are integers. These problems are known as mixed-integer nonlinear problems (MINLPs), and surrogate-based optimization for MINLPs has only recently been studied. Existing algorithms have only been tested on low-dimensional problems with only a few nonlinear constraints, and they do not handle discreteness directly when fitting surrogate models. Instead, these algorithms relax the integrality constraint to construct smooth surrogate models [4-6]. An alternative approach for optimization of mixed-variable surrogate-based systems is a brute-force approach, which implies fitting different surrogate models for each discrete realization. This approach is computationally intractable (i.e., a problem with 10 binary variables would require the fitting and optimization of 1024 surrogates). While sampling-based algorithms, such as genetic algorithms and mesh adaptive direct search, can also be used as an alternative, these algorithms tend to require many samples to converge; thus, they may not be suitable for simulation-based problems with high computational cost.

In this work, we extend our previous work both with respect to methodology and algorithmic implementation and propose a novel surrogate-based algorithm for constrained MINLPs that can handle discrete variables efficiently. Both neural network and Gaussian process models are used as surrogates, and one-hot encoding is used to construct a tractable mixed-variable surrogate model without relaxing integrality constraints. The proposed algorithm can handle both black-box (i.e., all constraints and objective are unknown) and gray-box (i.e., some constraints or objective are known) problems. The proposed algorithm is tested on a set of benchmark MINLP problems and process synthesis and design case studies. The proposed methodology outperforms existing algorithms and is able to locate global optima more efficiently.

References:

1. Boukouvala, F. and C.A. Floudas, ARGONAUT: AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems. Optimization Letters, 2017. 11(5): p. 895-913.
2. Henao, C.A. and C.T. Maravelias, Surrogate-based superstructure optimization framework. AIChE Journal, 2011. 57(5): p. 1216-1232.
3. Jones, D.R., M. Schonlau, and W.J. Welch, Efficient Global Optimization of Expensive Black-Box Functions. Journal of Global Optimization, 1998. 13(4): p. 455-492.
4. Holmström, K., N.-H. Quttineh, and M.M. Edvall, An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization. Optimization and Engineering, 2008. 9(4): p. 311-339.
5. Mueller, J., MISO - Mixed Integer Surrogate Optimization. 2016: United States.
6. Rashid, K., S. Ambani, and E. Cetinkaya, An adaptive multiquadric radial basis function method for expensive black-box mixed-integer nonlinear constrained optimization. Engineering Optimization, 2013. 45(2): p. 185-206.