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In this paper, we propose a novel class of Nash problems for cognitive radio (CR) networks, modeled as Gaussian frequency-selective interference channels, wherein each secondary user (SU) competes against the others to maximize his own opportunistic throughput by choosing jointly the sensing duration, the detection thresholds, and the vector power allocation. The proposed general formulation allows us to accommodate several (transmit) power and (deterministic/probabilistic) interference constraints, such as constraints on the maximum individual and/or aggregate (probabilistic) interference tolerable at the primary receivers. To keep the optimization as decentralized as possible, global (coupling) interference constraints are imposed by penalizing each SU with a set of time-varying prices based upon his contribution to the total interference; the prices are thus additional variable to optimize. The resulting players' optimization problems are nonconvex; moreover, there are possibly price clearing conditions associated with the global constraints to be satisfied by the solution. All this makes the analysis of the proposed games a challenging task; none of classical results in the game theory literature can be successfully applied. The main contribution of this paper is to develop a novel optimization-based theory for studying the proposed nonconvex games; we provide a comprehensive analysis of the existence and uniqueness of a standard Nash equilibrium, devise alternative best-response based algorithms, and establish their convergence. Some of the proposed algorithms are totally distributed and asynchronous, whereas some others require limited signaling among the SUs (in the form of consensus algorithms) in favor of better performance; overall, they are thus applicable to a variety of CR scenarios, either cooperative or noncooperative, which allows the SUs to explore the existing tradeoff between signaling and performance.