Multiple Kernel Learning-Based Uncertainty Set Construction for Robust optimization

Abstract

In robust optimization (RO), a focal point is the design of uncertainty set that delineates possible realizations of uncertainty since it heavily impacts the robustness of solutions. We propose in this paper a multiple kernel learning (MKL) based support vector clustering (SVC) method for polytopic uncertainty set construction in data-driven RO. By assuming a set of candidate piecewise linear kernel functions, the MKL framework not only derives an enclosing sphere in the input space, but also automatically derives optimal coefficients of kernel functions by only solving a quadratically constrained quadratic program. The learnt sphere turns out to be a compact polyhedral uncertainty set to be used in RO, which helps reducing the conservatism of robust solutions. Meanwhile, although massive data samples and kernel functions are used in MKL, the induced polytopic uncertainty set tends to have a succinct expression, thereby well preserving the tractability of the induced optimization problem. It also allows a decisionmaker to conveniently adjust the conservatism of the data-driven uncertainty set by manipulating only one parameter, which is user-friendly in practice. Numerical case studies are carried out to demonstrate the potential advantages of the proposed method in promoting the practicability of RO techniques.