Safe Dual Gradient Method for Network Utility Maximization Problems

TL;DR

This paper presents a safe dual gradient algorithm for network utility maximization that guarantees primal feasibility at all iterations and achieves sublinear regret, suitable for safety-critical resource allocation.

Contribution

Introduces the safe dual gradient method (SDGM), a novel first-order algorithm ensuring safety constraints are met at every step in network utility maximization.

Findings

SDGM guarantees primal feasibility at all iterations.

The algorithm achieves a sublinear static regret of O(√T).

It effectively manages safety constraints without real-time customer communication.

Abstract

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 ${\cal O}(\sqrt{T})$.