Error Bounds for Linear Matrix Inequalities

For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most O(2 ?d). The nonnegative integer d is the so{called degree of singularity of the linear matrix inequality, and denotes the amount of constraint violation in the iterate. For infeasible linear matrix inequalities, we show that the minimal norm of {approximate primal solutions is at least 1=O(1=(2 d ?1)), and the minimal norm of {approximate Farkas{ type dual solutions is at most O(1== 2 d ?1). As an application of these error bounds, we show that for any bounded sequence of {approximate solutions to a semi-deenite programming problem, the distance to the optimal solution set is at most O(2 ?k), where k is the degree of singularity of the optimal solution set.