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This paper investigates the asymptotic and non-asymptotic behavior of thequantized primal dual algorithm in network utility maximization problems, inwhich a group of agents maximize the sum of their individual concave objectivefunctions under linear constraints. In the asymptotic scenario, we use theinformation theoretic notion of differential entropy power to establishuniversal lower bounds on the exponential convergence rates of joint primaldual, primal and dual variables under optimum achieving quantization schemes.These results provide trade offs between the speed of exponential convergence,the agents objective functions, the communication bit rates, and the number ofagents and constraints. In the non-asymptotic scenario, we obtain lower boundson the mean square distance of joint primal dual, primal and dual variablesfrom the optimal solution for any finite time instance. These bounds holdregardless of the quantization scheme used.

Convergence Analysis of Quantized Primal-dual Algorithm in Network Utility Maximization Problems