  We study the problem of differentially private optimization with linear
constraints when the right-hand-side of the constraints depends on private
data. This type of problem appears in many applications, especially resource
allocation. Previous research provided solutions that retained privacy but
sometimes violated the constraints. In many settings, however, the constraints
cannot be violated under any circumstances. To address this hard requirement,
we present an algorithm that releases a nearly-optimal solution satisfying the
constraints with probability 1. We also prove a lower bound demonstrating that
the difference between the objective value of our algorithm's solution and the
optimal solution is tight up to logarithmic factors among all differentially
private algorithms. We conclude with experiments demonstrating that our
algorithm can achieve nearly optimal performance while preserving privacy.
