  Our model is a generalized linear programming relaxation of a much studied
random K-SAT problem. Specifically, a set of linear constraints C on K
variables is fixed. From a pool of n variables, K variables are chosen
uniformly at random and a constraint is chosen from C also uniformly at random.
This procedure is repeated m times independently. We ask the following
question: is the resulting linear programming problem feasible? We show that
the feasibility property experiences a linear phase transition, when n diverges
to infinity and m=cn for some constant c. Namely, there exists a critical value
c* such that, when c<c*, the system is feasible or is asymptotically almost
feasible, as n increases, but, when c>c*, the "distance" from feasibility is at
least a positive constant independent of n. Our results are obtained using
powerful local weak convergence methods developed by Aldous and Steele.
  By exploiting a linear programming duality, our theorem implies the following
result in the context of sparse random graphs G(n, cn) on n nodes with cn
edges, where edges are equipped with randomly generated weights. Let M(n,c)
denote maximum weight matching in G(n, cn). We prove that when c is a constant
and n\to\infty, the limit \lim_n M(n,c)/n exists, with high probability. We
further extend this result to maximum weight b-matchings also in G(n,cn).
