Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

• 成果类型: Article
• 署名作者: Adachi, Satoru; Nakatsukasa, Yuji
• 署名单位: University of Tokyo; University of Oxford
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1206-8
• 发表日期: 2019
• 页码: 79-116
• 关键词: trust-region subproblem; quadratic constraint; optimization; software; bounds

摘要

A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or More's algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists.