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Cognitive radio makes it possible for an unlicensed user to access a licensed spectrum opportunistically on the basis of non-interfering. This paper addresses the problem of joint route selection and resource allocation in OFDMA-based multihop cognitive radio networks, in the objective of optimizing different types of end-to-end performance. Aiming to solve it optimally, we first show that this problem of optimal resource allocation can be formulated as a convex optimization problem and identify its necessary and sufficient conditions. Based on this conclusion, we propose an iterative algorithm that can be implemented in a distributed manner. This algorithm applies Lagrangian duality theory and the Frank-Wolfe method. The scheme thus converges to a globally optimal solution. We present numerical results from using the algorithm to provide insight into the optimal cross-layer design, e.g., the relationship between bottleneck throughput and hops, and the effect of interference temperature constraints.