We seek distributed protocols that attain the global optimum allocation of link transmitter powers and source rates in a cross-layer design of a mobile ad-hoc network. Although the underlying network utility maximization is nonconvex, convexity plays a major role in our development. We provide new convexity results surrounding the Shannon capacity formula, allowing us to abandon suboptimal high-SIR approximations that have almost become entrenched in the literature. More broadly, these new results can be back-substituted into many existing problems for similar benefit. Three protocols are developed. The first is based on a convexification of the underlying problem, relying heavily on our new convexity results. We provide conditions under which it produces a globally optimum resource allocation. We show how it may be distributed through message passing for both rate- and power-allocation. Our second protocol relaxes this requirement and involves a novel sequence of convex approximations, each exploiting existing TCP protocols for source rate allocation. Message passing is only used for power control. Our convexity results again provide sufficient conditions for global optimality. Our last protocol, motivated by a desire of power control devoid of message passing, is a near optimal scheme that makes use of noise measurements and enjoys a convergence rate that is orders of magnitude faster than existing methods.