An equivalent nonlinear optimization model with triangular low-rank factorization for semidefinite programs

• Published in: Optimization methods & software Vol. ahead-of-print; no. ahead-of-print; pp. 1 - 15
• Main Authors: Yamakawa, Yuya, Ikegami, Tetsuya, Fukuda, Ellen H., Yamashita, Nobuo
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.11.2023; Taylor & Francis Ltd
• Subjects: Algorithms; Factorization; nonlinear optimization problems; Nonlinear programming; Optimization models; Semidefinite optimization problems; sequential quadratic programming method; triangular low-rank factorization
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2023.2222434

In this paper, we propose a new nonlinear optimization model to solve semidefinite optimization problems (SDPs), providing some properties related to local optimal solutions. The proposed model is based on another nonlinear optimization model given by [S. Burer and R. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program. Ser. B 95 (2003), pp. 329-357], but it has several nice properties not seen in the existing one. Firstly, the decision variable of the proposed model is a triangular low-rank matrix. Secondly, the existence of a strict local optimum of the proposed model is guaranteed under some conditions, whereas the existing model has no strict local optimum. In other words, it is difficult to construct solution methods equipped with fast convergence using the existing model. We also present some numerical results, showing that the use of the proposed model allows to deliver highly accurate solutions.