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APPROXIMATE 1-NORM MINIMIZATION AND MINIMUM-RANK STRUCTURED SPARSITY FOR VARIOUS GENERALIZED INVERSES VIA LOCAL SEARCH.
- Source :
-
SIAM Journal on Optimization . 2021, Vol. 31 Issue 3, p1722-1747. 26p. - Publication Year :
- 2021
-
Abstract
- Fundamental in matrix algebra and its applications, a generalized inverse of a real matrix A is a matrix H that satisfies the Moore--Penrose (M--P) property AH A = A. If H also satisfies the additional useful M--P property, H AH = H, it is called a reflexive generalized inverse. Reflexivity is equivalent to minimum rank, so we are particularly interested in reflexive generalized inverses. We consider aspects of symmetry related to the calculation of a sparse reflexive generalized inverse of A. As is common, and following Fampa and Lee [Oper. Res. Lett., 46 (2018), pp. 605-610] for calculating sparse generalized inverses, we use (vector) 1-norm minimization for inducing sparsity and for keeping the magnitude of entries under control. When A is symmetric, we may naturally desire a symmetric H, while generally such a restriction on H may not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically efficient and practical local-search algorithm to block construct an approximate 1-norm minimizing symmetric reflexive generalized inverse. Another aspect of symmetry that we consider relates to another M--P property: H is ah-symmetric if AH is symmetric. The ah-symmetry property is the key one for solving least-squares problems using H. Here we do not assume that A is symmetric, and we do not impose symmetry on H. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically efficient and practical local-search algorithm to column block construct an approximate 1-norm minimizing ah-symmetric reflexive generalized inverse. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10526234
- Volume :
- 31
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- SIAM Journal on Optimization
- Publication Type :
- Academic Journal
- Accession number :
- 153117949
- Full Text :
- https://doi.org/10.1137/19M1281514