Artikel
On nondegenerate M-stationary points for sparsity constrained nonlinear optimization
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 NC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^C$$\end{document}-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.
- Sprache
-
Englisch
- Erschienen in
-
Journal: Journal of Global Optimization ; ISSN: 1573-2916 ; Volume: 82 ; Year: 2021 ; Issue: 2 ; Pages: 219-242 ; New York, NY: Springer US
- Klassifikation
-
Mathematik
- Thema
-
Sparsity constraint
M-stationarity
M-index
Nondegeneracy
Genericity
Morse theory
Saddle points
- Ereignis
-
Geistige Schöpfung
- (wer)
-
Lämmel, S.
Shikhman, V.
- Ereignis
-
Veröffentlichung
- (wer)
-
Springer US
- (wo)
-
New York, NY
- (wann)
-
2021
- DOI
-
doi:10.1007/s10898-021-01070-7
- Letzte Aktualisierung
-
10.03.2025, 11:42 MEZ
Datenpartner
ZBW - Deutsche Zentralbibliothek für Wirtschaftswissenschaften - Leibniz-Informationszentrum Wirtschaft. Bei Fragen zum Objekt wenden Sie sich bitte an den Datenpartner.
Objekttyp
- Artikel
Beteiligte
- Lämmel, S.
- Shikhman, V.
- Springer US
Entstanden
- 2021
Ähnliche Objekte (12)
Proximity measures based on KKT points for constrained multi-objective optimization
A polyhedral study of event-based models for the resource-constrained project scheduling problem
On mixed-integer optimal control with constrained total variation of the integer control
Lagrangian relaxation based heuristics for a chance-constrained optimization model of a hybrid solar-battery storage system
Nonconvex constrained optimization by a filtering branch and bound
Sequential optimality conditions for cardinality-constrained optimization problems with applications
A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws
A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
Group Interventions in Schools : Promoting Mental Health for At-Risk Children
Mountains and minerals, rivers and rocks
The Neural Crest and Neural Crest Cells in Vertebrate Development and Evolution
Algebraic combinatorics
Proximity measures based on KKT points for constrained multi-objective optimization
A polyhedral study of event-based models for the resource-constrained project scheduling problem
On mixed-integer optimal control with constrained total variation of the integer control
Lagrangian relaxation based heuristics for a chance-constrained optimization model of a hybrid solar-battery storage system
Nonconvex constrained optimization by a filtering branch and bound
Sequential optimality conditions for cardinality-constrained optimization problems with applications
A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws
A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
Group Interventions in Schools : Promoting Mental Health for At-Risk Children
Mountains and minerals, rivers and rocks
The Neural Crest and Neural Crest Cells in Vertebrate Development and Evolution
Algebraic combinatorics
Proximity measures based on KKT points for constrained multi-objective optimization
A polyhedral study of event-based models for the resource-constrained project scheduling problem
On mixed-integer optimal control with constrained total variation of the integer control
Lagrangian relaxation based heuristics for a chance-constrained optimization model of a hybrid solar-battery storage system
Nonconvex constrained optimization by a filtering branch and bound
Sequential optimality conditions for cardinality-constrained optimization problems with applications
A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws
A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
Group Interventions in Schools : Promoting Mental Health for At-Risk Children
Mountains and minerals, rivers and rocks
The Neural Crest and Neural Crest Cells in Vertebrate Development and Evolution