Optimality conditions for group sparsity constrained optimization problems with equality and inequality constraints

The group sparsity constrained optimization problem with equality and inequality constraints (GSOP) refers to minimizing a given objective function while satisfying group sparsity constraints as well as equality and inequality constraints. By utilizing the constraint qualifications, we first derive the separability property of the Fr & eacute;chet, Mordukhovich and Clarke normal cones for the constraint set. Then, we introduce three categories of Karush-Kuhn-Tucker (KKT) points applicable to GSOP and obtain the corresponding optimality conditions using the separability of normal cones. Finally, the second-order optimality conditions for GSOP are provided. These results may provide some theoretical basis for analysing and solving the GSOPs.