Quasi-Nash Equilibria for Non-Convex Distributed Power Allocation Games in Cognitive Radios

In this paper, we consider a sensing-based spectrum sharing scenario in cognitive radio networks where the overall objective is to maximize the sum-rate of each cognitive radio user by optimizing jointly both the detection operation based on sensing and the power allocation, taking into account the influence of the sensing accuracy and the interference limitation to the primary users. The resulting optimization problem for each cognitive user is non-convex, thus leading to a non-convex game, which presents a new challenge when analyzing the equilibria of this game where each cognitive user represents a player. In order to deal with the non-convexity of the game, we use a new relaxed equilibria concept, namely, quasi-Nash equilibrium (QNE). A QNE is a solution of a variational inequality obtained under the first-order optimality conditions of the player´s problems, while retaining the convex constraints in the variational inequality problem. In this work, we state the sufficient conditions for the existence of the QNE for the proposed game. Specifically, under the so-called linear independent constraint qualification, we prove that the achieved QNE coincides with the NE. Moreover, a distributed primal-dual interior point optimization algorithm that converges to a QNE of the proposed game is provided in the paper, which is shown from the simulations to yield a considerable performance improvement with respect to an alternating direction optimization algorithm and a deterministic game.