Strong Duality in Robust Semi - Definite Linear Programming under Data Uncertainty ∗

This paper develops the deterministic approach to duality for semi-definite linear programming problems in the face of data uncertainty. We establish strong duality between the robust counterpart of an uncertain semi-definite linear programming model problem and the optimistic counterpart of its uncertain dual. We prove that strong duality between the deterministic counterparts holds under a characteristic cone condition. We also show that the characteristic cone condition is also necessary for the validity of strong duality for every linear objective function of the original model problem. In addition, we derive that a robust Slater condition alone ensures strong duality for uncertain semi-definite linear programs with affinely parameterized data. Finally, we present strong duality with computationally easily tractable dual programs for two classes of uncertainties: the scenario uncertainty and the spectral norm uncertainty.