(264b) Rank-Filter-Based Algorithm to Stochastic Minlp Optimization

Many chemical engineering problems can be formulated as a stochastic mixed integer nonlinear programming (MINLP) problem [1-5]. To find the global optimum, both the discrete decision variables and continuous design/control variables are required to be optimized simultaneously under uncertainty. One key problem to the stochastic MINLPs is the evaluation and optimization of the expected performance for a fixed decision variable.

In this study, an approximation method is developed for evaluating a decision's expectation. The coarse expected performance is calculated using a short Monte Carlo simulation and intermediate optimization results of nonlinear programming (NLP). The short simulation uses a small sample size, which consists of a small number of realizations for random variables, and the intermediate results are obtained by the truncated iteration procedures of an NLP solver.

A rank-based filter is proposed, which fully utilizes the information of relative order stored in the coarse expectations of different decisions. This filter separates the ?good? decisions from the others by ranking the coarse expectations. The rigorous calculations of the expectations are performed for the selected ?good? decisions. The accurate expected performances require a long Monte Carlo simulation and the precise optimal results of NLP solvers. The optimum of the original stochastic MINLP problem is obtained by comparing the accurate expectations of ?good? decisions. The rank-filter-based algorithm guarantees extra high computation efficiency because the rigorous calculation of the expected performances is performed for the set of ?good? decisions instead of that of all the decisions. The reduction in the number of samplings and NLP optimizations leads to a significant saving in computation time.

Three benchmark problems are illustrated to show the computation efficiency of this proposed algorithm. The first two benchmarks [6-7] are the process synthesis/planning problems. Although these two examples share the same superstructure of process network, they have different number of random variables and different probability/cumulative distribution functions. The calculation results show that this proposed algorithm is robust to the number and distribution functions of random variables in a stochastic MINLP problem.

The last example [8-9] is challenging because it is formulated as a non-convex MINLP problem with a huge decision space. This example consists of 32000 combinations of discrete decision variables, 15 continuous operation variables and 4 random variables. The calculation results demonstrate that the rank-filter-based algorithm can reduce several orders of magnitude of computational expense in the design and optimization of complicated chemical process systems.

[1] Lorenz T. Biegler, & Ignacio E. Grossmann, "Retrospective on optimization", Computers & Chemical Engineering, 28(8), 1169-1192, 2004.

[2] Ignacio E. Grossmann, & Lorenz T. Biegler, "Future perspective on optimization", Computers & Chemical Engineering, 28(8), 1193-1218, 2004.

[3] Stacy L Janak, & Christodoulos A. Floudas, "Advances in robust optimization approaches for scheduling under uncertainty", Computer Aided Chemical Engineering, 20(2), 1051-1056, 2005.

[4] Joaquín Acevedo, & Efstratios N. Pistikopoulos, "Stochastic optimization based algorithms for process synthesis under uncertainty", Computers & Chemical Engineering, 22 (4-5), 647-671, 1998.

[5] Nikolaos V. Sahinidis, "Optimization under uncertainty: state-of-the-art and opportunities", Computers & Chemical Engineering, 28 (6-7), 971-983, 2004.

[6] Marianthi G. Ierapetritou, Joaquín Acevedo, & Efstratios N. Pistikopoulos, "An optimization approach for process engineering problems under uncertainty", Computers & Chemical Engineering, 20 (6-7), 703-709, 1996.

[7] Vishal Goyal, Marianthi G. Ierapetritou, "Stochastic MINLP optimization using simplicial approximation.", Computers & Chemical Engineering, 31(9), 1081-1087, 2007.

[8] C. S. Adjiman, I. P. Androulakis, & C. A. Floudas, "Global Optimization of Mixed Integer Nonlinear Problems", AIChE Journal, 46, 1769-1797, 2000.

[9] T. Westerlund, F. Pettersson, & I. E. Grossmann, "Optimization of pump configurations as a MINLP problem.", Computers & Chemical Engineering,

18, 845-858, 1994.