2022 Fiscal Year Research-status Report
Solving ill-posed conic optimization problems
| Project/Area Number |
19K20217
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| Research Institution | The Institute of Statistical Mathematics |
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Principal Investigator |
ロウレンソ ブルノ・フィゲラ 統計数理研究所, 数理・推論研究系, 准教授 (80778720)
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | error bounds / amenable cones / facial residual function / hyperbolicity cones / self-duality / automorphism group |
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| Outline of Annual Research Achievements |
(a) We completed several preprints on the following topics: geometry of hyperbolicity cones, error bounds for power cones and self-duality of polyhedral cones. Among our results, we were able to show that all hyperbolicity cones are amenable and we also investigated their automorphism group under certain conditions. For power cones, we completely determined their error bounds and automorphisms. Finally, we showed that self-duality for a polyhedral cone can be completely detected through the positive semidefiniteness of one of its slack matrices and we showed a surprising connection between slack matrices of irreducible self-dual polyhedral cones and extreme rays of doubly nonnegative matrices.
(b) Following the revision of papers that were in peer-review in the previous fiscal year, several of those papers were finally accepted at important journals in optimization and neighbouring areas. This includes papers in SIAM Journal on Optimization, SIAM Journal on Applied Algebra and Geometry, SIAM Journal on Matrix Analysis, Mathematical Programming and Foundations of Computational Mathematics.
(c) We presented our results in workshops and conferences both online and in-person. There were also research visits to collaborators in Australia and Brazil.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Several preprints were completed and several papers were accepted at important journals in optimization and neighbouring areas.
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| Strategy for Future Research Activity |
(a) We will explore the computation of error bounds for other families of conic feasibility problems not yet covered by our past results.
(b) Explore further applications of error bounds in convergence analysis of algorithms.
(c) We will continue the investigation of geometric aspects of convex cones.
(d) Present our findings at conferences and visit research collaborators for in-depth discussions.
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| Causes of Carryover |
Although the pandemic situation considerably improved in comparison with the previous fiscal years, travelling was still not easy and some of our research meetings got postponed. For the next fiscal year, we plan to use the remaining budget to visit research collaborators in order to advance the topics of this project.
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