2022 Fiscal Year Final Research Report
A study of manifolds of optimization problems via convex algebraic geometry
| Project/Area Number |
19K03631
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
|
|---|
| Research Institution | Tokyo University of Marine Science and Technology |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2019-04-01 – 2023-03-31
|
| Keywords | 最適化理論 / 半正定値計画問題 / 凸代数幾何 / 交互射影法 |
|---|
| Outline of Final Research Achievements |
(1) We showed that strictly feasibility of the primal and the dual problem of SDP is equivalent to existence of nontrivial solution to the homogenized KKT system. (2) We obtained sufficient conditions for the optimal value of a singular SDP to change continuously, and a perturbing direction in which the optimal value changes continuously by using a facial reduction sequence. (3) When a semialgebraic set intersects a line non-transversely, we expressed the exact convergence rate of alternating projections with the multiplicity of a polynomial. We also obtained the polynomial defining the boundary which
determines the behavior of alternating projections.
|
| Free Research Field |
最適化理論
|
| Academic Significance and Societal Importance of the Research Achievements |
半正定値計画問題に射影幾何のアイデアを応用し，最適値が不連続に変化する現象に幾何的な意味を与え，strict feasibility と KKT 条件の関係を明らかにした．また，イデアル論を交互射影法の解析に初めて応用し，厳密収束レートの公式と，挙動の変化する境界の定義方程式を求めた． これらの結果は， 半正定値計画問題と最適化アルゴリズムに対する新しい見方を与え， 最適化理論そのものの新しい展開に貢献するものである．
|
