Semidefinite Programs with Interval Uncertainty: Reduced Vertex Results

In this paper, we derive a reduced vertex result for robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n times n, a direct vertex approach would require satisfaction of 2^{n(m+1)(n+1)/2} vertex constraints: a huge number, even for small values of n and m. The conditions obtained here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2^{n-1}, thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty.