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