Development of approximation methods for cone optimization problems using semidefinite bases

• Project/Area Number: 17K18946
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Social systems engineering, Safety engineering, Disaster prevention engineering, and related fields
• Research Institution: University of Tsukuba
• Principal Investigator: Yoshise Akiko 筑波大学, システム情報系, 教授 (50234472)
• Project Period (FY): 2017-06-30 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥6,240,000 (Direct Cost: ¥4,800,000、Indirect Cost: ¥1,440,000); Fiscal Year 2019: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2018: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2017: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: 錐最適化 / 半正定値最適化 / 二重非負値最適化 / 共正値最適化 / 線形計画問題 / 半正定値錐 / 優対角錐 / 二重非負値錐 / 半正定値基 / 半正定値緩和 / 二重非負値緩和 / 錐最適化問題 / 半正定値最適化問題 / 二重非負値最適化問題 / 共正値最適化問題 / 完全正値最適化問題

Outline of Final Research Achievements

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

Academic Significance and Societal Importance of the Research Achievements

「数理最適化」は，生産システムやサービス事業の効率化や，さらに最近では人工知能を支える要素技術として，社会に定着している．本研究では，この数理最適化分野で近年特に盛んに研究されている錐最適化の，とりわけ解くことが困難と言われている問題群に対して，応募者らが発案した「半正定値基」を用いて，新たな発想に基づく解法を提案している．さらに「拡張半正定値基」を新規に提案し，それらの凸包が，線形計画問題で判定できる新たな行列の集合を与えていることを理論的に示し，さらにそれらを用いて求解が困難とされる最大安定集合問題に対する，計算効率性に優れた解法を提案したことは，学術的にも，また社会的にも意義は高い．

