Enhancing robustness and sparsity via mathematical optimization

Resumo

This thesis is focused on deriving robust or sparse approaches under an optimization perspective for problems that have traditionally fell into the Operations Research or the Statisics fields. In particular, the aim of this Ph.D. dissertation is to merge optimization techniques with statistical concepts, leading to novel methods that may outperform the classic approaches and bridge theoretical mathematics with real life problems. On one hand, the proposed robust approaches will provide new insights into the modelling and interpretation of classic problems in the Operations Research area, yielding solutions that are resilient to uncertainty of various kinds. On the other hand, the sparse approaches derived to address some up-to-the-minute topics in Statistics will take the form of Mixed Integer Non-Linear Programs (i.e. optimization problems with some integer or binary variables and non linear objective function, denoted as MINLP thereafter). The proposed methods will be shown to be computationally tractable and to enhance interpretability while attaining a good predictive quality. More specifically, Chapter 1 is focused on discovering potential causalities in multivariate time series. This is undertaken by formulating the estimation problem as a MINLP in which the constraints model different aspects of the sparsity, including constraints that do not allow spurious relationships to appear. The method shows a good performance in terms of forecasting power and recovery of the original model. Analogously, in Chapter 2 the aim is to discover the relevant predictors in a linear regression context without carrying out significance tests, since they may fail in the presence of strong collinearity. To this aim, the preferred estimation method is tightened, deriving MINLPs in which the constraints measure the significance of the predictors and are designed to avoid collinearity issues. The tightened approaches attain a good trade-off between interpretability and accuracy. In contrast, in Chapter 3 the classic newsvendor problem is generalized by assuming correlated demands. In particular, a robust inventory approach with distribution-free autoregressive demand is formulated as an optimization problem, using techniques that merge statistical concepts with uncertainty sets. The obtained solutions are robust against the presence of noises with high variability in the data while avoiding overconservativeness. In Chapter 4 this formulation is extended to multivariate time series in a more complex setting, where decisions over the location-allocation of facilities and their production levels are sought. Empirically, we illustrate that, in order to design an efficient supply chain or to improve an existent one, it is important to take into account the correlation and variability of the multivariate data, developing data-driven techniques which make use of robust forecasting methods. A closer examination of the specific characteristics of the problem and the uncertainty sets is undertaken in Chapter 5, where the portfolio selection problem with transaction costs is considered. In this chapter, theoretical results that relate transaction costs with different ways of protection against uncertainty of the returns are derived. As a consequence, the numerical experiments show that calibrating the transaction costs term yields to results that are resilient to estimation error.