2020 Fiscal Year Final Research Report
Duality and error analysis on conic programming through facial reduction algorithms
| Project/Area Number |
17K00031
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical informatics
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| Research Institution | The University of Electro-Communications |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
高橋 里司 電気通信大学, 大学院情報理工学研究科, 准教授 (40709193)
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Keywords | 対称錐計画 / 半正定値計画 / ２次錐計画 / 面削減法 / 凸最適化 |
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| Outline of Final Research Achievements |
There are two major achievements in this project. The first one is to show that the upper bound of the iterations of Facial Reduction Algorithms (FRA) applied to a (nonlinear) cone containing polyhedral cones, depends only on that of the nonlinear part; in other words, we can safely ignore the polyhedral cones at least from the viewpoint of FRA. The other is to extend two of Chubanov's algorithms proposed for homogeneous LPs using projection and rescaling to symmetric cones and semi-infinite polyhedral cones, respectively. For these algorithms, we implemented them only for the semidefinite programming problem and conducted numerical experiments. The latter algorithm evaluates the number of iterations using the "volume" of the feasible region as the conditional number. Especially if if the condition number is zero, then FRA, the main theme of this project, should be applied to the conic programming problem.
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| Free Research Field |
錐線形計画
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| Academic Significance and Societal Importance of the Research Achievements |
最初の成果（多面推を含む非線形錐に対するFRA反復回数の評価）は、特に最近注目を浴びている非負正定値錐に関してそのFRA 反復回数のオーダーを変えるというインパクトがあった。また、もう１つの Chubanov のアルゴリズムの拡張に関しては、錐線形計画に関する我々の理解を深めるとともに、「悪条件」にも様々な定義が様々あり、どの定義の悪条件かに依存して選択するべきアルゴリズムが異なってくることを提示した。これらはまだ直接的に社会的インパクトがある成果とは言えないかもしれないが、今後の進展の余地が多く、学術的意義は大きいと考えられる。
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