Robust Solutions to Uncertain Semidefinite Programs

TLDR

This work studies semidefinite programs whose data depend on unknown bounded perturbations. The goal is to find robust solutions that minimize the worst‑case objective while satisfying constraints for all admissible perturbations. Assuming data matrices are rational functions of the perturbations, the authors formulate sufficient conditions for robust solutions as semidefinite programs. For full perturbations, the conditions are necessary and sufficient, guaranteeing unique and Hölder‑stable robust solutions that can regularize ill‑conditioned SDPs, as illustrated on examples from linear programming, maximum‑norm minimization, polynomial interpolation, and integer programming.

Abstract

In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.