Interference due to the sharing of common spectrum band among links and congestion due to the contention among flows sharing the same link have become obstacles to good performance in wireless networks, especially in multi-hop cognitive radio networks (MHCRNs). Consequently, the high end-to-end throughput for MHCRNs calls for a framework of cross-layer optimization design. In this paper, by taking into account the problem of joint optimal rate, power, and spectrum allocation (JORPS), we propose a new cross-layer optimization framework for MHCRNs under spectrum underlay manner using orthogonal frequency division multiple access (OFDMA). The formulation is shown to be a mix-integer non-linear non-convex optimization problem, which is μμ-hard in general. To tackle this cumbersome, we firstly applied a partially distributed solution. Then, we showed that this approach fast converges to the global optimum at a cost of computational complexity.