(194h) Model Reduction-Based Global Optimisation for Large-Scale Steady State Nonlinear Systems

PDE-constrained black/grey box modelling and optimisation methodologies have wide applicability in industrial engineering areas where many finite approximation-based generic commercial simulators or input-output data from real complex systems are available [1]. However, many simulators solely provide input-output results of simulation process and cannot perform optimisation tasks. Hence, the optimisation of real complex systems is problematic. Even if the high dimensional complex model codes are available, the cost of computing derivatives is often unacceptable. One of the most effective techniques to reduce computational costs for input/output systems is the use of equation-free methods [2]. Recently, equation-free based reduced SQP algorithms have been exploited for large-scale local optimisation with black-box steady-state simulators [3, 4]. With a small set of input-output data from the simulator, matrix-free reduced Jacobian and Hessian matrix techniques can be employed. Moreover, high nonlinearity of such large-scale distributed parameter system decides existence of multiple local optimums. Deterministic global optimisation algorithms (GOP) can compute global optimal solutions offering theoretical guarantees on the global optimality. However, distributed parameter systems pose computational challenges for these optimisation methods [5].

The aim of this work is to construct deterministic global optimisation methods for large-scale input/output simulators. Deterministic global optimisation methods are usually computationally intensive due to the repeated utilisation of branch-and-bound algorithms. Hence, in terms of computation cost, detailed models of large-scale problems are very hard or even impossible to deal with. Model reduction techniques can produce low-order systems that are computationally amenable. In this work, a combined principal component analysis (PCA) and artificial neural networks (ANNs)-based model reduction methodology is employed for the global optimisation of large-scale distributed steady state systems. The basic PCA-ANN-GOP framework performs well for small-scale problems. However, the optimisation problem is hard to solve due to the high non-convexity of activation functions in the reduced ANN structure [6]. To enlarge capability of our optimisation framework, two kinds of improvements has been made: The first, is a novel piece-wise linear approximation (PWA) reformulation for the nonlinear activation function. The computation results show this PWA model can reduce the complexity and ensure the accuracy. The second improvement is the replacement of the nonlinear activation function with a continuous piece-wise linear function. The performance of the improved PCA-ANN-GOP framework is demonstrated through an illustrative nonlinear large-scale chemical engineering example: a complex large-scale combustion process.

[1] Boukouvala, F., Hasan, M.F. and Floudas, C.A., 2017. Global optimization of general constrained grey-box models: new method and its application to constrained PDEs for pressure swing adsorption. Journal of Global Optimization, 67(1-2), pp.3-42.

[2] Theodoropoulos, C., 2011. Optimisation and linear control of large scale nonlinear systems: a review and a suite of model reduction-based techniques. In Coping with Complexity: Model Reduction and Data Analysis (pp. 37-61).Springer, Berlin, Heidelberg.

[3] Bonis, I. and Theodoropoulos, C., 2012. Model reduction-based optimization using large-scale steady-state simulators. Chemical Engineering Science,69(1), pp.69-80.

[4] Petsagkourakis, P., Bonis, I. and Theodoropoulos, C., 2018. Reduced Order Optimization of Large-Scale Nonlinear Systems with Nonlinear Inequality Constraints Using Steady StateSimulators. Industrial & Engineering Chemistry Research, 57(30), pp.9952-9963.

[5] Houska, B. and Chachuat, B., 2019. Global optimization in Hilbert space. Mathematical programming,pp.1-29.

[6] Schweidtmann, A.M. and Mitsos, A., 2019. Deterministic global optimization with artificial neural networks embedded. Journal of Optimization Theory and Applications,180(3), pp.925-948.