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# Safe dynamic pricing for nonstationary network resource allocation

*Proceedings of the 6th Annual Learning for Dynamics & Control Conference*, PMLR 242:155-167, 2024.

#### Abstract

This paper introduces the Safe Pricing for NUM with Gradual Variations (SPNUM-GV) algorithm, addressing challenges in pricing-based distributed resource allocation for safety-critical systems with non-stationary utility functions. Focusing on domains where 1) users’ optimal demand can only be induced through posted prices, 2) real-time two-way communication with the users is not available, 3) the induced demand must always belong to an arbitrarily shaped convex and compact feasible set in spite of price response uncertainty, and 4) the users’ response to prices are evolving over time, we design SPNUM-GV to generate prices that ensure stage-wise safety of the induced demand while achieving sublinear regret. SPNUM-GV ensures safety by determining a “desired demand” within a shrunk feasible set using a projected gradient method and updating the prices to induce a demand close to the desired demand by leveraging an estimate of the users’ price response function. By tuning the amount of shrinkage to account for the error between the desired and the induced demand, we prove that the induced demand always belongs to the feasible set. In addition, we prove that the regret incurred by the induced demand is $O(\sqrt{T(1+V_T)})$ after $T$ iterations, where $V_T$ is an upper bound on the total gradual variations of the users’ utility functions. Numerical simulations demonstrate the efficacy of SPNUM-GV and support our theoretical findings.