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The cross-layer utility maximization problem subject to stability constraints is considered for a multi-hop wireless network. A time slotted network, where the channel gains are changing randomly from slot to slot is assumed. The optimal cross-layer network control policy can be decomposed into three subproblems: 1) flow control, 2) next-hop routing and in-node scheduling, and 3) power and rate control, also known as resource allocation (RA). In the case of multi-hop networks, RA subproblem is particularly difficult to solve due to the self interference problem which arises when a node simultaneously transmits and receives in the same channel. According to relative distances between networks nodes, the self interference coefficients can be several order of magnitude larger than the power gains between distinct nodes. Thus, standard RA methods for bipartite networks can not be applied directly. The main contribution of this paper is to derive a novel RA algorithm for multi-hop wireless networks which is capable to deal with the self interference problem and does not rely on combinatorial constraints for finding the set of links which can be simultaneously activated. The numerical results show that the proposed RA algorithm can provide significant gains at network layer in terms of end-to-end rates and network congestion, even though the solution is local.