Data-Driven Optimization under Uncertainty
Creators: Yang, Shu-Bo
Publisher: name: University of Alberta Library
Year: 2023

<p>With the advances made in machine learning and data science, data-driven modeling and optimization techniques have garnered significant attention in recent years. However, despite the availability of various data-driven methods for addressing optimization problems under uncertainty, their practical applicability is often constrained by their own limitations such as high computational costs and reliance on non-trivial assumptions. This thesis aims to investigate and develop novel data-driven approaches to address various optimization problems involving uncertainty and overcome the limitations of existing methods.  </p>

<p>First, we propose a novel framework to address process optimization under surrogate model prediction uncertainty. The framework involves approximating an ensemble surrogate model with a computationally efficient mixture density network (MDN) and embedding the MDN into a chance-constrained optimization problem with a mean-variance-type objective. This method reduces the high computational cost seen in other existing methods for optimization considering surrogate model prediction uncertainty. This approach is demonstrated through a numerical example and two case studies, showcasing its capability to solve various optimization problems under surrogate model prediction uncertainty. </p>

<p>Second, we develop a new neural network (NN)-based approach for solving the uncertain optimization problems in joint chance-constrained formulations (joint chance-constrained optimization problems, JCCPs). The approach involves approximating a joint chance constraint (JCC) with a NN and incorporating the NN into the optimization model. This method makes NP-hard JCCPs tractable and deterministically solvable. This method is applied to a process optimization problem to show its performance in solving a nonlinear JCCP. Furthermore, we extend the above method to handle joint chance-constrained stochastic optimal control problems (JCC-SOCPs). We replace the NN with the recurrent neural network (RNN) for the approximation of the JCC in a JCC-SOCP. This method has a much lower computational burden than other commonly used stochastic optimal control approaches. This approach is applied to a numerical SOCP example and a case study to demonstrate its efficacy.</p>

<p>Third, we propose a novel distributionally robust chance-constrained optimization (DRCCP) method to handle JCCPs without knowing true uncertainty distributions. This DRCCP method is based on the kernel ambiguity set established by utilizing the maximum mean discrepancy (MMD). This approach overcomes the restrictions of the popular Wasserstein DRCCP which necessitates complicated assumptions on uncertain constraints. A numerical example and a nonlinear process optimization problem are studied to demonstrate the efficacy of the presented DRCCP method. Subsequently, this DRCCP approach is further combined with a neural network-like deep kernel to enhance its performance. The effectiveness of this deep kernel-based DRCCP is demonstrated by applying it to a case study. </p>

<p>Fourth, we develop another novel DRCCP method to further overcome more limitations of the popular Wasserstein DRCCP. This method is based on the Sinkhorn ambiguity set constructed by using the Sinkhorn distance. This approach outperforms the Wasserstein DRCCP by being assumption-free on uncertain constraints and being able to hedge against more general families of uncertainty distributions. The performance of this method is evaluated through a numerical example and a nonlinear process optimization.</p>

<p>Fifth, we develop an innovative iterative algorithm that can remove outliers and extreme data samples leading to overly conservative DRCCP solutions. The presented algorithm can significantly improve the DRCCP solution quality, and it can simultaneously ensure the feasibility of the DRCCP solution. Moreover, the proposed algorithm is compatible with various DRCCP models such as Wasserstein DRCCP, kernel DRCCP, Sinkhorn DRCCP, etc. </p>

<p>Overall, we make several contributions through this research. From a methodological perspective, we develop several novel data-driven approaches for optimization under uncertainty and significantly overcome the limitations of existing popular methods. From an application perspective, our proposed methods can be applied to various real-world optimization problems such as process optimization and optimal control.</p>