On nondegenerate M-stationary points for sparsity constrained nonlinear optimization

Abstract We study sparsity constrained nonlinear optimization (SCNO) from a topological point of view. Special focus will be on M-stationary points Burdakov et al. (SIAM J Optim 26:397–425, 2016), also introduced as $$N^C$$ N C -stationary in Pan (J Oper Res Soc China 3:421–439, 2015). introduce nondegenerate and define their M-index. show that all are generically nondegenerate. In particular, the constraint is active at local minimizers generic SCNO. Some relations to other stationarity concepts, such S-stationarity, basic feasibility, CW-minimality, discussed detail. By doing so, issues instability degeneracy due different concepts highlighted. The concept M-stationarity allows adequately describe global structure SCNO along lines Morse theory. For that, we changes lower level sets while passing an point. As novelty for SCNO, multiple cells dimension equal M-index needed attached. This intriguing fact strong contrast with problems considered before, where just one cell suffices. consequence, derive relation which relates numbers one. appearance saddle cannot thus neglected perspective optimization. Due multiplicity phenomenon cell-attachment, may lead more than two minimizers. conclude relatively involved source well-known difficulty if solving optimality.