Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone

• 成果类型: Article
• 署名作者: Permenter, Frank; Parrilo, Pablo
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1169-9
• 发表日期: 2018
• 页码: 1-54
• 关键词: half-plane property; semidefinite programming relaxations; symmetric h-matrices; Strong Duality; optimization; POLYNOMIALS

摘要

We develop a practical semidefinite programming (SDP) facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone. The proposed method simplifies SDPs with no strictly feasible solution (a frequent output of parsers) by solving a sequence of easier optimization problems and could be a useful pre-processing technique for SDP solvers. We demonstrate effectiveness of the method on SDPs arising in practice, and describe our publicly-available software implementation. We also show how to find maximum rank matrices in our PSD cone approximations (which helps us find maximal simplifications), and we give a post-processing procedure for dual solution recovery that generally applies to facial-reduction-based pre-processing techniques. Finally, we show how approximations can be chosen to preserve problem sparsity.