  In this paper, we introduce a novel first-order dual gradient algorithm for
solving network utility maximization problems that arise in resource allocation
schemes over networks with safety-critical constraints. Inspired by
applications where customers' demand can only be affected through posted prices
and real-time two-way communication with customers is not available, we require
an algorithm to generate \textit{safe prices}. This means that at no iteration
should the realized demand in response to the posted prices violate the safety
constraints of the network. Thus, in contrast to existing first-order methods,
our algorithm, called the safe dual gradient method (SDGM), is guaranteed to
produce feasible primal iterates at all iterations. We ensure primal
feasibility by 1) adding a diminishing safety margin to the constraints, and 2)
using a sign-based dual update method with different step sizes for plus and
minus directions. In addition, we prove that the primal iterates produced by
the SDGM achieve a sublinear static regret of ${\cal O}(\sqrt{T})$.
