An Efficient and Numerically Robust Method for Polynomial Optimization Problems and their Extensions

• Project/Area Number: 19560063
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Engineering fundamentals
• Research Institution: The University of Electro-Communications
• Principal Investigator: MASAKAZU Muramatsu The University of Electro-Communications, 電気通信学部, 教授 (70266071)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 多項式計画 / 半正定値計画 / 大域的最適化 / 錐線形計画問題 / 面的削減 / 錐的拡大 / 半正定値計画問題 / 数値的安定性 / 多項式最適化問題 / 非線形最適化 / 多項式計画問題 / 数理計画 / 対称錐計画

Research Abstract

多項式計画問題および対称錐上の多項式計画問題に対する半正定値計画緩和の研究を行った。特に多項式の疎性や等式条件を利用して、半正定値計画緩和の問題のサイズを劇的に小さくできることを示した。この手法を利用したソフトウェア、SparsePOPを開発し、公開した。SparsePOPにより、これまでは解けなかったような多数の変数を持つ多項式計画問題を解く事ができるようになった。

Research Products

• [Journal Article] Invariance under affine transformation in semidefinite programming relaxation for polynomial optimization problems (2009)
  • Author(s): Hayato Waki, Masakazu Muramatsu, Masakazu Kojima
  • Journal Title: Pacific Journal of Optimization 5 Pages: 297-312
  • Peer Reviewed
• [Journal Article] A Note on Sparse SOS and SDP Relaxations for Polynomial Optimization Problems over Symmetric Cones (2009)
  • Author(s): Masakazu Kojima, Masakazu Muramatsu
  • Journal Title: Computational Optimization and Applications 42 Pages: 31-41
  • Peer Reviewed
• [Journal Article] Invariance under affine transformation in semidefinite programming relaxation for polynomial optimization problems (2009)
  • Author(s): H.Waki, et.al.
  • Journal Title: Pacific Journal of Optimization 5巻 Pages: 297-312
  • Peer Reviewed
• [Journal Article] Equality based contraction of semidefinite programming relaxation in polynomial optimization (2008)
  • Author(s): Cong Vo, Masakazu Muramatsu, Masakazu Kojima
  • Journal Title: Journal of Operations Research Society of Japan 52 Pages: 111-125
  • NAID: 110006632508
  • Peer Reviewed
• [Journal Article] SparsePOP: a sparse semidefinite programming relaxation of polynomial optimization problems (2008)
  • Author(s): Hayato Waki, Hiroshi Sugimoto, Masakazu Kojima, Masakazu Muramatsu
  • Journal Title: ACM Transaction of Mathematical Software 25
  • Peer Reviewed
• [Journal Article] SparsePOP : a Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems. (2008)
  • Author(s): Hayato Waki Sunyoung Kimiroshi Suglmoto Masakazu Kojima Mazakazu Muramatsu
  • Journal Title: ACM Transactions on Mathematical Software 35
  • Peer Reviewed
• [Journal Article] ″Equality based contraction of semidefinite programming relaxation in polynomial optimization (2008)
  • Author(s): Cong Vo, Masakazu Muramatsu, Masakazu Kojima
  • Journal Title: Journal of Operations Research Society of Japan 52 Pages: 111-125
  • NAID: 110006632508
  • Peer Reviewed
• [Journal Article] Equality based contraction of semidefinite pro-gramming relaxations in polynomial optimization (2008)
  • Author(s): C. Vo, M. Muramatsu and M. Kojima
  • Journal Title: Journal of Operations Research Societyof Japan 51 Pages: 111-125
  • NAID: 110006632508
  • Peer Reviewed
• [Journal Article] Towards a pivoting procedure for a class of second-order cone program-ming problems having multiple cone constraints (2007)
  • Author(s): Masakazu Muramatsu
  • Journal Title: Pacific Journal of Optimization 3 Pages: 87-98
  • Peer Reviewed
• [Journal Article] An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems over Symmetric Cones (2007)
  • Author(s): Masakazu Kojima, Masakazu Muramatsu
  • Journal Title: Mathematical Programming 110 Pages: 315-336
