The distributed resource allocation problem aims to allocate a mount of resources under some local constraints while minimizing the total cost function. First, based on the Karush-Kuhn-Tucker conditions, an initialization-free distributed optimization algorithm is proposed for second-order multi-agent systems over an undirected connected network. The global equality constraint dual variable is developed with a proportional-integral control, and the local convex function inequality constraint dual variable is sought adaptively. Based on the set-value LaSalle's invariance principle, it is shown that the designed algorithm asymptotically converges to the optimal point if the global cost function is nonsmooth convex. Then, the proposed algorithm is extended to Euler-Lagrange multi-agent systems over an undirected connected network. Furthermore, by virtual of the Barbalat's lemma, it is shown that the proposed algorithm asymptotically converges to the optimal solution if the global cost function is nonsmooth convex. Finally, several numerical examples are used to illustrate the performance of the proposed algorithms.