Abstract
We study sparsity constrained nonlinear optimization (SCNO) from a topological point of view. Special focus will be on M-stationary points from Burdakov et al. (SIAM J Optim 26:397–425, 2016), also introduced as N^C-stationary points in Pan et al. (J Oper Res Soc China 3:421–439, 2015). We introduce nondegenerate M-stationary points and define their M-index. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic SCNO. Some relations to other stationarity concepts, such as S-stationarity, basic feasibility, and CW-minimality, are discussed in detail. By doing so, the issues of instability and degeneracy of points due to different stationarity concepts are highlighted. The concept of M-stationarity allows to adequately describe the global structure of SCNO along the lines of Morse theory. For that, we study topological changes of lower level sets while passing an M-stationary point. As novelty for SCNO, multiple cells of dimension equal to the M-index are needed to be attached. This intriguing fact is in strong contrast with other optimization problems considered before, where just one cell suffices. As a consequence, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one. The appearance of such saddle points cannot be thus neglected from the perspective of global optimization. Due to the multiplicity phenomenon in cell-attachment, a saddle point may lead to more than two different local minimizers. We conclude that the relatively involved structure of saddle points is the source of well-known difficulty if solving SCNO to global optimality.
Highlights
We consider the sparsity constrained nonlinear optimization: SCNO: min f (x) s. t.x ∈Rn x 0 ≤ s, where the so-called 0 "norm" counts non-zero entries of x:x 0 = |{i ∈ {1, . . . , n} | xi = 0 }|, the objective function f ∈ C2 (Rn, R) is twice continuously differentiable, and s ∈ {0, 1, . . . , n − 1} is an integer
The difficulty of solving SCNO comes from the combinatorial nature of the sparsity constraint x 0 ≤ s
As a consequence of proposed Morse theory, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one
Summary
We consider the sparsity constrained nonlinear optimization: SCNO: min f (x) s. t. M-stationarity corresponds to the standard Karush–Kuhn–Tucker condition of the tightened program, where zero entries of an MPCC feasible point remain locally vanishing. Further research in this direction is presented in a series of subsequent papers [4,6]. We prove that all M-stationary points are generically nondegenerate It follows that all local minimizers of SCNO are nondegenerate with vanishing M-index, the sparsity constraint is active. As a consequence of proposed Morse theory, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one The appearance of such saddle points cannot be neglected from the perspective of global optimization. Given a twice continuously differentiable function f : Rn → R, ∇ f denotes its gradient, and D2 f stands for its Hessian
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