# (456c) Nonlinear Robust Optimization for Process Design and Operations

Authors:
University of Alberta
University of Alberta
University of Alberta
Robust optimization has become an active research area for addressing uncertainty in optimization problems and significant progress has been made in the past. However, most of the past works are limited to linear optimization (or linear in uncertain parameters) problems [1-3]. On the other hand, many realistic problems are formulated as nonlinear optimization problems and only very few work has studied the corresponding robust optimization technique [4]. To address this issue, we propose a novel robust optimization framework that can be used to address general nonlinear problems those are commonly seen in process design and process operations optimization. The major techniques used in the proposed method include local linearization with respect to the uncertain parameters, multiple local approximation and local affinely adjustable decision rule. The problems to be addressed can be classified as the following categories based on the complexity level of the nonlinear problems.

In the first case, the uncertainty is only involved in the inequality constraints. This is the simplest case of nonlinear optimization problems and it is often seen in process design or operations optimization problem with only static decisions. Here the objective is to find a robust decision that is feasible to all the possible uncertainty realizations in an uncertainty set. Linearization can be directly applied to the nonlinear inequality constraint and robust counterpart optimization formulation is then applied. In the second case, the nonlinear optimization problem involves design variables and state variables coupled by equality constraints, and inequality constraints are enforced for some state variables. In such optimization problems, there will be equality constraints containing uncertain parameters and the traditional robust optimization method cannot be directly applied. However, the state variables can be determined after the design variables as well as the uncertain parameters are fixed. Using the implicit function theorem, the state variables can be replaced by a function of uncertain parameters and design variables. Then the robust optimization formulation can be applied through linearization of the inequality constraints. Third, the nonlinear problem involves design variables, operation variables and state variables. For this type of problem, the operation variables can be adjusted based on the realization of the uncertainty. Correspondingly, a local affinely adjustable decision rule is adopted for the operation variables (i.e., an affine function of the uncertain parameter). The decision rule will be applied in the nonlinear problem and the problem can be reduced to the second case.

For all the above problems, the linearization and robust optimization formulation will be enforced around different samples of the uncertain parameters. The problem is solved by an iterative algorithm. Samples are randomly generated in each iteration, and the optimal solution obtained will be tested to find a new sample point that violates the constraints. The new sample violating the constraints is added to the robust optimization model, and the algorithm stops until no new violating sample is found. Application of the proposed method in various process design and process operations examples will be investigated to verify the efficacy of the nonlinear robust optimization algorithm.

Reference:
[1] Ben-Tal, A., Nemirovski, A. (1999). Robust solutions of uncertain linear programs.Â Operations research letters,Â 25(1), 1-13.

[2] Li, Z., Ding, R., Floudas, C.A. (2011). A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization.Â Industrial & engineering chemistry research,Â 50(18), 10567-10603.

[3] Bertsimas D., Sim M. (2004). The Price of Robustness. Operations Research, 52(1):35â??53.

[4] Zhang, Y. (2007). General robust-optimization formulation for nonlinear programming.Â Journal of optimization theory and applications,Â 132(1), 111-124.