Information Geometry of Convex Optimization : Extension and Applications

• Project/Area Number: 20340024
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: National Graduate Institute for Policy Studies (2010); The Institute of Statistical Mathematics (2008-2009)
• Principal Investigator: TSUCHIYA Takashi 政策研究大学院大学, 政策研究科, 教授 (00188575)
• Co-Investigator(Kenkyū-buntansha): OHARA Atsumi 福井大学, 工学研究科, 教授 (90221168) ; MURAMATSU Masakazu 電気通信大学, 情報理工学研究科, 教授 (70266071) ; FUKUDA Mituhiro 東京工業大学, 情報理工学研究科, 准教授 (80334548)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥9,360,000 (Direct Cost: ¥7,200,000、Indirect Cost: ¥2,160,000); Fiscal Year 2010: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2009: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2008: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
• Keywords: 凸最適化 / 情報幾何 / 内点法 / 数理計画 / 計算複雑度 / 制御理論 / グラフィカルモデル / 微分幾何 / 対称錐計画 / 線形計画 / 半正定値計画

Research Abstract

Interior-point algorithms for semidefinite programs and symmetric cone programs are analyzed in view of information geometry to show that the iteration complexity of primal-dual interior-point algorithms is approximately represented as a infomration geometric integral over central trajectory. Through extensive numerical experiments we demonstrated that the integral very accurately predict iteration-complexity of interior-point algorithms. One of the largest Gaussian graphical models in the world are successfully solved with super computer. Primal-dual interior-pont algorithms for large-scale Gaussian graphical models are developed. Regularization and facial reduction approaches for ill-conditioned semidefinite programs are developed.

