Solving ill-posed conic optimization problems
| Project/Area Number |
19K20217
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 60020:Mathematical informatics-related
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| Research Institution | The Institute of Statistical Mathematics (2020-2022)
The University of Tokyo (2019) |
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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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| Project Status |
Granted (Fiscal Year 2022)
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| Budget Amount *help |
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2022: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | error bounds / amenable cones / facial residual function / hyperbolicity cones / self-duality / automorphism group / exponential cone / p-cones / 連続最適化 / 錐最適化 / 恭順錐 / 面縮小法 / 錐線形計画問題 / 悪条件問題 / 射影縮尺法 |
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| Outline of Research at the Start |
Conic optimization corresponds to a large class of mathematical problems that have both practical and theoretical relevance. However, sometimes those problems have unfavorable theoretical properties, i.e., they are “ill-posed”. The goal of this project is to analyze mathematical aspects related to ill-posed problems and to research effective methods for solving them.
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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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Report
(4 results)
Research Products
(45 results)
