On robust numerical methods of semidefinite relaxation for polynomial optimization problems

• Project/Area Number: 22560061
• 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: MURAMATSU Masakazu 電気通信大学, 情報理工学(系)研究科, 教授 (70266071)
• Project Period (FY): 2010-04-01 – 2013-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 最適化 / 錐線形計画 / 多項式計画 / 半正定値計画 / 頑健性 / 面的削減法

Research Abstract

We established the theory of facial reduction algorithm (FRA) for conic linear optimization problems, and prove that the conic expansion approach is dual to FRA. For SDP relaxation for polynomial optimization problems (POPs), sometimes we can compute the optimal value of POP by solving the SDP relaxation problem although the SDP problem is infeasible. We elucidated the reason of this phenomena, and proposed a new SDP relaxation scheme using this property.

Research Products

• [Journal Article] Facial reduction algorithms for conic optimization problems (2013)
  • Author(s): H. Waki, and M. Muramatsu
  • Journal Title: Journal of Optimization Theory and Applications Volume: 158 Issue: 1 Pages: 188-215
  • DOI: 10.1007/s10957-012-0219-y
  • Peer Reviewed
• [Journal Article] Strange behaviors of interior-point methods for solving semidefinite pro- gramming problems in polynomial optimization (2012)
  • Author(s): H. Waki, M. Nakata, and M. Muramatsu
  • Journal Title: Computational Optimization and Applications Volume: 53 Issue: 3 Pages: 823-844
  • DOI: 10.1007/s10589-011-9437-8
  • Peer Reviewed
• [Journal Article] Strange Behaviors of Interior-point Methods for Solving Semidefinite Programming Problems in Polynomial Optimization (2012)
  • Author(s): Hayato Waki, Maho Nakata and Masakazu Muramatsu
  • Journal Title: Computational Optimization and Applications Volume: 53 Pages: 823-844
  • Peer Reviewed
• [Journal Article] An extension of the elimination method for a sparse SOS polynomial (2011)
  • Author(s): H. Waki and M. Muramatsu
  • Journal Title: Journal of the Operations Research Society of Japan Volume: 54 Pages: 161-190
  • URL: http://www.orsj.or.jp/~archive/pdf/e_mag/Vol.54_4_161.pdf
  • Peer Reviewed
• [Journal Article] An Extension of the Elimination Method for a Sparse SOS Polynomial (2011)
  • Author(s): H.Waki, M.Muramatsu
  • Journal Title: Journal of Operations Research Society of Japan Volume: 54 Pages: 161-190
  • Peer Reviewed
• [Journal Article] A facial reduction algorithm for finding sparse SOS representations (2010)
  • Author(s): H. Waki and M. Muramatsu
  • Journal Title: Operations Research Letters Volume: 38 Issue: 5 Pages: 361-365
  • DOI: 10.1016/j.orl.2010.05.011
  • Peer Reviewed
• [Journal Article] Afacial reduction algorithm for fording sparse SOS representations (2010)
  • Author(s): H.Waki, M.Muramatsu
  • Journal Title: Operations Research Letters Volume: 38 Pages: 361-365
  • Peer Reviewed
• [Presentation] 低ランク整数完全正値行列の判定について (2014)
  • Author(s): 川瀬弘樹,村松正和,高橋里司
  • Organizer: 2014年日本オペレーションズ・リサーチ学会春季研究発表会
  • Place of Presentation: 大阪
• [Presentation] 2次元トーラスグラフ上での情報拡散ゲームにおけるナッシュ均衡 (2013)
  • Author(s): 祐成友樹,村松正和
  • Organizer: 2013年日本オペレーションズ・リサーチ学会秋季研究発表会
  • Place of Presentation: 徳島
• [Presentation] Adaptive SDP relaxation for polynomial optimization (2013)
  • Author(s): M. Muramatsu
  • Organizer: International Conference on Continuous Optimization (ICCOPT 2013)
  • Place of Presentation: リスボン
• [Presentation] A perturbed sums of squares theorem for polynomial optimization and its applications (2012)
  • Author(s): Masakazu Muramatsu, Levent Tuncel, and Hayato Waki
  • Organizer: International Symposium on Mathematical Programming (ISMP2012)
  • Place of Presentation: ベルリン
• [Presentation] 多項式計画に対する半正定値緩和 (2012)
  • Author(s): 村松正和
  • Organizer: 組合せ最適化セミナー(第9回)
  • Place of Presentation: 京都大学
  • Invited
• [Presentation] 多項式計画に対する半正定値緩和の不思議な性質 (2011)
  • Author(s): 村松正和
  • Organizer: 最適化理論の産業・諸科学への応用
  • Place of Presentation: 九州大学マスフォアインダストリ研究所(招待講演)
  • Year and Date: 2011-10-14
• [Presentation] Strange Behaviors of Interior-point Methods for Solving Semidefinite Programming Problems in Polynomial Optimization and their Reason (2011)
  • Author(s): 村松正和
  • Organizer: 最適化手法の深化と広がり
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-07-21
• [Presentation] The Facial Reduction Algorithm for Semidefuute Programming Relaxation of Polynomial Optimization Problems (2010)
  • Author(s): Masakazu Muramatsu
  • Organizer: HPOPT 2010
  • Place of Presentation: Tilburg University(TheNetherlands)
  • Year and Date: 2010-06-16
• [Presentation] The facial reduction algorithm for semidefinite programming relaxation of polynomial optimization problems (2010)
  • Author(s): M. Muramatsu
  • Organizer: 11th International Workshop on High Performance Optimization Techniques (HPOPT2010)
  • Place of Presentation: Tilburg, the Netherlands
  • Invited
• [Presentation] 多項式計画に対する半正定値緩和
  • Author(s): 村松正和
  • Organizer: 組み合わせ最適化セミナー
  • Place of Presentation: 京都大学
  • Invited
• [Book] あたらしい数理最適化-GurobiとPython言語で解く- (2012)
  • Author(s): 久保幹雄, J. Pedro Pedroso,村松正和, Abdur Rais
  • Publisher: 近代科学社
• [Book] あたらしい数理最適化-Gurobi と Python 言語で解く- (2012)
  • Author(s): 久保幹雄, J. Pedro Pedroso, 村松正和, Abdur Rais
  • Total Pages: 250
  • Publisher: 近代科学社