  In this paper, we describe a new active-set algorithmic framework for
minimizing a non-convex function over the unit simplex. At each iteration, the
method makes use of a rule for identifying active variables (i.e., variables
that are zero at a stationary point) and specific directions (that we name
active-set gradient related directions) satisfying a new "nonorthogonality"
type of condition. We prove global convergence to stationary points when using
an Armijo line search in the given framework. We further describe three
different examples of active-set gradient related directions that guarantee
linear convergence rate (under suitable assumptions). Finally, we report
numerical experiments showing the effectiveness of the approach.
