We consider the problem of minimizing a general continuously differentiable function over symmetric sets under sparsity constraints. These type of problems are generally hard to solve as the sparsity constraint induces a combinatorial constraint into the problem, rendering the feasible set to be nonconvex. We begin with a study of the properties of the orthogonal projection operator onto sparse symmetric sets, which also results with efficient computation methods. We then introduce and study three types of optimality conditions: basic feasibility, $L$-stationarity and coordinate-wise optimality. A hierarchy between the optimality conditions is established by using the results derived on the orthogonal projection operator. Methods for generating points satisfying the various optimality conditions are presented, analyzed, and finally tested on specific applications.
Accepted for publication in MOR