A minimum spanning tree problem is normally described as the minimization of a linear objective over a set of linear constraints, plus integrality restrictions on the variables. Why is it nonlinear? A fun trial would then perhaps be to use a heuristic Frank-Wolfe style of method, in which you solve linearized MST problems (which can be done very fast), and to use a heuristic line search that "projects" infeasible trees onto spanning trees.
I really do not understand your problem statement. You first speak about Lagrangian relaxation applied to alinear problem but your problem is an MST and then when asked by Hossein you say that your problem is nonlinear. Can you please make a real statement of your true problem? Otherwise we cannot help you but only confuse you even more.
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