Interior-point methods on manifolds: theory and applications
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909230745780224 |
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| author | Hirai, Hiroshi Nieuwboer, Harold Walter, Michael |
| author_facet | Hirai, Hiroshi Nieuwboer, Harold Walter, Michael |
| contents | Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_04771 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Interior-point methods on manifolds: theory and applications Hirai, Hiroshi Nieuwboer, Harold Walter, Michael Optimization and Control Data Structures and Algorithms Differential Geometry Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature. |
| title | Interior-point methods on manifolds: theory and applications |
| topic | Optimization and Control Data Structures and Algorithms Differential Geometry |
| url | https://arxiv.org/abs/2303.04771 |