## 1 Introduction
### Problem Definition  
A cross-layer algorithm with optimal channel selection for power optimization (at the physical layer), rate allocation (transport layer) and delay minimization is designed with QoS constraints. QoS is defined by PU interference, BER, SNR, SINR, delay and rate requirements. We employ a multi-objective optimization function with these constraints to solve the resource allocation problem. We maximize transmission rate, minimize transmission power and minimize transmission delay in a constrained environment. We propose a power/energy efficient algorithm with optimum channel selection meant to jointly optimize transmission power, transmission delay and the transmission rate. This leads to power/energy efficiency; improves network QoS by regulating interference and BER; and reduces power wasted by individual nodes during spectrum sensing. To maximize transmission rate, we enable the SU access more than one channel at a time. Since we jointly optimize transmission power, delay and rate with controlled channel selection, this translates to multi-objective optimization. To solve this multi-objective optimization problem, we combine the multiple objectives into a single weighted objective function solved as a non-linear mixed binary optimization problem. We then employ a branch-and-cut algorithm that uses polyhedral cutting planes, to solve the resulting optimization problem. This is an exact optimization technique that guarantees an optimal solution, whenever it exists. The resulting optimal solution then enables us to investigate the rate-power-delay trade-offs by tuning the priority weights associated with each objective. Thus, we study the optimal values against the BER, interference, packet size and network size.  
Earlier works have concentrated on bi-objective models. They minimize the power and delay [7] or minimize the power and maximize the transmission rate [1, 8, 9], but not all of them simultaneously. The authors in [1, 9] design a QoS-constrained bi-objective optimization model in order to minimize transmission power and maximize rate. As we design a QoS-constrained multi-objective optimization problem, we minimize the power consumption and transmission delay and maximize transmission rate as we observe their variation with network size and packet size. This is done amidst QoS constraints to improve the user-perceived QoS, which to the best of our knowledge, no prior work has been done combining all these three objectives jointly. Most of the previous works focus on either minimizing power or energy without considering the delay minimization [10] or on delay minimization ignoring power minimization and also do not consider Cognitive Radio Sensor Networks [11].  
The rest of this paper is structured as follows: In Section 2, we review the related work; in Section 3, we handle the network model that is used in the study; in Section 4, we formulate the problem whereby we deal with the QoS constraints, that is BER and PU interference, and also we introduce multi-objective optimization for rate, power and delay; in Section 5, we provide the results of our model and the conclusions and recommendations for future work are handled in Section 6.