  • Peer Reviewed
• [Journal Article] An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones (2007)
  • Author(s): M. Kojima and M. Muramatsu
  • Journal Title: Mathematical Programming 110 Pages: 315-336
  • Peer Reviewed
• [Presentation] 半正定値計画問題に対する単体法の実装報告 (2009)
  • Author(s): 加藤誉運, ほか
  • Organizer: 日本OR学会秋季研究発表会
  • Place of Presentation: 長崎
  • Year and Date: 2009-09-10
• [Presentation] 半正定値計画問題に対する単体法の実装報告 (2009)
  • Author(s): 加藤誉運、田村明久、山崎康史、脇隼人、村松正和
  • Organizer: 日本オペレーションズ・リサーチ学会秋期研究発表会
  • Year and Date: 2009-09-09
• [Presentation] Facial Reduction Algorithm and Conic Expansion Algorithm (2009)
  • Author(s): Masakazu Muramatsu, Hayato Waki
  • Organizer: International Symposium on Mathematical Programming
  • Place of Presentation: Chicago
  • Year and Date: 2009-08-28
• [Presentation] A Facial Reduction Algorithm for Semidefinite Programming Problems in Polynomial Optimization (2009)
  • Author(s): Hayato Waki, Masakazu Muramatsu
  • Organizer: International Symposium on Mathematical Programming
  • Place of Presentation: Chicago
  • Year and Date: 2009-08-28
• [Presentation] Facial Reduction Algorithm and Conic Expansion Algorithm (2009)
  • Author(s): M.Muramatsu, H.Waki
  • Organizer: ISMP 2009
  • Place of Presentation: Chicago
  • Year and Date: 2009-08-28
• [Presentation] A Facial Reduction Algorithm for Semidefinite Programming Problems in Polynomial Optimization (2009)
  • Author(s): H.Waki, M.Muramatsu
  • Organizer: ISMP 2009
  • Place of Presentation: Chicago
  • Year and Date: 2009-08-28
• [Presentation] A Facial Reduction Approach in Conic Programming (2009)
  • Author(s): 村松正和脇隼人
  • Organizer: SCOPE研究部会
  • Place of Presentation: 中央大学
  • Year and Date: 2009-03-21
• [Presentation] A Facial Reduction Approach in Conic Programming (2009)
  • Author(s): 村松正和、脇隼人
  • Organizer: 日本オペレーションズ・リサーチ学会研究部会SCOPE
  • Year and Date: 2009-03-18
• [Presentation] 錐線形計画における錐の面的縮小と正則拡大およびその応用 (2009)
  • Author(s): 村松正和、脇隼人
  • Organizer: 日本オペレーションズ・リサーチ学会春期研究発表会
  • Year and Date: 2009-03-18
• [Presentation] 錐線形計画における面的縮小と正則拡大およびその応用 (2009)
  • Author(s): 村松正和脇隼人
  • Organizer: 日本オペレーションズリサーチ学会春期研究発表会
  • Place of Presentation: 筑波大学
  • Year and Date: 2009-03-18
• [Presentation] 計算と最適化の新展開に向けて (2009)
  • Author(s): 久野誉人、藤澤克樹、村松正和
  • Organizer: 日本オペレーションズ・リサーチ学会春期研究発表会
  • Year and Date: 2009-03-17
• [Presentation] Strange behaviors of interior-point mehtods for solving semidefinite programming problems in polynomial optimization (2008)
  • Author(s): Masakazu Muramatsu, Hayato Waki
  • Organizer: HPOPT 2008
  • Year and Date: 2008-06-13
• [Presentation] Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization (2008)
  • Author(s): Hayato Waki, Masakazu Muramatsu
  • Organizer: HPOPT 2008
  • Place of Presentation: Tilburg University, Netherlands
  • Year and Date: 2008-06-13
• [Presentation] Semide fi nite programming relaxation for polynomial optimization problems (2008)
  • Author(s): Masakazu Muramatsu
  • Organizer: International Conference on Modeling, Computation and Optimization
  • Year and Date: 2008-01-09
• [Presentation] Semidefinite programming relaxation for polynomial opti-mization problems (2008)
  • Author(s): M. Muramatsu
  • Organizer: International Conference on Modeling, Computation and Optimization
  • Place of Presentation: Delhi, India
• [Remarks]
  • URL: http://jsb.cs.uec.ac.jp/~muramatu/
• [Remarks]
  • URL: http://jsb.cs.uec.ac.jp/~muramatu/
• [Remarks]
  • URL: http://jsb.cs.uec.ac.jp/~muramatu/publications.html