  We consider the problem of minimizing a Lipschitz differentiable function
over a class of sparse symmetric sets that has wide applications in engineering
and science. For this problem, it is known that any accumulation point of the
classical projected gradient (PG) method with a constant stepsize $1/L$
satisfies the $L$-stationarity optimality condition that was introduced in [3].
In this paper we introduce a new optimality condition that is stronger than the
$L$-stationarity optimality condition. We also propose a nonmonotone projected
gradient (NPG) method for this problem by incorporating some support-changing
and coordintate-swapping strategies into a projected gradient method with
variable stepsizes. It is shown that any accumulation point of NPG satisfies
the new optimality condition and moreover it is a coordinatewise stationary
point. Under some suitable assumptions, we further show that it is a global or
a local minimizer of the problem. Numerical experiments are conducted to
compare the performance of PG and NPG. The computational results demonstrate
that NPG has substantially better solution quality than PG, and moreover, it is
at least comparable to, but sometimes can be much faster than PG in terms of
speed.
