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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Vydáno v:	Journal of global optimization Ročník 64; číslo 2; s. 399 - 416
Hlavní autoři:	Nguyen, Van-Bong, Sheu, Ruey-Lin, Xia, Yong
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.02.2016 Springer Springer Nature B.V
Témata:	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
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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 ( 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.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).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 ... and generate a family of quadratic subproblems ... subject to two quadratic constraints. Each ..., if the problem dimension ..., 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 ... We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find ... numerically. Computational experiments show that our method solves the problem correctly and efficiently. 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. 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 [Formula omitted] and generate a family of quadratic subproblems [Formula omitted] subject to two quadratic constraints. Each [Formula omitted], if the problem dimension [Formula omitted], 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 [Formula omitted]. We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find [Formula omitted] numerically. Computational experiments show that our method solves the problem correctly and efficiently. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Issue Title: Special Issue on MAGO 2014 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 ... and generate a family of quadratic subproblems ... subject to two quadratic constraints. Each ..., if the problem dimension ..., 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 ... We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find ... numerically. Computational experiments show that our method solves the problem correctly and efficiently.
Audience	Academic
Author	Sheu, Ruey-Lin Xia, Yong Nguyen, Van-Bong
Author_xml	– sequence: 1 givenname: Van-Bong surname: Nguyen fullname: Nguyen, Van-Bong organization: Department of Mathematics, National Cheng Kung University – sequence: 2 givenname: Ruey-Lin surname: Sheu fullname: Sheu, Ruey-Lin email: rsheu@mail.ncku.edu.tw organization: Department of Mathematics, National Cheng Kung University – sequence: 3 givenname: Yong surname: Xia fullname: Xia, Yong organization: State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University
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CitedBy_id	crossref_primary_10_1016_j_orl_2017_11_010 crossref_primary_10_1145_3638048 crossref_primary_10_1177_02783649231186165 crossref_primary_10_1109_TWC_2018_2808955 crossref_primary_10_1007_s10898_019_00835_5 crossref_primary_10_1007_s11042_016_4087_6 crossref_primary_10_1007_s40314_018_0575_9 crossref_primary_10_1137_18M1164639
Cites_doi	10.1023/B:JOTA.0000026129.07165.5a 10.1137/S003614450444614X 10.1007/s10898-008-9285-y 10.1287/moor.28.2.246.14485 10.1023/A:1008314922240 10.1137/07070601X 10.1007/s10898-008-9378-7 10.1093/bioinformatics/btp019 10.1023/A:1013072027218 10.1023/A:1013869015288 10.1016/j.cam.2013.08.005 10.6026/97320630002230 10.1090/S0002-9939-1961-0122827-1 10.1023/A:1008316327038 10.1023/A:1013807129844 10.1137/S105262340139001X 10.1002/0471787779 10.1007/s10589-012-9479-6 10.1007/b102138 10.1137/1.9780898718829 10.1109/ISSSTA.2006.311820 10.1137/1.9781611972757.44 10.1007/978-0-387-74503-9
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Keywords	Semidefinite programming Quadratic fit line search S-Lemma 90C25 90C22 Fractional programming Quadratically constrained quadratic programming (Generalized) Rayleigh quotient
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Snippet	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... 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... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Issue Title: Special Issue on MAGO 2014 The problem is a type of "sum-of-ratios"... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).The problem is a type of "sum-of-ratios" fractional programming and is known to be...
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SubjectTerms	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
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Title	Maximizing the sum of a generalized Rayleigh quotient and another Rayleigh quotient on the unit sphere via semidefinite programming
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