On Computing the Nonlinearity Interval in Parametric Semidefinite Optimization

• 成果类型: Article; Early Access
• 署名作者: Hauenstein, Jonathan D.; Mohammad-Nezhad, Ali; Tang, Tingting; Terlaky, Tamas
• 署名单位: University of Notre Dame; Purdue University System; Purdue University; California State University System; San Diego State University; Lehigh University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1234
• 发表日期: 2022
• 关键词: sensitivity-analysis; CONVERGENCE; path; set

摘要

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painleve-Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.