  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 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.
