Study on globally convergent algorithms for solving nonlinear systems using mathematical techniques

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K04151
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 21020:Communication and network engineering-related
• Research Institution: Chuo University
• Principal Investigator: Yamamura Kiyotaka 中央大学, 理工学部, 教授 (30182603)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 非線形理論・回路 / 非線形数値解析 / 大規模集積回路 / 全解探索 / 数理計画法 / 整数計画法 / ホモトピー法 / 区間解析

Outline of Final Research Achievements

In this project, we developed efficient and globally-convergent algorithms for solving large-scale nonlinear circuits and systems. We first proposed an efficient homotopy method for solving nonlinear circuits, and proved its global convergence property. By this method, bipolar analog integrated circuits with more than 20,000 elements could be solved with the theoretical guarantee of global convergence. We next proposed an efficient algorithm for finding all solutions of nonlinear circuits using linear programming. By this algorithm, all solutions of large-scale systems where the number of variables is several thousands could be found in practical computation time. We further proposed an efficient method for finding all solution sets of nonlinear circuits using integer programming. By this method, complete analysis of nonlinear circuits can be performed easily without making complicated programs. These algorithms are excellent at both efficiency and practicality.

Free Research Field

工学

Academic Significance and Societal Importance of the Research Achievements

大規模集積回路をはじめとする非線形システムの解析問題は科学技術における重要かつ難しい問題の一つで、現在でも「解に収束しない」「効率的な解法が存在しない」「実用化が難しい」などの困難が生じている。本研究では、非線形システムの解析問題の中でも特に難しいとされる「解に収束することが理論的に保証された解法」「すべての解を確実に求めることができる解法」などの大域的求解法の分野を対象に、数理的手法（ホモトピー法、線形計画法、整数計画法など）を用いた非線形システムの“効率的で実用的な”大域的求解法の開発を行い、集積回路設計、ニューラルネットワーク、ＡＩなどの分野の発展に貢献した。