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2023 Fiscal Year Final Research Report

Theory and algorithms for ill-conditioned conic linear programming

Research Project

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Project/Area Number 20H04145
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 60020:Mathematical informatics-related
Research InstitutionThe University of Electro-Communications

Principal Investigator

Muramatsu Masakazu  電気通信大学, 大学院情報理工学研究科, 教授 (70266071)

Co-Investigator(Kenkyū-buntansha) 山下 真  東京工業大学, 情報理工学院, 教授 (20386824)
奥野 貴之  成蹊大学, 理工学部, 准教授 (70711969)
蛯原 義雄  九州大学, システム情報科学研究院, 教授 (80346080)
Project Period (FY) 2020-04-01 – 2024-03-31
Keywords半正定値計画 / 錐線形計画 / 2次錐計画 / 数理最適化
Outline of Final Research Achievements

A small-gain theorem leveraging the characteristics of the Rectified Linear Unit, commonly used in machine learning, was derived. Next, the semidefinite programming (SDP) relaxation conditions for quadratic constrained quadratic programming (QCQP) were analyzed, demonstrating applicability to forest structures and simultaneously tridiagonalizable cases. Additionally, an algorithm guaranteeing convergence to points satisfying strong optimality conditions (SOSP points) under ill-conditioned SDP was developed, enhancing local convergence compared to traditional methods. Other research also included stability analysis of recurrent neural networks, improvements in constrained optimization problems on the positive semidefinite cone, and QCQP analysis using bipartite graphs. Furthermore, a theory for the complete solving ill-conditioned SDPs was established.

Free Research Field

連続最適化

Academic Significance and Societal Importance of the Research Achievements

本研究は、機械学習や最適化理論において進展をもたらすものである。例えば、Rectified Linear Unitを用いたスモールゲイン定理の導出により、安定性解析がより正確かつ効率的になる。また、QCQPに関する研究は、従来より大きなQCQPを高速に安定的に解くことを可能にしたので、応用の幅が広がった。SOSPへの収束の証明や悪条件の錐線形計画問題を厳密に解く研究は、理論を深めるものとして重要なものであり、今後の進展が待たれる。全体として、錐線形計画に関連するこれらの技術は、最適化問題の理論的な深まりと実践的応用を促進するものである。

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Published: 2025-01-30  

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