In this paper, we mainly study the existence of solutions to sparsity constrained optimization (SCO). Based on the expressions of tangent cone and normal cone of sparsity constraint, we present and characterize two first-order necessary optimality conditions for SCO: N-stationarity and T-stationarity. Then we give the second-order necessary and sufficient optimality conditions for SCO. At last, we extend these results to SCO with nonnegative constraint.