  To use the spectrum efficiently, cognitive radios leverage knowledge of the
channel state information (CSI) to optimize the performance of the secondary
users (SUs) while limiting the interference to the primary users (PUs). The
algorithms in this paper are designed to maximize the weighted ergodic
sum-capacity of SUs, which transmit orthogonally and adhere simultaneously to
constraints limiting: i) the long-term (ergodic) capacity loss caused to each
PU receiver; ii) the long-term interference power at each PU receiver; and iii)
the long-term power at each SU transmitter. Formulations accounting for
short-term counterparts of i) and ii) are also discussed. Although the
long-term capacity constraints are non-convex, the resultant optimization
problem exhibits zero-duality gap and can be efficiently solved in the dual
domain. The optimal allocation schemes (power and rate loadings, frequency
bands to be accessed, and SU links to be activated) are a function of the CSI
of the primary and secondary networks as well as the Lagrange multipliers
associated with the long-term constraints. The optimal resource allocation
algorithms are first designed under the assumption that the CSI is perfect,
then the modifications needed to accommodate different forms of imperfect CSI
(quantized, noisy, and outdated) are analyzed.
