Robust Optimization in High-Dimensional Data Space with Support Vector Clustering

Abstract

Data-driven robust optimization has attracted immense attentions. In this work, we propose a data-driven uncertainty set for robust optimization under high-dimensional uncertainty. We propose to first decompose the high-dimensional data space into the principal subspace and the residual subspace by employing principal component analysis, and then adopt support vector clustering and classic polyhedral uncertainty set to describe the intricate geometry in the principal subspace and the tiny variations in the residual subspace, respectively, giving rise to a new data-driven uncertainty set. Similar to classic uncertainty sets, the proposed data-driven uncertainty set can also preserve the tractability of robust optimization problems. In addition, we establish the probabilistic guarantee theoretically by further calibrating the uncertainty set with an independent dataset, which ensures that the data-driven uncertainty set covers a portion of uncertainty with a given confidence level. Numerical results show the effectiveness of the proposed uncertainty set in reducing conservatism of robust optimization problems as well as the fidelity of the established probabilistic guarantee.