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In this paper, we introduce a novel first-order dual gradient algorithm forsolving network utility maximization problems that arise in resource allocationschemes over networks with safety-critical constraints. Inspired byapplications where customers' demand can only be affected through posted pricesand real-time two-way communication with customers is not available, we requirean algorithm to generate \textit{safe prices}. This means that at no iterationshould the realized demand in response to the posted prices violate the safetyconstraints of the network. Thus, in contrast to existing first-order methods,our algorithm, called the safe dual gradient method (SDGM), is guaranteed toproduce feasible primal iterates at all iterations. We ensure primalfeasibility 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 andminus directions. In addition, we prove that the primal iterates produced bythe SDGM achieve a sublinear static regret of ${\cal O}(\sqrt{T})$.

Safe Dual Gradient Method for Network Utility Maximization Problems