Research Products

• [Journal Article] SD基に基づく半正定値行列錐の凸多面錐近似 (2020)
  • Author(s): 汪玉柱，吉瀬章子
  • Journal Title: 統計数理研究所共同研究リポート「最適化：モデリングとアルゴリズム」 Volume: 428 Pages: 106-113
• [Journal Article] QAPの半正定値緩和問題を解くためのセンタリングADMM (2020)
  • Author(s): 加納伸一，吉瀬章子
  • Journal Title: 統計数理研究所共同研究リポート「最適化：モデリングとアルゴリズム」 5) 巻 428 Volume: 428 Pages: 114-129
• [Journal Article] Centering ADMM for the semidefinite relaxation of the QAP (2020)
  • Author(s): Shin-ichi Kanoh and Akiko Yoshise
  • Journal Title: Discussion Paper Series, Department of Policy and Planning Sciences, University of Tsukuba Volume: 1368 Pages: 1-16
• [Journal Article] Polyhedral approximations of the semidefinite cone and their application (2019)
  • Author(s): Yuzhu Wang, Akihiro Tanaka and Akiko Yoshise
  • Journal Title: Discussion Paper Series, Department of Policy and Planning Sciences, University of Tsukuba Volume: 1359 Pages: 1-20
• [Journal Article] LP-based tractable subcones of the semidefinite plus nonnegative cone (2018)
  • Author(s): Tanaka Akihiro、Yoshise Akiko
  • Journal Title: Annals of Operations Research Volume: 265 Issue: 1 Pages: 155-182
  • DOI: 10.1007/s10479-017-2720-z
  • Peer Reviewed / Open Access
• [Journal Article] Bounds for two static optimization problems on routing and spectrum allocation of anycasting (2018)
  • Author(s): Yasutaka Miyagawa, Yosuke Watanabe, Maiko Shigeno, Kiyo Ishii, Atsuko Takehusa and Akiko Yoshise
  • Journal Title: Optical Switching and Networking Volume: 31 Pages: 144-161
  • DOI: 10.1016/j.osn.2018.10.008
  • Peer Reviewed / Open Access
• [Journal Article] Optimization-based analysis of last-mile one-way mobility sharing (2018)
  • Author(s): Masaki Yamada, Masashi Kimura, Naoki Takahashi and Akiko Yoshise
  • Journal Title: Discussion Paper Series Volume: 1353 Pages: 1-35
• [Journal Article] 半正定値基底を用いた錐最適化問題の近似について (2018)
  • Author(s): 成島大吾，田中彰浩，吉瀬章子
  • Journal Title: 日本オペレーションズリサーチ学会春季研究発表会アブストラクト集 Volume: 2018 Pages: 168-169
• [Journal Article] ワンウェイ型ラストマイルモビリティシェアリングのオペレーション最適化及び新規戦略の検討 (2018)
  • Author(s): 山田匡規，木村雅志，高橋直希，吉瀬章子
  • Journal Title: 日本オペレーションズリサーチ学会春季研究発表会アブストラクト集 Volume: 2018 Pages: 88-89
• [Presentation] ガソリンスタンドにおける多段階勤務スケジューリング作成モデルの改良 (2020)
  • Author(s): 山口大輔, 大原敬之, 近藤大祐, 廣田雄介, 吉瀬章子
  • Organizer: 日本オペレーションズ・リサーチ学会春季発表会
• [Presentation] モビリティシェアリングにおけるユーザーの利用実態を考慮したオペレーション最適化モデルの改善 (2020)
  • Author(s): 小市敦也, 吉瀬章子
  • Organizer: 日本オペレーションズ・リサーチ学会春季発表会
• [Presentation] 筑波大学の総合選抜入試導入に伴う専門導入科目の時間割作成 (2020)
  • Author(s): 小代圭祐, 本多里紗, 久富彩香, 黒田翔, 吉瀬章子
  • Organizer: 日本オペレーションズ・リサーチ学会春季発表会
• [Presentation] 総合選抜入試における専門導入科目の時間割スケジューリング (2019)
  • Author(s): 小代圭祐, 黒田翔, 吉瀬章子
  • Organizer: SS2019
• [Presentation] SD基に基づく半正定値行列錐の凸多面錐近似 (2019)
  • Author(s): 汪玉柱, 吉瀬章子
  • Organizer: 日本オペレーションズ・リサーチ学会秋季発表会
• [Presentation] A strategic optimization model for one-way carsharing systems (2019)
  • Author(s): 張凱, 吉瀬章子
  • Organizer: 日本オペレーションズ・リサーチ学会秋季発表会
• [Presentation] Two approaches for solving hard conic optimization problems (2019)
  • Author(s): Yuzhu Wang, Shin-ichi Kanoh and Akiko Yoshise
  • Organizer: NACA-ICOTA2019
  • Int'l Joint Research / Invited
• [Presentation] Polyhedral approximations of the semidefinite cone and their applications (2019)
  • Author(s): Yuzhu Wang, Akihiro Tanaka and Akiko Yoshise
  • Organizer: NACA-ICOTA2019
  • Int'l Joint Research
• [Presentation] Polyhedral approximations of the semidefinite cone and their applications (2019)
  • Author(s): Shin-ichi Kanoh and Akiko Yoshise
  • Organizer: ICCOPT 2019
  • Int'l Joint Research
• [Presentation] Polyhedral approximations of the semidefinite cone and their applications (2019)
  • Author(s): Yuzhu Wang, Akihiro Tanaka and Akiko Yoshise
  • Organizer: ICCOPT 2019
  • Int'l Joint Research
• [Presentation] Polyhedral approximations of the semidefinite cone and their applications (2018)
  • Author(s): Yuzhu wang, Akihiro Tanaka and Akiko Yoshise
  • Organizer: The Sixth Asian Conference on Nonlinear Analysis and Optimization
  • Int'l Joint Research / Invited
• [Presentation] Acceleration of the Lagrangian-DNN method for a class of QOPs (2018)
  • Author(s): Yuzhu wang and Akiko Yoshise
  • Organizer: International Symposium on Mathematical Programming
  • Int'l Joint Research
• [Presentation] ワンウェイ型ラストマイルモビリティシェアリングのオペレーション最適化及び新規戦略の検討 (2018)
  • Author(s): 山田匡規，木村雅志，高橋直希，吉瀬章子
  • Organizer: 日本オペレーションズリサーチ学会春季研究発表会
• [Presentation] 半正定値基底を用いた錐最適化問題の近似について (2018)
  • Author(s): 成島大吾，田中彰浩，吉瀬章子
  • Organizer: 日本オペレーションズリサーチ学会春季研究発表会
• [Presentation] Inner and Outer Approximations of the Semidefinite Cone Using SD Bases and their Applications to Some NP-Hard Problems (2017)
  • Author(s): Daigo Narushima, Akihiro Tanaka, and Akiko Yoshise
  • Organizer: SIAM Optimization 2017
  • Int'l Joint Research