(520d) A Chance-Constrained Framework for Optimization Under Uncertainty

Nowadays, the process industry is highly competitive and has to fulfill high quality standards for their products. Therefore, the use of computer-aided optimization software became an integral part of today’s planning in process industries. The increase of available computing power and important advancements in algorithms led to more and more complex models of real-world problems. In general those models are characterized by deterministic assumptions. Probabilities and uncertainties are only rarely considered as stochastic influences but as expected values. Anyway, since several years, uncertain parameters can be taken into account by implementing so called chance constraints [1]. Since chemical processes are subject to multiple uncertainties, the use of chance constraints can lead to more realistic models and give further useful information, as shown in several examples.

Even though chance constraints are known for several years, only few models using them were developed so far [2]. The major reason for the little use of chance constraints might be the absence of support of chance constraints by usual optimization software. Therefore the evaluation of chance constraints requires external processing, including multivariate numerical integration, root-finding and the use of further mathematical software. Those algorithms had to be individually implemented for each model so far. Furthermore, the evaluation of the chance constraints becomes often the most time consuming part of the optimization process, therefore a not-optimized implementation might lead to unacceptable high computation time.

This work presents a framework for implementing chance constraints in optimization models. It can be easily linked to numerous different solvers and provides interfaces to adopt further mathematical software or tools. The use of this framework should significantly reduce the effort to implement uncertain parameters into own optimization models.

Beside the usability, the improvement of the computing time was an essential part of the framework design. Therefore, the calculating algorithm was analyzed and key elements identified. Beside the provision of second derivatives for the solver, major computational time reduction could be realized by improving the integration procedure and the root-finding algorithm. The integration procedure was accelerated using fewer integration points without losing numerical precision [3]. The root-finding step could be improved by using higher derivatives for the Newton’s method and implementing a data analysis for better starting points. Those improvements led to a significant lower computation time.

The talk will present the use of the framework on the basis of an example and give an insight in the algorithmic improvements.

[1] H. Arellano-Garcia, G. Wozny. (2009) Chance Constrained Optimization of Process Systems under Uncertainty: Strict Monotonicity. Computers and Chemical Engineering. 33 (1568–1583)

[2] T. Barz, H. Arellano-Garcia, G. Wozny. (2011) Robust Implementation of Optimal Decisions using a Two-Layer Chance Constrained Approach. Industrial and Engineering Chemistry Research. doi:10.1021/ie1014525

[3] T. Barz, G. Wozny, H. Arellano-Garcia. (2011) An Efficient Sparse Approach to Sensitivity Generation for Large-Scale Dynamic Optimization. Computers and Chemical Engineering. doi:10.1016/j.compchemeng.2010.10.008