Research Products

• [Journal Article] Conformal geometry of escort probability and its Applications (2012)
  • Author(s): Atsumi Ohara, Hiroshi Matsuzoe and Shun-ichi Amari
  • Journal Title: Modern Physics Letters B Volume: 26 Pages: 10-10
• [Journal Article] Latest developments in the SDPA family for solving large-scale SDPs. in Handbook on Semidefinite, Conic and Polynomial Optimization (2012)
  • Author(s): M. Yamashita, K. Fujisawa, M. Fukuda, K. Kobayashi, K. Nakata, and M. Nakata
  • Journal Title: M. F. Anjos and J. B. Lasserre, eds. Pages: 687-713
• [Journal Article] An extension of the elimination method for a sparse SOS polynomial (2011)
  • Author(s): Hayato Waki, Masakazu Muramatsu
  • Journal Title: Journal of Operations Research Society of Japan Pages: 161-190
• [Journal Article] Geometry of q-exponential family of probability distributions (2011)
  • Author(s): Shun-ichi Amari and Atsumi Ohara
  • Journal Title: Entropy Volume: 13 Pages: 1170-1185
• [Journal Article] Information geometry of q-Gaussian densities and behaviors of solutionsto related diffusion equations (2010)
  • Author(s): Atsumi Ohara and Tatsuaki Wada
  • Journal Title: Journal of Physics A : Mathematical and Theoretical Volume: 43 Pages: 18-18
• [Journal Article] A facial reduction algorithm for finding sparse SOS representations (2010)
  • Author(s): Hayato Waki and Masakazu Muramatsu
  • Journal Title: Operations Research Letters Volume: 38 Pages: 361-365
• [Journal Article] Proximity of weighted and layered least squares solutions (2009)
  • Author(s): Tomonari Kitahara and Takashi Tsuchiya
  • Journal Title: SIAM Journal on Matrix Analysis and Applications Volume: 31 Pages: 1172-1186
• [Journal Article] Covariance regularization in inverse space (2009)
  • Author(s): Genta Ueno and Takashi Tsuchiya
  • Journal Title: Quarterly Journal of the Royal Meteorological Society Volume: 135 Pages: 1133-1156
• [Journal Article] Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation (2009)
  • Author(s): Atsumi Ohara
  • Journal Title: European Physical Journal B Volume: 70 Pages: 15-28
• [Journal Article] Invariance under affine transformation in semidefinite programmingrelaxation for polynomial optimization problems (2009)
  • Author(s): H. Waki, M. Muramatsu and M. Kojima
  • Journal Title: Pacific Journal of Optimization Volume: 5 Pages: 297-312
• [Journal Article] Proximity of Weighted and Layered Least Squares Solutions (2009)
  • Author(s): T.Kitahara, T.Tsuchiya
  • Journal Title: SIAM Journal on Matrix Analysis and Applications 31 Pages: 1172-1186
  • Peer Reviewed
• [Journal Article] Invariance under affine transformation in semidefinite programming relaxation for polynomial optimization problems" (2009)
  • Author(s): Hayato Waki, Masakazu Muramatsu, Masakazu Koiima
  • Journal Title: Pacific Journal of Optimization 5 Pages: 297-312
  • Peer Reviewed
• [Journal Article] Covariance regularization in inverse space (2009)
  • Author(s): Genta Ueno, Takashi Tsuchiya
  • Journal Title: Quarterly Journal of Royal Meteorological Society (掲載予定)
  • Peer Reviewed
• [Journal Article] SparsePOP : a sparse semidefinite programming relaxation of polynomial optimization problems (2008)
  • Author(s): Hayato Waki and Sunyoung Kim and Masakazu Kojima and Mazakazu Muramatsu and H. Sugimoto
  • Journal Title: ACM Transactions on Mathematical Software Volume: 35 Pages: 2-12
• [Journal Article] A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms (2008)
  • Author(s): R. D. C. Monteiro, Takashi Tsuchiya
  • Journal Title: Mathematical Programming 115 Pages: 105-149
  • Peer Reviewed
• [Journal Article] Parallel solver for semidefinite programming problem having sparse Schur complement matrix.
  • Author(s): M. Yamashita, K. Fujisawa, M. Fukuda, K. Nakata, and M. Nakata
  • Journal Title: Transactions on Mathematical Software Volume: (To appear)
• [Presentation] Study of iteration-complexity for the central trajectory via curvature of information geometry (2011)
  • Author(s): A. Ohara, S. Kakihara and T. Tsuchiya
  • Organizer: INFORMS 2011, Charlotte
  • Place of Presentation: Charlotte, NC, USA
• [Presentation] Information geometry (2010)
  • Author(s): Takashi Tsuchiya
  • Organizer: interior-point algorithms and graphical modeling. Long Term Workshop : Mathematical Sciences and their Applications
  • Place of Presentation: Kamisuwa, Nagano, Japan
  • Year and Date: 2010-09-10
• [Presentation] Geometry for q-exponential families (2010)
  • Author(s): Hiroshi Matsuzoe and Atsumi Ohara
  • Organizer: 2nd International Colloquium on Differential Geometry and its Related Fields
  • Place of Presentation: Veliko Tarnovo, Bulgaria
• [Presentation] On the implementation of the primal-dual interior-point method for semidefinite programs with log-determinants (2009)
  • Author(s): M. Fukuda
  • Organizer: 20th International Symposium on Mathematical Programming(ISMP)
  • Place of Presentation: Chicago IL, USA
  • Year and Date: 2009-08-27
• [Presentation] Information geometric approach to interior-point algorithms in LP and SDP (2009)
  • Author(s): Satoshi Kakihara, Renato Monteiro, Atsumi Ohara and Takashi Tsuchiya
  • Organizer: International Symposium on Mathematical Programming
  • Place of Presentation: Chicago
• [Presentation] A dually flat structure on the space of escort distributions (2009)
  • Author(s): Atsumi Ohara, Hiroshi Matsuzoe and Shun-ichi Amari
  • Organizer: Mathematical Aspects of Generalized Entropies and their Applications
  • Place of Presentation: Kyoto
• [Presentation] A Facial reduction algorithm for semidefinite programming problemsin polynomial optimization (2009)
  • Author(s): Hayato Waki and Masakazu Muramatsu
  • Organizer: International Symposium on Mathematical Programming
  • Place of Presentation: Chicago
• [Presentation] Facial reduction algorithm and conic expansion algorithm (2009)
  • Author(s): Masakazu Muramatsu and Hayato Waki
  • Organizer: International Symposium on Mathematical Programming
  • Place of Presentation: Chicago
• [Presentation] Information geometric approach to interior-point algorithms in LP and SDP (2009)
  • Author(s): S.Kakihara, R.D.C.Monteiro, A.Ohara, T.Tsuchiya
  • Organizer: The 20^<th> International Symposium on Mathematical Programming
  • Place of Presentation: 米国シカゴ
• [Presentation] An information geometric approach to polynomial-time interior-point algorithms : complexity bound via curvature integral (2008)
  • Author(s): Takashi Tsuchiya and Atsumi Ohara
  • Organizer: SIAM Conference on Optimization
  • Place of Presentation: Boston MA, USA
  • Year and Date: 2008-05-11
• [Presentation] An Information Geometric Approach to Polynomial-Time Interior-Point Algorithms : Complexity Bound Via Curvature Integral (2008)
  • Author(s): Takashi Tsuchiya
  • Organizer: The 8^<th> SIAM Conference on Optimization
  • Place of Presentation: Boston, USA
  • Year and Date: 2008-05-11
• [Presentation] Applications of semidefinite programs with log-determinant terms (2008)
  • Author(s): M. Fukuda
  • Organizer: SIAM Conference on Optimization
  • Place of Presentation: Boston MA, USA
  • Year and Date: 2008-05-10
• [Presentation] A simple variant of the MTY-PC algorithm and its objective-function-free polynomial-time convergence for bounded LPs (2008)
  • Author(s): Tomonari Kitahara and Takashi Tsuchiya
  • Organizer: SIAM Conference on Optimization
  • Place of Presentation: Boston MA, USA
  • Year and Date: 2008-05-10
• [Presentation] Geometric aspects of a certain type of nonlinear diffusion equations (2008)
  • Author(s): Atsumi Ohara and Tatsuaki Wada
  • Organizer: International Symposium on Nonlinear Theory and its Applications
  • Place of Presentation: Budapest
• [Presentation] Strange behaviors of interior-point mehtods for solving semidefiniteprogramming problems in polynomial optimization (2008)
  • Author(s): Hayato Waki and Masakazu Muramatsu
  • Organizer: 10th International Workshop on High Performance Optimization
  • Place of Presentation: Techniques