Projections onto Spectral Matrix Cones
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911246913110016 |
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| author | Cederberg, Daniel Boyd, Stephen |
| author_facet | Cederberg, Daniel Boyd, Stephen |
| contents | Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projections onto Spectral Matrix Cones Cederberg, Daniel Boyd, Stephen Optimization and Control 90C25, 65K05 G.1.6 Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning. |
| title | Projections onto Spectral Matrix Cones |
| topic | Optimization and Control 90C25, 65K05 G.1.6 |
| url | https://arxiv.org/abs/2511.01089 |