For most optimisation problems, the uncertainty in the output results, due to errors arising from the estimation of the system parameters used in the optimisation is usually difficult to measure. We propose a strategy which combines the ability to approximate the experimental errors and evaluate the system parameters based on these errors with the ability to solve the optimisation problem in the presence of these errors. The proposed strategy transforms the data into intervals using statistical methods and then uses these intervals to evaluate worst case scenarios of the constraints and objective functions of the optimisation problem. The strategy employs a posteriori error estimation methods which produce bounds on the constraints and objective functions in order to obtain the “best’’ worst case scenarios. These worst case scenarios reflect the propagation of error from the data and ensure the feasibility of the results of the optimisation problems.
The evaluation of worst case scenarios results in a bilevel optimisation formulation of the actual optimisation model. This resulting bilevel optimisation problem is then solved using a nested-level approach where the inner level is solved for each iteration of the outer level for the case where the outer level problems are assumed to be differentiable. For the case in which the outer level problems are non-differentiable, the strategy uses a semi-infinite formulation which overcomes the evaluation of the worst case scenarios thereby rendering the problems differentiable if the functions are differentiable. The semi-infinite formulation is then solved using a discretisation algorithm. Log-barrier functions and Newton methods, which are known for their rapid convergence to the optimal points, are used as part of the algorithms developed to solve the optimisation problem. The strategy also employs the Reduced-Basis method to solve the functions and/or physical quantities used in the optimisation problem. The method is efficient and computationally economic while maintaining a certain level of accuracy in approximating the functions and/or physical quantities. This efficiency complements the interior point algorithms in obtaining real-time solutions for the optimisation problems.
We then apply the strategy to an irrigation design model whose physical system is governed by the equation for the
conservation of water. The results have provided insights into the internal workings of the proposed strategy, as well as the algorithms developed in solving the resulting formulations.
Doing research under both Associate Professor Murali and Professor Patera, I had a good grounding in terms of the theory and practical aspects of the strategy. I was also fortunate to have the guidance and input from Professor Robert Freund and Professor Sun Jie in the field of optimisation. For the most part of my Ph.D. project, I had the opportunity
interact with the Reduced-Basis research group under Professor Patera where we had fortnightly video conferencing sessions to discuss our various projects. These sessions were very useful in generating ideas and exposing me to other research work.