2019/12/09 by Sebastian Lämmel, Lämmel, Sebastian, Vladimir Shikhman +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Advanced Optimization Algorithms Research #Algebraic Topology (math.AT) #Chemokine receptors and signaling #FOS: Mathematics #Optimization and Control (math.OC) #Sphingolipid Metabolism and Signaling
paper · pdf · doi:10.48550/arxiv.1912.04087
openalex publication_date 2019/12/09 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study sparsity constrained nonlinear optimization (SCNO) from a\ntopological point of view. Special focus will be on M-stationary points from\nBurdakov et al. (2016). We introduce nondegenerate M-stationary points and\ndefine their M-index. We show that all M-stationary points are generically\nnondegenerate. In particular, the sparsity constraint is active at all local\nminimizers of a generic SCNO. Some relations to other stationarity concepts,\nsuch as S-stationarity, basic feasibility, and CW-minimality, are discussed in\ndetail. By doing so, the issues of instability and degeneracy of points due to\ndifferent stationarity concepts are highlighted. The concept of M-stationarity\nallows to adequately describe the global structure of SCNO along the lines of\nMorse theory. For that, we study topological changes of lower level sets while\npassing an M-stationary point. As novelty for SCNO, multiple cells of dimension\nequal to the M-index are needed to be attached. This intriguing fact is in\nstrong contrast with other optimization problems considered before, where just\none cell suffices. As a consequence, we derive a Morse relation for SCNO, which\nrelates the numbers of local minimizers and M-stationary points of M-index\nequal to one. The appearance of such saddle points cannot be thus neglected\nfrom the perspective of global optimization. Due to the multiplicity phenomenon\nin cell-attachment, a saddle point may lead to more than two different local\nminimizers. We conclude that the relatively involved structure of saddle points\nis the source of well-known difficulty if solving SCNO to global optimality.\n