Development of Global Optimization Methods by Generalized Eigenvalue Computation

• Project/Area Number: 17H01699
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Mathematical informatics
• Research Institution: The University of Tokyo
• Principal Investigator: Iwata Satoru 東京大学, 大学院情報理工学系研究科, 教授 (00263161)
• Co-Investigator(Kenkyū-buntansha): 武田 朗子 東京大学, 大学院情報理工学系研究科, 教授 (80361799) ; 中務 佑治 国立情報学研究所, 情報学プリンシプル研究系, 准教授 (10723554)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥7,280,000 (Direct Cost: ¥5,600,000、Indirect Cost: ¥1,680,000); Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 数理最適化 / 大域最適化 / 一般化固有値 / アルゴリズム / 機械学習 / 一般化固有値計算

Outline of Final Research Achievements

While non-convex optimization problems are hard to solve in general, this research has focused on those problems that have geometrical backgrounds and investigated a methodology to design efficient algorithms for obtaining globally optimal solutions. In particular, we have developed an efficient algorithm for non-convex quadratic constraint quadratic programming (QCQP) with one constraint. In addition, we exploit this algorithm to obtain relaxation solutions for general non-convex QCQP with the aid of the Lagrange multipliers.

Academic Significance and Societal Importance of the Research Achievements

これまで厳密な最適解を効率的に計算することは殆ど不可能だろうと思われていた非凸最適化問題に対して，幾何的な背景を有する特殊な問題であれば，一般化固有値計算を用いて，大域最適化を得られることを明らかにした．特に，一般の非凸無制約最適化問題の解法として広く使われている信頼領域法の中で繰り返し解く必要が生じる信頼領域部分問題(TRS)の高速解法を発表した論文は，比較的多くの研究に引用されている．その中には，３次正則化に対して手法を拡張したドイツの研究者による論文など，興味深い後続研究が現れている．

Research Products

• [Int'l Joint Research] ブルッセル自由大学(ベルギー)
• [Int'l Joint Research] 香港理工大学(中国)
• [Journal Article] An inexact successive quadratic approximation method for a class of difference-of-convex optimization problems (2022)
  • Author(s): Tianxiang Liu, Akiko Takeda
  • Journal Title: Computational Optimization and Applications Volume: 82 Issue: 1 Pages: 141-173
  • DOI: 10.1007/s10589-022-00357-z
  • Peer Reviewed
• [Journal Article] Primal-dual subgradient method for constrained convex optimization problems (2021)
  • Author(s): Michael R. Metel, Akiko Takeda
  • Journal Title: Optimization Letters Volume: 15 Issue: 4 Pages: 1491-1504
  • DOI: 10.1007/s11590-021-01728-x
  • Peer Reviewed
• [Journal Article] Stochastic Proximal Methods for Non-Smooth Non-Convex Constrained Sparse Optimization (2021)
  • Author(s): Michael Metel, Akiko Takeda,
  • Journal Title: Journal of Machine Learning Research Volume: 22 (115) Pages: 1-36
  • Peer Reviewed
• [Journal Article] A Projected Gradient Method for Opinion Optimization with Limited Changes of Susceptibility to Persuasion (2021)
  • Author(s): Naoki Marumo, Atsushi Miyauchi, Akiko Takeda, Akira Tanaka
  • Journal Title: CIKM Volume: 30 Pages: 1274-1283
  • DOI: 10.1145/3459637.3482408
  • Peer Reviewed
• [Journal Article] A Gradient Method for Multilevel Optimization (2021)
  • Author(s): Ryo Sato, Mirai Tanaka, Akiko Takeda
  • Journal Title: Advances in Neural Information Processing Systems Volume: 34 Pages: 7522-7533
  • Peer Reviewed
• [Journal Article] A weighted linear matroid parity algorithm (2021)
  • Author(s): Satoru Iwata, Yusuke Kobayashi
  • Journal Title: SIAM Journal on Computing Volume: - Issue: 2 Pages: STOC17-238-STOC17-280
  • DOI: 10.1137/17m1141709
• [Journal Article] Data-driven structured noise filtering via common dynamics estimation (2020)
  • Author(s): Ivan Markovsky, Tianxiang Liu, Akiko Takeda
  • Journal Title: IEEE Transactions on Signal Processing Volume: 68 Pages: 3064-3073
  • DOI: 10.1109/tsp.2020.2993676
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A hybrid penalty method for a class of optimization problems with multiple rank constraints (2020)
  • Author(s): Tianxiang Liu, Ivan Markovsky, Ting Kei Pong, Akiko Takeda
  • Journal Title: SIAM Journal on Matrix Analysis and Applications Volume: 41 Issue: 3 Pages: 1260-1283
  • DOI: 10.1137/19m1269919
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Convex fairness constrained model using causal effect estimators (2020)
  • Author(s): Hikaru Ogura, Akiko Takeda
  • Journal Title: WWW '20: Companion Proceedings of the Web Conference 2020 Volume: - Pages: 723-732
  • DOI: 10.1145/3366424.3383556
  • Peer Reviewed
• [Journal Article] Robust Bayesian Model Selection for Variable Clustering with the Gaussian Graphical Model (2020)
  • Author(s): Daniel Andrade, Akiko Takeda, Kenji Fukumizu
  • Journal Title: Statistics and Computing Volume: 30 Issue: 2 Pages: 351-376
  • DOI: 10.1007/s11222-019-09879-9
  • Peer Reviewed / Open Access
• [Journal Article] Estimation of Gaussian Mixture Models via Tensor Moments with Application to Online Learning (2020)
  • Author(s): Donya Rahmani, Mahesan Niranjan, Damien Fay, Akiko Takeda, Jacek Brodzki
  • Journal Title: Pattern Recognition Letters Volume: 131 Pages: 285-292
  • DOI: 10.1016/j.patrec.2020.01.001
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Finding a Stable Allocation in Polymatroid Intersection (2020)
  • Author(s): Satoru Iwata, Yu Yokoi
  • Journal Title: Mathematics of Operations Research Volume: 45 Issue: 1 Pages: 63-85
  • DOI: 10.1287/moor.2018.0976
  • Peer Reviewed
• [Journal Article] A Blossom Algorithm for Maximum Edge-Disjoint T-Paths (2020)
  • Author(s): Iwata Satoru、Yokoi Yu
  • Journal Title: Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms Volume: － Pages: 1933-1944
  • DOI: 10.1137/1.9781611975994.119
  • ISBN: 9781611975994
  • Peer Reviewed
• [Journal Article] Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint (2019)
  • Author(s): Satoru Adachi, Yuji Nakatsukasa
  • Journal Title: Mathematical Programming Volume: 173 Issue: 1-2 Pages: 79-116
  • DOI: 10.1007/s10107-017-1206-8
  • Peer Reviewed / Open Access
• [Journal Article] Subspace methods for multi-channel sum-of-exponentials common dynamics estimation (2019)
  • Author(s): Ivan Markovsky,Tianxiang Liu, Akiko Takeda
  • Journal Title: Proceedings of the 58th IEEE Conference on Decision and Control Volume: - Pages: 2672-2675
  • DOI: 10.1109/cdc40024.2019.9029261
• [Journal Article] Equivalences and differences in conic relaxations of combinatorial quadratic optimization problems (2018)
  • Author(s): Ito N.、Kim S.、Kojima M.、Takeda A.、Toh K.-C.
  • Journal Title: Journal of Global Optimization Volume: 72 Issue: 4 Pages: 619-653
  • DOI: 10.1007/s10898-018-0676-4
  • NAID: 40021559420
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Index Reduction via Unimodular Transformations (2018)
  • Author(s): Iwata Satoru, Takamatsu Mizuyo
  • Journal Title: SIAM Journal on Matrix Analysis and Applications Volume: 39 Issue: 3 Pages: 1135-1151
  • DOI: 10.1137/17m111794x
  • Peer Reviewed
• [Journal Article] Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization (2018)
  • Author(s): Yamada Shinji、Takeda Akiko
  • Journal Title: Journal of Global Optimization Volume: On-line First Issue: 2 Pages: 1-38
  • DOI: 10.1007/s10898-018-0617-2
  • Peer Reviewed
• [Presentation] Weighted Linear Matroid Parity (2021)
  • Author(s): Satoru Iwata
  • Organizer: SIAM Conference on Optimization
  • Int'l Joint Research / Invited
• [Presentation] Deterministic and Stochastic Gradient Methods for Non-Smooth Non-Convex Regularized Optimization (2020)
  • Author(s): Akiko Takeda
  • Organizer: Variational Analysis and Optimisation Webinar series
  • Int'l Joint Research / Invited
• [Presentation] Pfaffian and Matroid Matching (2020)
  • Author(s): Satoru Iwata
  • Organizer: TUNGA (Theory Underlying Algorithms)
  • Int'l Joint Research / Invited
• [Presentation] Matroid Parity (2019)
  • Author(s): Satoru Iwata
  • Organizer: The 12th Annual Meeting of the Asian Association for Algorithms and Computation
  • Int'l Joint Research / Invited
• [Presentation] Mader's Theorem on Edge-disjoint T-paths (2019)
  • Author(s): Satoru Iwata
  • Organizer: Cargese Workshop on Combinatorial Optimization
  • Int'l Joint Research / Invited
• [Presentation] Group Lasso for Household Energy Consumption Prediction and Toward Nonconvex Regularizer (2019)
  • Author(s): Akiko Takeda
  • Organizer: The PolyU AMA - RIKEN AIP Joint Workshop on Optimization and Machine Learning
  • Int'l Joint Research
• [Presentation] Efficient DC algorithm for nonconvex sparse optimization problems (2019)
  • Author(s): kiko Takeda, Jun-ya Gotoh, Katsuya Tono
  • Organizer: International Conference on Optimization: Techniques and Applications
  • Int'l Joint Research
• [Presentation] Nonconvex Quadratic Optimization with One or Two Constraints (2019)
  • Author(s): Akiko Takeda
  • Organizer: National Taiwan Normal University
  • Invited
• [Presentation] Hyperparameter Learning via Bilevel Nonsmooth Optimization (2018)
  • Author(s): Akiko Takeda
  • Organizer: The Workshop on Variational Analysis and Stochastic Optimization
  • Int'l Joint Research
• [Presentation] On the roles of eigenvalues in optimization (2018)
  • Author(s): Yuji Nakatsukasa
  • Organizer: 3rd International Symposium on Research and Education of Computational Science
  • Int'l Joint Research / Invited
• [Presentation] Index Reduction via Unimodular Transformation (2018)
  • Author(s): Satoru Iwata
  • Organizer: International Symposium on Mathematical Programmig
• [Presentation] Global optimization via eigenvalues (2018)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Theory and Algorithms in Data Science Seminar, Alan Turing Institute (UK)
  • Invited
• [Presentation] Global optimization via eigenvalues (2017)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Householder Symposium
  • Int'l Joint Research / Invited
• [Presentation] Nonconvex Quadratic Optimization with One or Two Constraints (2017)
  • Author(s): Akiko Takeda
  • Organizer: Colloquium at Department of Applied Mathematics, The Hong Kong Polytechnic University
  • Invited