On efficient algorithms for nonconvex feasibility problem and its applications

• Project/Area Number: 16K05280
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Akita Prefectural University
• Principal Investigator: Matsushita Shin-ya 秋田県立大学, システム科学技術学部, 准教授 (20435449)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Discontinued (Fiscal Year 2019)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2019: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 制約可能性問題 / 非凸集合 / 射影法 / 非凸性 / 最良近似問題

Outline of Final Research Achievements

This research aims to develop efficient methods for solving nonconvex feasibility problems. To deal with the non-convex constraint sets, we have investigated modification of a variety of optimization methods, such as fixed-point approximation methods, the proximal gradient method, the projection methods, the Douglas-Rachford method. In particular, we have presented some relevant convergence rate results for fixed-point approximation method and the proximal point algorithm. Moreover, we have investigated theoretical and numerical properties of the proposed methods.

Academic Significance and Societal Importance of the Research Achievements

非凸制約集合は，制御工学における安定性や画像復元技術など，様々な分野で現れる重要な概念である。近年，非線形解析学の分野で研究されていた射影法が非凸集合に関連する制約可能性問題に有効であることが明らかになっているが，その理論的な収束の保証や収束率については解明されていないのが課題となっていた。

Research Products

• [Journal Article] On the finite termination of the gradient projection method (2019)
  • Author(s): Shin-ya Matsushita
  • Journal Title: Journal of Nonlinear and Variational Analysis Volume: 3 Issue: 1 Pages: 19-26
  • DOI: 10.23952/jnva.3.2019.1.03
  • Peer Reviewed / Open Access
• [Journal Article] On the Haugazeau-like projective method for the sum problem (2018)
  • Author(s): Shin-ya Matsushita and Li Xu
  • Journal Title: Journal of Nonlinear and Convex Analysis Volume: 19 Pages: 1515-1523
  • Peer Reviewed
• [Journal Article] A convergence rate of the proximal point algorithm in Banach spaces (2018)
  • Author(s): Shin-ya Matsushita
  • Journal Title: Optimization Volume: 印刷中 Issue: 6 Pages: 881-888
  • DOI: 10.1080/02331934.2018.1432609
  • Peer Reviewed
• [Journal Article] On the convergence rate of the Krasnosel'skii-Mann iteration (2017)
  • Author(s): Shin-ya Matsushita
  • Journal Title: Bulletin of the Australian Mathematical Society Volume: 印刷中 Issue: 1 Pages: 162-170
  • DOI: 10.1017/s000497271600109x
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] On the finite termination of the Douglas-Rachford method for the convex feasibility problem (2016)
  • Author(s): S. Matsushita, Li Xu
  • Journal Title: Optimization Volume: 65 Issue: 11 Pages: 2037-2047
  • DOI: 10.1080/02331934.2016.1209674
  • Peer Reviewed
• [Presentation] Rates of asymptotic regularity for the forward-backward splitting algorithm (2019)
  • Author(s): Shin-ya Matsushita
  • Organizer: Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions, Workshop 2 on Numerical Algorithms in Nonsmooth Optimization
  • Int'l Joint Research / Invited
• [Presentation] 作用素分割法の収束の評価について (2019)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2019年度年会・実関数論分科会
• [Presentation] On a splitting method for maximal monotone operators (2018)
  • Author(s): Shin-ya Matsushita
  • Organizer: The Sixth Asian Conference on Nonlinear Analysis and Optimization (NAO-Asia2018)
  • Int'l Joint Research / Invited
• [Presentation] 不動点近似法に対する収束解析 (2018)
  • Author(s): 松下慎也
  • Organizer: 第57回実函数論・函数解析学合同シンポジウム
  • Invited
• [Presentation] 極大単調作用素の和のリゾルベントについて (2018)
  • Author(s): 松下慎也
  • Organizer: 京都大学数理解析研究所共同研究(公開型)「非線形解析学と凸解析学の研究」
• [Presentation] 作用素分割法の収束について (2018)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2018年度年会・実関数論分科会
• [Presentation] Convergence rates of the proximal point algorithm (2017)
  • Author(s): Shin-ya Matsushita
  • Organizer: The 10th Anniversary International Conference on Nonlinear Analysis and Convex Analysis
  • Int'l Joint Research
• [Presentation] On the convergence rate of the proximal point algorithm (2017)
  • Author(s): Shin-ya Matsushita
  • Organizer: The12th International Conference on Fixed Point Theory and Its Applications
  • Int'l Joint Research
• [Presentation] 近接点法の収束について (2017)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2017年度秋季総合分科会・実関数論分科会
• [Presentation] Krasnosel’skii-Mann iteration の収束について (2017)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2017年度年会, 実関数論分科会
  • Place of Presentation: 首都大学東京
• [Presentation] On convergence of the fixed point iterations (2016)
  • Author(s): 松下慎也
  • Organizer: RIMS共同研究「非線形解析学と凸解析学の研究」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Alternating direction method of multipliersの収束について (2016)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2016年度秋季総合分科会, 実関数論分科会
  • Place of Presentation: 関西大学
• [Presentation] On the existence of solutions to the sum problem (2016)
  • Author(s): Shin-ya Matsushita
  • Organizer: The fifth Asian conference on Nonlinear Analysis and Optimization
  • Place of Presentation: Toki Messe, Niigata, Japan
  • Int'l Joint Research
• [Presentation] On the convergence rate of the fixed point iterations (2016)
  • Author(s): Shin-ya Matsushita
  • Organizer: New Trends in Optimization and Variational Analysis for Applications
  • Place of Presentation: Quy Nhon University, Viet Nam
  • Int'l Joint Research