Maximizing the sum of a generalized Rayleigh quotient and another Rayleigh quotient on the unit sphere via semidefinite programming

Van Bong Nguyen, Ruey Lin Sheu, Yong Xia

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

The problem is a type of “sum-of-ratios” fractional programming and is known to be NP-hard. Due to many local maxima, finding the global maximizer is in general difficult. The best attempt so far is a critical point approach based on a necessary optimality condition. The problem therefore has not been completely solved. Our novel idea is to replace the generalized Rayleigh quotient by a parameter μ and generate a family of quadratic subproblems (Pμ)′s subject to two quadratic constraints. Each (Pμ), if the problem dimension n≥3, can be solved in polynomial time by incorporating a version of S-lemma; a tight SDP relaxation; and a matrix rank-one decomposition procedure. Then, the difficulty of the problem is largely reduced to become a one-dimensional maximization problem over an interval of parameters [μ̲,μ¯]. We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find μ numerically. Computational experiments show that our method solves the problem correctly and efficiently.

Original languageEnglish
Pages (from-to)399-416
Number of pages18
JournalJournal of Global Optimization
Volume64
Issue number2
DOIs
Publication statusPublished - 2016 Feb 1

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics

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