Development of approximation methods for cone optimization problems using semidefinite bases

2019 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

数理最適化

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．