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

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 be...

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Veröffentlicht in:Journal of global optimization Jg. 64; H. 2; S. 399 - 416
Hauptverfasser: Nguyen, Van-Bong, Sheu, Ruey-Lin, Xia, Yong
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.02.2016
Springer
Springer Nature B.V
Schlagworte:
Algorithms
Computer Science
Discriminant analysis
Eigenvalues
Intervals
Investment analysis
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Maximization
Operations Research/Decision Theory
Optimization
Polynomials
Programming
Quadratic programming
Quotients
Rankings
Ratios
Rayleigh number
Real Functions
Semidefinite programming
Studies
Texts
Semidefinite programming
Quadratic fit line search
S-Lemma
90C25
90C22
Fractional programming
Quadratically constrained quadratic programming
(Generalized) Rayleigh quotient
ISSN:0925-5001, 1573-2916
Online-Zugang:Volltext
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Zusammenfassung: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 ( P μ ) ′ s subject to two quadratic constraints. Each ( P μ ) , 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 [ μ ̲ , 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.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-015-0315-2