  A block decomposition method is proposed for minimizing a (possibly
non-convex) continuously differentiable function subject to one linear equality
constraint and simple bounds on the variables. The proposed method iteratively
selects a pair of coordinates according to an almost cyclic strategy that does
not use first-order information, allowing us not to compute the whole gradient
of the objective function during the algorithm. Using first-order search
directions to update each pair of coordinates, global convergence to stationary
points is established for different choices of the stepsize under an
appropriate assumption on the level set. In particular, both inexact and exact
line search strategies are analyzed. Further, linear convergence rate is proved
under standard additional assumptions. Numerical results are finally provided
to show the effectiveness of the proposed method.
