(288f) Global Optimization of Grey-Box Constrained Models

Grey-box optimization methods have a wide range of applicability in various fields which rely on expensive simulations or solely input-output data [1]. Consequently, there has been great interest in the development of methods for the optimization of models for which derivative information is either not available or prohibitively costly, such as expensive finite-element or partial-differential equation systems, flowsheet simulations, mechanical engineering design and molecular geometry design problems, to name a few [2, 3]. Most of the existing methodologies have been developed for unconstrained or bound constrained problems, while extensions to constrained cases have been performed through heuristic or penalty-type methods [3]. There are many opportunities in the development of global derivative-free optimization methods which can rigorously handle general non-linear constraints of unknown closed-form, with a limited number of samples. Motivated by our recent work for the constrained optimization of a complex, non-linear partial differential equation system for pressure swing adsorption [4], this work aims to introduce a general grey-box optimization algorithm which is suitable for a wide range of applications with multiple constraints.

The novel features of the presented methodology include: (a) rigorous selection of the sample data for model development and model updating though optimization of the probabilistic distance of the samples in both the input and the output spaces [5], (b) exploration of a diverse set of both interpolating and non-interpolating functional forms (quadratic, kriging, radial-basis functions) for representing the objective function and each of the constraints individually, (c) global optimization of both the surrogate-model parameters and the formulated surrogate problem though deterministic global optimization [6, 7], and finally (d) initial global search strategy followed by partitioning of the input space using clustering techniques.

Results are presented for a set of test problems for constrained global optimization as well as for a real application of a pressure swing adsorption system for a large set of materials, in order to illustrate the performance of the method for problems with increasing complexity in feasibility. Comparative analysis of the developed method with commercially available software for this class of problems is also presented.  Finally, the effect of the variability in the initial sampling on the consistency of the method will be discussed, while a new approach for exploiting information from simplified physics or mesh coarsening of the original expensive simulation to reduce this effect will be presented.

References:

1.         Forrester, A.I.J., A. Sóbester, and A.J. Keane, Engineering Design via Surrogate Modelling - A Practical Guide. 2008, John Wiley & Sons.

2.         Conn, A.R., K. Scheinberg, and L.N. Vicente, Introduction to derivative-free optimization. MPS-SIAM Series on Optimization. 2009, Philadelphia: SIAM.

3.         Rios, L. and N. Sahinidis, Derivative-free optimization: a review of algorithms and comparison of software implementations. Journal of Global Optimization, 2012: p. 1-47.DOI 10.1007/s10898-012-9951-y

4.         Hasan, M.M.F., et al., Modeling, Simulation, and Optimization of Postcombustion CO2 Capture for Variable Feed Concentration and Flow Rate. 2. Pressure Swing Adsorption and Vacuum Swing Adsorption Processes. Industrial & Engineering Chemistry Research. 51(48): p. 15665-15682

5.         Zukui Li and C.A. Floudas, Optimal Scenario Reduction Framework based on Distance of Uncertainty Distribution and Output Performance: I. Single Reduction via Mixed Integer Linear Optimization. In Preparation

6.         Misener, R. and C. Floudas, GloMIQO: Global mixed-integer quadratic optimizer. Journal of Global Optimization, 2012: p. 1-48.DOI:10.1007/s10898-012-9874-7

7.         Misener, R. and C. Floudas, ANTIGONE: Algorithms for coNTinuous/ Integer Global Optimization of Nonlinear Equations. Submitted, 2013