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Convex Analysis on the Hermitian Matrices
Abstract
There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues. A new approach to this characterization is given, via a simple Fenchel conjugacy formula. We then apply this formula to derive expressions for subdifferentials, and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables. Analogous results hold for Hermitian matrices.
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Cited by
Parikh, N., & Boyd, S. (2014) · Foundations and Trends® in Optimization · doi:10.1561/2400000003 · Cited by 4,102
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Boyd, S., Diaconis, P., & Xiao, L. (2004) · SIAM Review · doi:10.1137/s0036144503423264 · Cited by 744
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Boyd, S., Ghosh, A., Prabbakar, B., & Shah, D. (2005) · Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies. · doi:10.1109/infcom.2005.1498447 · Cited by 558
Metadata
- Authors
- Lewis, A. S.
- Venue
- SIAM Journal on Optimization
- Published
- 1996-02
- Volume
- 6
- Issue
- 1
- Pages
- 164-177
- ISSN
- 1052-6234 (ISSN-L)
- 1095-7189
- OpenAlex
- W2126119581
- Cited by
- 187
- References
- 0
- FWCI
- 7.73
- Type
- journal article
- Language
- en
- Open access
- closed
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- Lewis, A. S. (1996). Convex Analysis on the Hermitian Matrices. SIAM Journal on Optimization. https://doi.org/10.1137/0806009
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