Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

• 成果类型: Article
• 署名作者: Freund, Robert M.; Ordonez, Fernando; Toh, Kim-Chuan
• 署名单位: University of Southern California; Massachusetts Institute of Technology (MIT); National University of Singapore; Singapore-MIT Alliance for Research & Technology Centre (SMART); Massachusetts Institute of Technology (MIT); Nanyang Technological University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-006-0035-y
• 发表日期: 2007
• 页码: 445-475
• 关键词: interior-point algorithms; conic linear-system; superlinear convergence; local convergence; condition numbers; complexity; optimization; STABILITY

摘要

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.