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We propose a robust distributed uplink power allocation algorithm for underlay cognitive radio networks (CRNs) with a view to maximizing the social utility of secondary users (SUs) when channel gains from SUs to primary base stations, and interference caused by primary users (PUs) to the SUs' base station are uncertain. In doing so, we utilize the worst case robust optimization to keep the interference caused by SUs to each primary base station below a given threshold, and satisfy each SU's quality of service in terms of its required SINR for all realizations of uncertain parameters. We model each uncertain parameter by a bounded distance between its estimated and exact values, and formulate the robust power allocation problem via protection values for constraints. We demonstrate that the convexity of our problem is preserved, and converts into a geometric programming problem, which we solve via a …