2018 Fiscal Year Final Research Report
Convex Algebraic Geometry and Optimization Theory
| Project/Area Number |
15K04993
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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 |
Foundations of mathematics/Applied mathematics
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| Research Institution | Tokyo University of Marine Science and Technology |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | 最適化理論 / 半正定値計画問題 / 凸代数幾何 / 実代数幾何 |
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| Outline of Final Research Achievements |
We studied representation theorems for nonnegative polynomials, a finite convergence property of a semidefinite relaxation method for polynomial optimization problems and perturbation analysis of singular semidefinite programming problems. The main contributions of our work are the following:
1. We obtained sufficient conditions for a nonnegative polynomial to be a sum of squares of power series using optimality conditions with Newton diagrams. 2. We gave a geometric condition with real radicals for a semidefinite relaxation of a polynomial optimization problems to have a finite convergence property in the case that the objective function is not contained in the quadratic module generated by constraint polynomials. 3. We obtained sufficient conditions for optimal values of perturbed singular semidefinite programs to change continuously.
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| Free Research Field |
最適化理論
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| Academic Significance and Societal Importance of the Research Achievements |
多項式最適化問題に対する半正定値緩和法の理論的背景を強化し，最適性条件と多項式の表現定理との新たな関係を明らかにした．また今まで注目されていなかった facial reduction sequence を用いることにより，特異な半正定値計画問題の摂動理論を構築した．これらの研究は，最適化理論の研究において対象を多項式に限定することで，代数幾何のアイデアを用いており，最適化理論の新しい展開とより実用的な理論の構築に貢献するものである．
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