In this work we consider the problem of minimizing a continuously differentiable function over a feasible set defined by box constraints. We present a decomposition method based on the solution of a sequence of subproblems. In particular, we state conditions on the rule for selecting the subproblem variables sufficient to ensure the global convergence of the generated sequence without convexity assumptions. The conditions require to select suitable variables (related to the violation of the optimality conditions) to guarantee theoretical convergence properties, and leave the degree of freedom of selecting any other group of variables to accelerate the convergence.
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