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We study the cross-layer design of congestion control and power allocation with outage constraint in an interference-limited multihop wireless networks. Using a complete-convexification method, we first propose a message-passing distributed algorithm that can attain the global optimal source rate and link power allocation. Despite the attractiveness of its optimality, this algorithm requires larger message size than that of the conventional scheme, which increases network overheads. Using the bounds on outage probability, we map the outage constraint to an SIR constraint and continue developing a practical near-optimal distributed algorithm requiring only local SIR measurement at link receivers to limit the size of the message. Due to the complicated complete-convexification method, however the congestion control of both algorithms no longer preserves the existing TCP stack. To take into account the TCP stack preserving property, we propose the third algorithm using a successive convex approximation method to iteratively transform the original nonconvex problem into approximated convex problems, then the global optimal solution can converge distributively with message-passing. Thanks to the tightness of the bounds and successive approximations, numerical results show that the gap between three algorithms is almost indistinguishable. Despite the same type of the complete-convexification method, the numerical comparison shows that the second near-optimal scheme has a faster convergence rate than that of the first optimal one, which make the near-optimal scheme more favorable and applicable in practice. Meanwhile, the third optimal scheme also has a faster convergence rate than that of a previous work using logarithm successive approximation method.