WOMBAT 2024

Abstract: Over the past two decades, robust optimization techniques have efficiently addressed decision problems under uncertainty, offering high assurance of feasibility without being overly conservative. However, research on estimating parameters for these robust decision models from data has been lacking. In this paper, we focus on a unified framework for robust decision models that integrate robust optimization and robust satisficing paradigms. In particular, we identify two layers of uncertainty: outcome uncertainty, involving deviations from specified inputs based on historical data, and estimation uncertainty, addressing deviations from latent input vectors, such as the unobservable true mean. We introduce estimation and prediction procedures tailored to reduce conservativeness while enhancing feasibility within this unified robust decision framework. The concept of minimum volume confidence set is introduced, which minimizes the size of the outcome confidence set while considering the likelihood of infeasibility, thereby improving the precision of uncertainty characterization. This method also accommodates asymmetric uncertainty by adjusting the confidence set accordingly. Additionally, our framework incorporates an affine predictive model that leverages side information to improve input vector predictions, seamlessly integrating into robust decision models. Our method has been implemented in the algebraic modelling toolbox, RSOME, facilitating practical applications.

Abstract: Compared to first-order optimisation algorithms, Newton-type methods offer significant theoretical and empirical advantages, including enhanced resilience to ill-conditioning, reduced sensitivity to hyperparameter uncertainty, superior local convergence, improved communication efficiency in distributed applications, and affine invariance. However, unlike the simplicity of first-order methods, second-order algorithms often involve non-trivial subproblems that must be solved at each iteration. Although the appropriate resolution of these subproblems is crucial to the effectiveness of the optimisation algorithm, they are often treated as an afterthought. This series of lectures will explore various Newton-type methods, focusing on the role of subproblem solvers. It will demonstrate how effectively leveraging suitable solvers can eliminate unnecessary safeguards and assumptions, resulting in simpler algorithms and analyses. We will cover a range of optimisation settings, including convex and non-convex problems, as well as unconstrained and constrained scenarios, along with both shared memory and distributed environments.