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This paper focuses on the local optimality for the stationary points of the composite group zero-norm regularized problem and its equivalent surrogates. First, by using the structure of the composite group zero-norm and its second subderivative characterization, we achieve several local optimal conditions for a stationary point of the group zero-norm regularized problem. Then, we obtain a family of equivalent surrogates for the group zero-norm regularized problem from a class of global exact penalties of its MPEC reformulation, established under the calmness of a partial perturbation to the composite group zero-norm constraint system. For the stationary points of these surrogates, we study their local optimality to the surrogates themselves and the group zero-norm regularized problem. The local optimality conditions obtained in this work not only recover the existing ones for zero-norm regularized problems, but also provide new criteria to judge the local optimality of a stationary point yielded by an algorithm for solving the corresponding surrogate problems.

Local optimality for stationary points of group zero-norm regularized problems and equivalent surrogates